https://hilbert.math.wisc.edu/wiki/api.php?action=feedcontributions&user=Beltran&feedformat=atomUW-Math Wiki - User contributions [en]2021-04-14T23:37:43ZUser contributionsMediaWiki 1.30.1https://hilbert.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=21148Analysis Seminar2021-04-13T17:37:37Z<p>Beltran: /* David Beltran */</p>
<hr />
<div><br />
The 2020-2021 Analysis Seminar will be organized by David Beltran and Andreas Seeger.<br />
It will be online for the entire academic year. The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar at different times, to accommodate speakers).<br />
<br />
Zoom links will be sent to those who have signed up for the Analysis Seminar List. If you would like to subscribe to the Analysis seminar list, send a blank email to analysis+join (at) g-groups (dot) wisc (dot) edu as well as an additional email to David and Andreas (dbeltran, seeger at math (dot) wisc (dot) edu) to notify the request.<br />
<br />
If you'd like to suggest speakers for the spring semester please contact David and Andreas.<br />
<br />
<br />
<br />
=[[Previous_Analysis_seminars]]=<br />
<br />
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars<br />
<br />
= Current Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 22<br />
|Alexei Poltoratski<br />
|UW Madison<br />
|[[#Alexei Poltoratski | Dirac inner functions ]]<br />
| <br />
|-<br />
|September 29<br />
|Joris Roos<br />
|University of Massachusetts - Lowell<br />
|[[#Polona Durcik and Joris Rooslinktoabstract | A triangular Hilbert transform with curvature, I ]]<br />
| <br />
|-<br />
|Wednesday September 30, 4 p.m.<br />
|Polona Durcik<br />
|Chapman University<br />
|[[#Polona Durcik and Joris Roos | A triangular Hilbert transform with curvature, II ]]<br />
| <br />
|-<br />
|October 6<br />
|Andrew Zimmer<br />
|UW Madison<br />
|[[#Andrew Zimmer | Complex analytic problems on domains with good intrinsic geometry ]]<br />
| <br />
|-<br />
|October 13<br />
|Hong Wang <br />
|Princeton/IAS<br />
|[[#Hong Wang | Improved decoupling for the parabola ]]<br />
| <br />
|-<br />
|October 20<br />
|Kevin Luli<br />
|UC Davis<br />
|[[#Kevin Luli | Smooth Nonnegative Interpolation ]]<br />
| <br />
|-<br />
|October 21, 4.00 p.m.<br />
|Niclas Technau<br />
|UW Madison<br />
|[[#Niclas Technau | Number theoretic applications of oscillatory integrals ]]<br />
| <br />
|-<br />
|October 27<br />
|Terence Harris<br />
| Cornell University<br />
|[[#Terence Harris | Low dimensional pinned distance sets via spherical averages ]]<br />
| <br />
|-<br />
|Monday, November 2, 4 p.m.<br />
|Yuval Wigderson<br />
|Stanford University<br />
|[[#Yuval Wigderson | New perspectives on the uncertainty principle ]]<br />
| <br />
|-<br />
|November 10, 10 a.m. <br />
|Óscar Domínguez<br />
| Universidad Complutense de Madrid<br />
|[[#Oscar Dominguez | New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions ]]<br />
| <br />
|-<br />
|November 17<br />
|Tamas Titkos<br />
|BBS U of Applied Sciences and Renyi Institute<br />
|[[#Tamas Titkos | Isometries of Wasserstein spaces ]]<br />
| <br />
|-<br />
|November 24<br />
|Shukun Wu<br />
|University of Illinois (Urbana-Champaign)<br />
||[[#Shukun Wu | On the Bochner-Riesz operator and the maximal Bochner-Riesz operator ]] <br />
| <br />
|-<br />
|December 1<br />
| Jonathan Hickman<br />
| The University of Edinburgh<br />
|[[#Jonathan Hickman | Sobolev improving for averages over space curves ]]<br />
| <br />
|-<br />
|February 2, 7:00 p.m.<br />
|Hanlong Fang<br />
|UW Madison<br />
|[[#Hanlong Fang | Canonical blow-ups of Grassmann manifolds ]]<br />
| <br />
|-<br />
|February 9<br />
|Bingyang Hu<br />
|Purdue University<br />
|[[#Bingyang Hu | Some structure theorems on general doubling measures ]]<br />
| <br />
|-<br />
|February 16<br />
|Krystal Taylor<br />
|The Ohio State University<br />
|[[#Krystal Taylor | Quantifications of the Besicovitch Projection theorem in a nonlinear setting ]]<br />
|<br />
|-<br />
|February 23<br />
|Dominique Maldague<br />
|MIT<br />
|[[#Dominique Maldague | A new proof of decoupling for the parabola ]]<br />
|<br />
|-<br />
|March 2<br />
|Diogo Oliveira e Silva<br />
|University of Birmingham<br />
|[[#Diogo Oliveira e Silva | Global maximizers for spherical restriction ]]<br />
|<br />
|-<br />
|March 9<br />
|Oleg Safronov <br />
|University of North Carolina Charlotte<br />
|[[#Oleg Safronov | Relations between discrete and continuous spectra of differential operators ]]<br />
|<br />
|-<br />
|March 16<br />
|Ziming Shi<br />
|Rutgers University<br />
|[[#Ziming Shi | Sharp Sobolev 1/2-estimate for dbar equations on strictly pseudoconvex domains with C^2 boundary ]]<br />
|<br />
|-<br />
|March 23<br />
|Xiumin Du<br />
|Northwestern University<br />
|[[#Xiumin Du | Falconer's distance set problem ]]<br />
|<br />
|-<br />
|March 30, 10:00 a.m.<br />
|Etienne Le Masson<br />
|Cergy Paris University<br />
|[[#Etienne Le Masson | Quantum ergodicity for Eisenstein series on large genus hyperbolic surfaces ]]<br />
|<br />
|-<br />
|April 6<br />
|Theresa Anderson <br />
|Purdue University<br />
|[[#Theresa Anderson | Dyadic analysis (virtually) meets number theory ]]<br />
|<br />
|-<br />
|April 13<br />
|Nathan Wagner<br />
|Washington University St. Louis<br />
|[[#Nathan Wagner | Weighted Estimates for the Bergman and Szegő projections on strongly pseudoconvex domains with near minimal smoothness ]]<br />
|<br />
|-<br />
|April 20<br />
|David Beltran<br />
| UW Madison<br />
|[[#David Beltran | Sobolev improving for averages over curves in $\mathbb{R}^4$]]<br />
|<br />
|-<br />
|April 27<br />
|Yumeng Ou<br />
|University of Pennsylvania<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===Alexei Poltoratski===<br />
<br />
Title: Dirac inner functions<br />
<br />
Abstract: My talk will focus on some new (and old) complex analytic objects arising from Dirac systems of differential equations.<br />
We will discuss connections between problems in complex function theory, spectral and scattering problems for differential<br />
operators and the non-linear Fourier transform.<br />
<br />
===Polona Durcik and Joris Roos===<br />
<br />
Title: A triangular Hilbert transform with curvature, I & II.<br />
<br />
Abstract: The triangular Hilbert is a two-dimensional bilinear singular<br />
originating in time-frequency analysis. No Lp bounds are currently<br />
known for this operator.<br />
In these two talks we discuss a recent joint work with Michael Christ<br />
on a variant of the triangular Hilbert transform involving curvature.<br />
This object is closely related to the bilinear Hilbert transform with<br />
curvature and a maximally modulated singular integral of Stein-Wainger<br />
type. As an application we also discuss a quantitative nonlinear Roth<br />
type theorem on patterns in the Euclidean plane.<br />
The second talk will focus on the proof of a key ingredient, a certain<br />
regularity estimate for a local operator.<br />
<br />
===Andrew Zimmer===<br />
<br />
Title: Complex analytic problems on domains with good intrinsic geometry<br />
<br />
Abstract: In this talk, I will describe a new class of domains in complex Euclidean space which is defined in terms of the existence of a Kaehler metric with good geometric properties. This class is invariant under biholomorphism and includes many well-studied classes of domains such as strongly pseudoconvex domains, finite type domains in dimension two, convex domains, homogeneous domains, and embeddings of Teichmuller spaces. Further, certain analytic problems are tractable for domains in this family even when the boundary is non-smooth. In particular, it is possible to characterize the domains in this family where the dbar-Neumann operator on (0, q)-forms is compact (which generalizes an old result of Fu-Straube for convex domains).<br />
<br />
===Hong Wang===<br />
<br />
Title: Improved decoupling for the parabola<br />
<br />
Abstract: In 2014, Bourgain and Demeter proved the $l^2$ decoupling estimates for the paraboloid with constant $R^{\epsilon}$. <br />
We prove an $(l^2, L^6)$ decoupling inequality for the parabola with constant $(\log R)^c$. This is joint work with Larry Guth and Dominique Maldague.<br />
<br />
===Kevin Luli===<br />
<br />
Title: Smooth Nonnegative Interpolation<br />
<br />
Abstract: Suppose E is an arbitrary subset of R^n. Let f: E \rightarrow [0, \infty). How can we decide if f extends to a nonnegative function C^m function F defined on all of R^n? Suppose E is finite. Can we compute a nonnegative C^m function F on R^n that agrees with f on E with the least possible C^m norm? How many computer operations does this take? In this talk, I will explain recent results on these problems. Non-negativity is one of the most important shape preserving properties for interpolants. In real life applications, the range of the interpolant is imposed by nature. For example, probability density, the amount of snow, rain, humidity, chemical concentration are all nonnegative quantities and are of interest in natural sciences. Even in one dimension, the existing techniques can only handle nonnegative interpolation under special assumptions on the data set. Our results work without any assumptions on the data sets. <br />
<br />
===Niclas Technau===<br />
<br />
Title: Number theoretic applications of oscillatory integrals<br />
<br />
Abstract: We discuss how the analysis of oscillatory integrals can be used to solve number theoretic problems. More specifically, the focus will be on understanding fine-scale statistics of sequences on the unit circle. Further, we shall briefly explain a connection to quantum chaos.<br />
<br />
===Terence Harris===<br />
<br />
Title: Low dimensional pinned distance sets via spherical averages<br />
<br />
Abstract: An inequality is derived for the average t-energy of weighted pinned distance measures, where 0 < t < 1, in terms of the L^2 spherical averages of Fourier transforms of measures. This generalises the result of Liu (originally for Lebesgue measure) to pinned distance sets of dimension smaller than 1, and strengthens Mattila's result from 1987, originally for the full distance set.<br />
<br />
===Yuval Wigderson===<br />
<br />
Title: New perspectives on the uncertainty principle<br />
<br />
Abstract: The phrase ``uncertainty principle'' refers to a wide array of results in several disparate fields of mathematics, all of which capture the notion that a function and its Fourier transform cannot both be ``very localized''. The measure of localization varies from one uncertainty principle to the next, and well-studied notions include the variance (and higher moments), the entropy, the support-size, and the rate of decay at infinity. Similarly, the proofs of the various uncertainty principles rely on a range of tools, from the elementary to the very deep. In this talk, I'll describe how many of the uncertainty principles all follow from a single, simple result, whose proof uses only a basic property of the Fourier transform: that it and its inverse are bounded as operators $L^1 \to L^\infty$. Using this result, one can also prove new variants of the uncertainty principle, which apply to new measures of localization and to operators other than the Fourier transform. This is joint work with Avi Wigderson.<br />
<br />
===Oscar Dominguez===<br />
<br />
Title: New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions<br />
<br />
Abstract: The celebrated Bourgain--Brezis--Mironescu formula enables us to recover Sobolev spaces in terms of limits of Gagliardo seminorms. Very recently, Brezis, Van Schaftingen and Yung have proposed an alternative methodology to approach Sobolev spaces via limits of weak-type Gagliardo functionals. The goal of this talk is twofold. Firstly, we will show that the BvSY result is a special case of a more general phenomenon based on maximal inequalities. In particular, we shall derive not only analogs of the BvSY theorem for different kinds of function spaces (Lebesgue, Calderon, higher-order Sobolev, …), but also applications to ergodic theory, Fourier series, etc. In the second part of the talk, we shall investigate the fractional setting in the BvSY theorem. Our approach is based on new Garsia-type inequalities and an application of the Caffarelli--Silvestre extension. This is joint work with Mario Milman.<br />
<br />
===Tamas Titkos===<br />
<br />
Title: Isometries of Wasserstein spaces<br />
<br />
Abstract: Due to its nice theoretical properties and an astonishing number of<br />
applications via optimal transport problems, probably the most<br />
intensively studied metric nowadays is the p-Wasserstein metric. Given<br />
a complete and separable metric space $X$ and a real number $p\geq1$,<br />
one defines the p-Wasserstein space $\mathcal{W}_p(X)$ as the collection<br />
of Borel probability measures with finite $p$-th moment, endowed with a<br />
distance which is calculated by means of transport plans \cite{5}.<br />
<br />
The main aim of our research project is to reveal the structure of the<br />
isometry group $\mathrm{Isom}(\mathcal{W}_p(X))$. Although<br />
$\mathrm{Isom}(X)$ embeds naturally into<br />
$\mathrm{Isom}(\mathcal{W}_p(X))$ by push-forward, and this embedding<br />
turned out to be surjective in many cases (see e.g. [1]), these two<br />
groups are not isomorphic in general. Kloeckner in [2] described<br />
the isometry group of the quadratic Wasserstein space<br />
$\mathcal{W}_2(\mathbb{R}^n)$, and it turned out that the case of $n=1$<br />
is special in the sense that $\mathrm{Isom}(\mathcal{W}_2(\mathbb{R})$<br />
is extremely rich. Namely, it contains a large subgroup of wild behaving<br />
isometries that distort the shape of measures. Following this line of<br />
investigation, in \cite{3} we described<br />
$\mathrm{Isom}(\mathcal{W}_p(\mathbb{R}))$ and<br />
$\mathrm{Isom}(\mathcal{W}_p([0,1])$ for all $p\geq 1$.<br />
<br />
In this talk I will survey first some of the earlier results in the<br />
subject, and then I will present the key results of [3]. If time<br />
permits, I will also report on our most recent manuscript [4] in<br />
which we extended Kloeckner's multidimensional results. Joint work with Gy\"orgy P\'al Geh\'er (University of Reading)<br />
and D\'aniel Virosztek (IST Austria).<br />
<br />
[1] J. Bertrand and B. Kloeckner, \emph{A geometric study of Wasserstein<br />
spaces: isometric rigidity in negative curvature}, International<br />
Mathematics Research Notices, 2016 (5), 1368--1386.<br />
<br />
[2] B. Kloeckner, \emph{A geometric study of Wasserstein spaces: Euclidean<br />
spaces}, Annali della Scuola Normale Superiore di Pisa - Classe di<br />
Scienze, Serie 5, Tome 9 (2010) no. 2, 297--323.<br />
<br />
[3] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{Isometric study of<br />
Wasserstein spaces – the real line}, Trans. Amer. Math. Soc., 373<br />
(2020), 5855--5883.<br />
<br />
[4] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{The isometry group of<br />
Wasserstein spaces: The Hilbertian case}, submitted manuscript.<br />
<br />
[5] C. Villani, \emph{Optimal Transport: Old and New,}<br />
(Grundlehren der mathematischen Wissenschaften)<br />
Springer, 2009.<br />
<br />
===Shukun Wu===<br />
<br />
Title: On the Bochner-Riesz operator and the maximal Bochner-Riesz operator<br />
<br />
Abstract: The Bochner-Riesz problem is one of the most important problems in the field of Fourier analysis. It has a strong connection to other famous problems, such as the restriction conjecture and the Kakeya conjecture. In this talk, I will present some recent improvements to the Bochner-Riesz conjecture and the maximal Bochner-Riesz conjecture. The main methods we used are polynomial partitioning and the Bourgain Demeter l^2 decoupling theorem. <br />
<br />
<br />
===Jonathan Hickman===<br />
<br />
Title: Sobolev improving for averages over space curves<br />
<br />
Abstract: Consider the averaging operator given by convolution with arclength measure on compact piece of a smooth curve in R^n. A simple question is to precisely quantify the gain in regularity induced by this averaging, for instance by studying the L^p-Sobolev mapping properties of the operator. This talk will report on ongoing developments towards understanding this problem. In particular, we will explore some non-trivial necessary conditions on the gain in regularity. Joint with D. Beltran, S. Guo and A. Seeger.<br />
<br />
===Hanlong Fang===<br />
<br />
Title: Canonical blow-ups of Grassmann manifolds<br />
<br />
Abstract: We introduce certain canonical blow-ups \mathcal T_{s,p,n}, as well as their distinct submanifolds \mathcal M_{s,p,n}, of Grassmann manifolds G(p,n) by partitioning the Plücker coordinates with respect to a parameter s. Various geometric aspects of \mathcal T_{s,p,n} and \mathcal M_{s,p,n} are studied, for instance, the smoothness, the holomorphic symmetries, the (semi-)positivity of the anti-canonical bundles, the existence of Kähler-Einstein metrics, the functoriality, etc. In particular, we introduce the notion of homeward compactification, of which \mathcal T_{s,p,n} are examples, as a generalization of the wonderful compactification. <br />
<br />
===Bingyang Hu===<br />
<br />
Title: Some structure theorems on general doubling measures.<br />
<br />
Abstract: In this talk, we will first several structure theorems about general doubling measures. Secondly, we will include some main idea to prove one of these results. More precisely, we will focus on the construction of an explicit family of measures that are p-adic doubling for any finite set of primes, however, not doubling. This part generalizes the work by Boylan, Mills and Ward in 2019 in a highly non-trivial way. As some application, we apply these results (that is, the same construction) to show analogous statements for Muckenhoupt Ap weights and reverse Holder weights. This is a joint work with Tess Anderson.<br />
<br />
===Krystal Taylor===<br />
<br />
Title: Quantifications of the Besicovitch Projection theorem in a nonlinear setting <br />
<br />
Abstract: There are several classical results relating the geometry, dimension, and measure of a set to the structure of its orthogonal projections. <br />
It turns out that many nonlinear projection-type operators also have special geometry that allows us to build similar relationships between a set and its "projections", just as in the linear setting. We will discuss a series of recent results from both geometric and probabilistic vantage points. In particular, we will see that the multi-scale analysis techniques of Tao, as well as the energy techniques of Mattila, can be strengthened and generalized to projection-type operators satisfying a transversality condition. As an application, we address the Buffon curve problem, which is to find upper and lower bounds for the rate of decay of the Favard curve length of the four-corner Cantor set.<br />
<br />
===Dominique Maldague===<br />
<br />
Title: A new proof of decoupling for the parabola<br />
<br />
Abstract: Decoupling has to do with measuring the size of functions with specialized Fourier support (in our case, in a neighborhood of the truncated parabola). Bourgain and Demeter resolved the l^2 decoupling conjecture in 2014, using ingredients like the multilinear Kakeya inequality, L^2 orthogonality, and induction-on-scales. I will present the ideas that go into a new proof of decoupling and make some comparison between the two approaches. This is related to recent joint work with Larry Guth and Hong Wang, as well as forthcoming joint work with Yuqiu Fu and Larry Guth.<br />
<br />
===Diogo Oliveira e Silva===<br />
<br />
Title: Global maximizers for spherical restriction<br />
<br />
Abstract: We prove that constant functions are the unique real-valued maximizers for all $L^2-L^{2n}$ adjoint Fourier restriction inequalities on the unit sphere $\mathbb{S}^{d-1}\subset\mathbb{R}^d$, $d\in\{3,4,5,6,7\}$, where $n\geq 3$ is an integer. The proof uses tools from probability theory, Lie theory, functional analysis, and the theory of special functions. It also relies on general solutions of the underlying Euler--Lagrange equation being smooth, a fact of independent interest which we discuss. We further show that complex-valued maximizers coincide with nonnegative maximizers multiplied by the character $e^{i\xi\cdot\omega}$, for some $\xi$, thereby extending previous work of Christ & Shao (2012) to arbitrary dimensions $d\geq 2$ and general even exponents. This talk is based on results obtained with René Quilodrán.<br />
<br />
===Oleg Safronov===<br />
<br />
Title: Relations between discrete and continuous spectra of differential operators<br />
<br />
Abstract: We will discuss relations between different parts of spectra of differential operators. In particular, we will see that negative and positive spectra of Schroedinger operators are related to each other. However, there is a stipulation: one needs to consider two operators one of which is obtained from the other<br />
by flipping the sign of the potential at each point x. If one knows only that the negative spectra of the two operators are discrete, then their positive spectra do not have gaps. If one knows more about the rate of accumulation of the discrete negative eigenvalues to zero, then one can say more about the absolutely continuous component of the positive spectrum.<br />
<br />
===Ziming Shi===<br />
<br />
Title: Sharp Sobolev $1/2$-estimate for $\bar\partial$ equations on strictly pseudoconvex domains with $C^2$ boundary <br />
<br />
Abstract: We give a solution operator for $\bar\partial$ equation that gains the sharp $1/2$-derivative in the Sobolev space $H^{s,p}$ on any strictly pseudoconvex domain with $C^2$-boundary, for all $1< p < \infty$ and $s>1/p$. <br />
We also show that the same solution operator gains a $1/2$-derivative in the H\"older-Zygmund space $\Lambda^s$ for any $s>0$, where previously it was known for $s>1$ by work of X. Gong. <br />
The main ingredients used in our proof are a Hardy-Littlewood lemma of Sobolev type and a new commutator estimate. <br />
Joint work with Liding Yao.<br />
<br />
===Xiumin Du===<br />
<br />
Title: Falconer's distance set problem<br />
<br />
Abstract: A classical question in geometric measure theory, introduced by Falconer in the 80s is, how large does the Hausdorff dimension of a compact subset in Euclidean space need to be to ensure that the Lebesgue measure of its set of pairwise Euclidean distances is positive. In this talk, I'll report some recent progress on this problem, which combines several ingredients including Orponen's radial projection theorem, Liu's L^2 identity obtained using a group action argument, and the refined decoupling theory. This is based on joint work with Alex Iosevich, Yumeng Ou, Hong Wang, and Ruixiang Zhang.<br />
<br />
===Etienne Le Masson===<br />
<br />
Title: Quantum ergodicity for Eisenstein series on large genus hyperbolic surfaces<br />
<br />
Abstract: We will present a delocalisation result for eigenfunctions of the Laplacian on finite area hyperbolic surfaces of large genus. This is a quantum ergodicity result analogous to a theorem of Zelditch showing that the mass of most L2 eigenfunctions and Eisenstein series (eigenfunctions associated with the continuous spectrum) equidistributes when the eigenvalues tend to infinity. Here we will fix a bounded spectral window and look at a similar equidistribution phenomenon when the area/genus goes to infinity (more precisely the surfaces Benjamini-Schramm converge to the plane). The conditions we require on the surfaces are satisfied with high probability in the Weil-Petersson model of random surfaces introduced by Mirzakhani. They also apply to congruence covers of the modular surface, where we recover a result of Nelson on the equidistribution of Maass forms (with weaker convergence rate). The proof is based on ergodic theory methods.<br />
Joint work with Tuomas Sahlsten.<br />
<br />
===Theresa Anderson===<br />
<br />
Title: Dyadic analysis (virtually) meets number theory<br />
<br />
Abstract: In this talk we discuss two ways in which dyadic analysis and number theory share a rich interaction. The first, which we will spend the most time motivating and discussing, involves a complete classification of "distinct dyadic systems". These are sets of grids which allow one to compare any Euclidean ball nicely with any dyadic cube, and allow for showing that a large number of continuous objects and operators can be "replaced" with their easier dyadic counterparts. If time remains, secondly, we define and make progress on showing the (failure) of a "Hasse principle" in harmonic analysis; specifically, we discuss the interplay between number theory and dyadic analysis that allows us to construct a measure that is "p-adic" doubling for any prime p (in a finite set of primes), yet not doubling overall.<br />
<br />
===Nathan Wagner===<br />
<br />
Title: Weighted Estimates for the Bergman and Szegő projections on strongly pseudoconvex domains with near minimal smoothness <br />
<br />
Abstract: The Bergman and Szegő projections are fundamental operators in complex analysis in one and several complex variables. Consequently, the mapping properties of these operators on L^p and other function spaces have been extensively studied. In this talk, we discuss some recent results for these operators on strongly pseudoconvex domains with near minimal smoothness. In particular, weighted L^p estimates are obtained, where the weight belongs to a suitable generalization of the Békollé-Bonami or Muckenhoupt class. For these domains with less boundary regularity, we use an operator-theoretic technique that goes back to Kerzman and Stein. We also obtain weighted estimates for the endpoint p=1, including weighted weak-type (1,1) estimates. Here we use a modified version of singular-integral theory and a generalization of the Riesz-Kolmogorov characterization of precompact subsets of Lebesgue spaces. This talk is based on joint work with Brett Wick and Cody Stockdale.<br />
<br />
===David Beltran===<br />
<br />
Title: Sobolev improving for averages over curves in $\mathbb{R}^4$ <br />
<br />
Abstract: Given a smooth non-degenerate space curve (that is, a smooth curve whose n-1 curvature functions are non-vanishing), it is a classical question to study the smoothing properties of the averaging operators along a compact piece of such a curve. This question can be quantified, for example, by studying the $L^p$-Sobolev mapping properties of those operators. These are well understood in 2 and 3 dimensions, and in this talk, we present a new sharp result in 4 dimensions. We focus on the positive results; the non-trivial examples which show that our results are best possible were presented by Jonathan Hickman in December 1st. This is joint work with Shaoming Guo, Jonathan Hickman and Andreas Seeger.<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]<br />
<br />
<br />
Graduate Student Seminar:<br />
<br />
https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html</div>Beltranhttps://hilbert.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=21147Analysis Seminar2021-04-13T17:37:06Z<p>Beltran: /* Abstracts */</p>
<hr />
<div><br />
The 2020-2021 Analysis Seminar will be organized by David Beltran and Andreas Seeger.<br />
It will be online for the entire academic year. The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar at different times, to accommodate speakers).<br />
<br />
Zoom links will be sent to those who have signed up for the Analysis Seminar List. If you would like to subscribe to the Analysis seminar list, send a blank email to analysis+join (at) g-groups (dot) wisc (dot) edu as well as an additional email to David and Andreas (dbeltran, seeger at math (dot) wisc (dot) edu) to notify the request.<br />
<br />
If you'd like to suggest speakers for the spring semester please contact David and Andreas.<br />
<br />
<br />
<br />
=[[Previous_Analysis_seminars]]=<br />
<br />
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars<br />
<br />
= Current Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 22<br />
|Alexei Poltoratski<br />
|UW Madison<br />
|[[#Alexei Poltoratski | Dirac inner functions ]]<br />
| <br />
|-<br />
|September 29<br />
|Joris Roos<br />
|University of Massachusetts - Lowell<br />
|[[#Polona Durcik and Joris Rooslinktoabstract | A triangular Hilbert transform with curvature, I ]]<br />
| <br />
|-<br />
|Wednesday September 30, 4 p.m.<br />
|Polona Durcik<br />
|Chapman University<br />
|[[#Polona Durcik and Joris Roos | A triangular Hilbert transform with curvature, II ]]<br />
| <br />
|-<br />
|October 6<br />
|Andrew Zimmer<br />
|UW Madison<br />
|[[#Andrew Zimmer | Complex analytic problems on domains with good intrinsic geometry ]]<br />
| <br />
|-<br />
|October 13<br />
|Hong Wang <br />
|Princeton/IAS<br />
|[[#Hong Wang | Improved decoupling for the parabola ]]<br />
| <br />
|-<br />
|October 20<br />
|Kevin Luli<br />
|UC Davis<br />
|[[#Kevin Luli | Smooth Nonnegative Interpolation ]]<br />
| <br />
|-<br />
|October 21, 4.00 p.m.<br />
|Niclas Technau<br />
|UW Madison<br />
|[[#Niclas Technau | Number theoretic applications of oscillatory integrals ]]<br />
| <br />
|-<br />
|October 27<br />
|Terence Harris<br />
| Cornell University<br />
|[[#Terence Harris | Low dimensional pinned distance sets via spherical averages ]]<br />
| <br />
|-<br />
|Monday, November 2, 4 p.m.<br />
|Yuval Wigderson<br />
|Stanford University<br />
|[[#Yuval Wigderson | New perspectives on the uncertainty principle ]]<br />
| <br />
|-<br />
|November 10, 10 a.m. <br />
|Óscar Domínguez<br />
| Universidad Complutense de Madrid<br />
|[[#Oscar Dominguez | New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions ]]<br />
| <br />
|-<br />
|November 17<br />
|Tamas Titkos<br />
|BBS U of Applied Sciences and Renyi Institute<br />
|[[#Tamas Titkos | Isometries of Wasserstein spaces ]]<br />
| <br />
|-<br />
|November 24<br />
|Shukun Wu<br />
|University of Illinois (Urbana-Champaign)<br />
||[[#Shukun Wu | On the Bochner-Riesz operator and the maximal Bochner-Riesz operator ]] <br />
| <br />
|-<br />
|December 1<br />
| Jonathan Hickman<br />
| The University of Edinburgh<br />
|[[#Jonathan Hickman | Sobolev improving for averages over space curves ]]<br />
| <br />
|-<br />
|February 2, 7:00 p.m.<br />
|Hanlong Fang<br />
|UW Madison<br />
|[[#Hanlong Fang | Canonical blow-ups of Grassmann manifolds ]]<br />
| <br />
|-<br />
|February 9<br />
|Bingyang Hu<br />
|Purdue University<br />
|[[#Bingyang Hu | Some structure theorems on general doubling measures ]]<br />
| <br />
|-<br />
|February 16<br />
|Krystal Taylor<br />
|The Ohio State University<br />
|[[#Krystal Taylor | Quantifications of the Besicovitch Projection theorem in a nonlinear setting ]]<br />
|<br />
|-<br />
|February 23<br />
|Dominique Maldague<br />
|MIT<br />
|[[#Dominique Maldague | A new proof of decoupling for the parabola ]]<br />
|<br />
|-<br />
|March 2<br />
|Diogo Oliveira e Silva<br />
|University of Birmingham<br />
|[[#Diogo Oliveira e Silva | Global maximizers for spherical restriction ]]<br />
|<br />
|-<br />
|March 9<br />
|Oleg Safronov <br />
|University of North Carolina Charlotte<br />
|[[#Oleg Safronov | Relations between discrete and continuous spectra of differential operators ]]<br />
|<br />
|-<br />
|March 16<br />
|Ziming Shi<br />
|Rutgers University<br />
|[[#Ziming Shi | Sharp Sobolev 1/2-estimate for dbar equations on strictly pseudoconvex domains with C^2 boundary ]]<br />
|<br />
|-<br />
|March 23<br />
|Xiumin Du<br />
|Northwestern University<br />
|[[#Xiumin Du | Falconer's distance set problem ]]<br />
|<br />
|-<br />
|March 30, 10:00 a.m.<br />
|Etienne Le Masson<br />
|Cergy Paris University<br />
|[[#Etienne Le Masson | Quantum ergodicity for Eisenstein series on large genus hyperbolic surfaces ]]<br />
|<br />
|-<br />
|April 6<br />
|Theresa Anderson <br />
|Purdue University<br />
|[[#Theresa Anderson | Dyadic analysis (virtually) meets number theory ]]<br />
|<br />
|-<br />
|April 13<br />
|Nathan Wagner<br />
|Washington University St. Louis<br />
|[[#Nathan Wagner | Weighted Estimates for the Bergman and Szegő projections on strongly pseudoconvex domains with near minimal smoothness ]]<br />
|<br />
|-<br />
|April 20<br />
|David Beltran<br />
| UW Madison<br />
|[[#David Beltran | Sobolev improving for averages over curves in $\mathbb{R}^4$]]<br />
|<br />
|-<br />
|April 27<br />
|Yumeng Ou<br />
|University of Pennsylvania<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===Alexei Poltoratski===<br />
<br />
Title: Dirac inner functions<br />
<br />
Abstract: My talk will focus on some new (and old) complex analytic objects arising from Dirac systems of differential equations.<br />
We will discuss connections between problems in complex function theory, spectral and scattering problems for differential<br />
operators and the non-linear Fourier transform.<br />
<br />
===Polona Durcik and Joris Roos===<br />
<br />
Title: A triangular Hilbert transform with curvature, I & II.<br />
<br />
Abstract: The triangular Hilbert is a two-dimensional bilinear singular<br />
originating in time-frequency analysis. No Lp bounds are currently<br />
known for this operator.<br />
In these two talks we discuss a recent joint work with Michael Christ<br />
on a variant of the triangular Hilbert transform involving curvature.<br />
This object is closely related to the bilinear Hilbert transform with<br />
curvature and a maximally modulated singular integral of Stein-Wainger<br />
type. As an application we also discuss a quantitative nonlinear Roth<br />
type theorem on patterns in the Euclidean plane.<br />
The second talk will focus on the proof of a key ingredient, a certain<br />
regularity estimate for a local operator.<br />
<br />
===Andrew Zimmer===<br />
<br />
Title: Complex analytic problems on domains with good intrinsic geometry<br />
<br />
Abstract: In this talk, I will describe a new class of domains in complex Euclidean space which is defined in terms of the existence of a Kaehler metric with good geometric properties. This class is invariant under biholomorphism and includes many well-studied classes of domains such as strongly pseudoconvex domains, finite type domains in dimension two, convex domains, homogeneous domains, and embeddings of Teichmuller spaces. Further, certain analytic problems are tractable for domains in this family even when the boundary is non-smooth. In particular, it is possible to characterize the domains in this family where the dbar-Neumann operator on (0, q)-forms is compact (which generalizes an old result of Fu-Straube for convex domains).<br />
<br />
===Hong Wang===<br />
<br />
Title: Improved decoupling for the parabola<br />
<br />
Abstract: In 2014, Bourgain and Demeter proved the $l^2$ decoupling estimates for the paraboloid with constant $R^{\epsilon}$. <br />
We prove an $(l^2, L^6)$ decoupling inequality for the parabola with constant $(\log R)^c$. This is joint work with Larry Guth and Dominique Maldague.<br />
<br />
===Kevin Luli===<br />
<br />
Title: Smooth Nonnegative Interpolation<br />
<br />
Abstract: Suppose E is an arbitrary subset of R^n. Let f: E \rightarrow [0, \infty). How can we decide if f extends to a nonnegative function C^m function F defined on all of R^n? Suppose E is finite. Can we compute a nonnegative C^m function F on R^n that agrees with f on E with the least possible C^m norm? How many computer operations does this take? In this talk, I will explain recent results on these problems. Non-negativity is one of the most important shape preserving properties for interpolants. In real life applications, the range of the interpolant is imposed by nature. For example, probability density, the amount of snow, rain, humidity, chemical concentration are all nonnegative quantities and are of interest in natural sciences. Even in one dimension, the existing techniques can only handle nonnegative interpolation under special assumptions on the data set. Our results work without any assumptions on the data sets. <br />
<br />
===Niclas Technau===<br />
<br />
Title: Number theoretic applications of oscillatory integrals<br />
<br />
Abstract: We discuss how the analysis of oscillatory integrals can be used to solve number theoretic problems. More specifically, the focus will be on understanding fine-scale statistics of sequences on the unit circle. Further, we shall briefly explain a connection to quantum chaos.<br />
<br />
===Terence Harris===<br />
<br />
Title: Low dimensional pinned distance sets via spherical averages<br />
<br />
Abstract: An inequality is derived for the average t-energy of weighted pinned distance measures, where 0 < t < 1, in terms of the L^2 spherical averages of Fourier transforms of measures. This generalises the result of Liu (originally for Lebesgue measure) to pinned distance sets of dimension smaller than 1, and strengthens Mattila's result from 1987, originally for the full distance set.<br />
<br />
===Yuval Wigderson===<br />
<br />
Title: New perspectives on the uncertainty principle<br />
<br />
Abstract: The phrase ``uncertainty principle'' refers to a wide array of results in several disparate fields of mathematics, all of which capture the notion that a function and its Fourier transform cannot both be ``very localized''. The measure of localization varies from one uncertainty principle to the next, and well-studied notions include the variance (and higher moments), the entropy, the support-size, and the rate of decay at infinity. Similarly, the proofs of the various uncertainty principles rely on a range of tools, from the elementary to the very deep. In this talk, I'll describe how many of the uncertainty principles all follow from a single, simple result, whose proof uses only a basic property of the Fourier transform: that it and its inverse are bounded as operators $L^1 \to L^\infty$. Using this result, one can also prove new variants of the uncertainty principle, which apply to new measures of localization and to operators other than the Fourier transform. This is joint work with Avi Wigderson.<br />
<br />
===Oscar Dominguez===<br />
<br />
Title: New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions<br />
<br />
Abstract: The celebrated Bourgain--Brezis--Mironescu formula enables us to recover Sobolev spaces in terms of limits of Gagliardo seminorms. Very recently, Brezis, Van Schaftingen and Yung have proposed an alternative methodology to approach Sobolev spaces via limits of weak-type Gagliardo functionals. The goal of this talk is twofold. Firstly, we will show that the BvSY result is a special case of a more general phenomenon based on maximal inequalities. In particular, we shall derive not only analogs of the BvSY theorem for different kinds of function spaces (Lebesgue, Calderon, higher-order Sobolev, …), but also applications to ergodic theory, Fourier series, etc. In the second part of the talk, we shall investigate the fractional setting in the BvSY theorem. Our approach is based on new Garsia-type inequalities and an application of the Caffarelli--Silvestre extension. This is joint work with Mario Milman.<br />
<br />
===Tamas Titkos===<br />
<br />
Title: Isometries of Wasserstein spaces<br />
<br />
Abstract: Due to its nice theoretical properties and an astonishing number of<br />
applications via optimal transport problems, probably the most<br />
intensively studied metric nowadays is the p-Wasserstein metric. Given<br />
a complete and separable metric space $X$ and a real number $p\geq1$,<br />
one defines the p-Wasserstein space $\mathcal{W}_p(X)$ as the collection<br />
of Borel probability measures with finite $p$-th moment, endowed with a<br />
distance which is calculated by means of transport plans \cite{5}.<br />
<br />
The main aim of our research project is to reveal the structure of the<br />
isometry group $\mathrm{Isom}(\mathcal{W}_p(X))$. Although<br />
$\mathrm{Isom}(X)$ embeds naturally into<br />
$\mathrm{Isom}(\mathcal{W}_p(X))$ by push-forward, and this embedding<br />
turned out to be surjective in many cases (see e.g. [1]), these two<br />
groups are not isomorphic in general. Kloeckner in [2] described<br />
the isometry group of the quadratic Wasserstein space<br />
$\mathcal{W}_2(\mathbb{R}^n)$, and it turned out that the case of $n=1$<br />
is special in the sense that $\mathrm{Isom}(\mathcal{W}_2(\mathbb{R})$<br />
is extremely rich. Namely, it contains a large subgroup of wild behaving<br />
isometries that distort the shape of measures. Following this line of<br />
investigation, in \cite{3} we described<br />
$\mathrm{Isom}(\mathcal{W}_p(\mathbb{R}))$ and<br />
$\mathrm{Isom}(\mathcal{W}_p([0,1])$ for all $p\geq 1$.<br />
<br />
In this talk I will survey first some of the earlier results in the<br />
subject, and then I will present the key results of [3]. If time<br />
permits, I will also report on our most recent manuscript [4] in<br />
which we extended Kloeckner's multidimensional results. Joint work with Gy\"orgy P\'al Geh\'er (University of Reading)<br />
and D\'aniel Virosztek (IST Austria).<br />
<br />
[1] J. Bertrand and B. Kloeckner, \emph{A geometric study of Wasserstein<br />
spaces: isometric rigidity in negative curvature}, International<br />
Mathematics Research Notices, 2016 (5), 1368--1386.<br />
<br />
[2] B. Kloeckner, \emph{A geometric study of Wasserstein spaces: Euclidean<br />
spaces}, Annali della Scuola Normale Superiore di Pisa - Classe di<br />
Scienze, Serie 5, Tome 9 (2010) no. 2, 297--323.<br />
<br />
[3] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{Isometric study of<br />
Wasserstein spaces – the real line}, Trans. Amer. Math. Soc., 373<br />
(2020), 5855--5883.<br />
<br />
[4] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{The isometry group of<br />
Wasserstein spaces: The Hilbertian case}, submitted manuscript.<br />
<br />
[5] C. Villani, \emph{Optimal Transport: Old and New,}<br />
(Grundlehren der mathematischen Wissenschaften)<br />
Springer, 2009.<br />
<br />
===Shukun Wu===<br />
<br />
Title: On the Bochner-Riesz operator and the maximal Bochner-Riesz operator<br />
<br />
Abstract: The Bochner-Riesz problem is one of the most important problems in the field of Fourier analysis. It has a strong connection to other famous problems, such as the restriction conjecture and the Kakeya conjecture. In this talk, I will present some recent improvements to the Bochner-Riesz conjecture and the maximal Bochner-Riesz conjecture. The main methods we used are polynomial partitioning and the Bourgain Demeter l^2 decoupling theorem. <br />
<br />
<br />
===Jonathan Hickman===<br />
<br />
Title: Sobolev improving for averages over space curves<br />
<br />
Abstract: Consider the averaging operator given by convolution with arclength measure on compact piece of a smooth curve in R^n. A simple question is to precisely quantify the gain in regularity induced by this averaging, for instance by studying the L^p-Sobolev mapping properties of the operator. This talk will report on ongoing developments towards understanding this problem. In particular, we will explore some non-trivial necessary conditions on the gain in regularity. Joint with D. Beltran, S. Guo and A. Seeger.<br />
<br />
===Hanlong Fang===<br />
<br />
Title: Canonical blow-ups of Grassmann manifolds<br />
<br />
Abstract: We introduce certain canonical blow-ups \mathcal T_{s,p,n}, as well as their distinct submanifolds \mathcal M_{s,p,n}, of Grassmann manifolds G(p,n) by partitioning the Plücker coordinates with respect to a parameter s. Various geometric aspects of \mathcal T_{s,p,n} and \mathcal M_{s,p,n} are studied, for instance, the smoothness, the holomorphic symmetries, the (semi-)positivity of the anti-canonical bundles, the existence of Kähler-Einstein metrics, the functoriality, etc. In particular, we introduce the notion of homeward compactification, of which \mathcal T_{s,p,n} are examples, as a generalization of the wonderful compactification. <br />
<br />
===Bingyang Hu===<br />
<br />
Title: Some structure theorems on general doubling measures.<br />
<br />
Abstract: In this talk, we will first several structure theorems about general doubling measures. Secondly, we will include some main idea to prove one of these results. More precisely, we will focus on the construction of an explicit family of measures that are p-adic doubling for any finite set of primes, however, not doubling. This part generalizes the work by Boylan, Mills and Ward in 2019 in a highly non-trivial way. As some application, we apply these results (that is, the same construction) to show analogous statements for Muckenhoupt Ap weights and reverse Holder weights. This is a joint work with Tess Anderson.<br />
<br />
===Krystal Taylor===<br />
<br />
Title: Quantifications of the Besicovitch Projection theorem in a nonlinear setting <br />
<br />
Abstract: There are several classical results relating the geometry, dimension, and measure of a set to the structure of its orthogonal projections. <br />
It turns out that many nonlinear projection-type operators also have special geometry that allows us to build similar relationships between a set and its "projections", just as in the linear setting. We will discuss a series of recent results from both geometric and probabilistic vantage points. In particular, we will see that the multi-scale analysis techniques of Tao, as well as the energy techniques of Mattila, can be strengthened and generalized to projection-type operators satisfying a transversality condition. As an application, we address the Buffon curve problem, which is to find upper and lower bounds for the rate of decay of the Favard curve length of the four-corner Cantor set.<br />
<br />
===Dominique Maldague===<br />
<br />
Title: A new proof of decoupling for the parabola<br />
<br />
Abstract: Decoupling has to do with measuring the size of functions with specialized Fourier support (in our case, in a neighborhood of the truncated parabola). Bourgain and Demeter resolved the l^2 decoupling conjecture in 2014, using ingredients like the multilinear Kakeya inequality, L^2 orthogonality, and induction-on-scales. I will present the ideas that go into a new proof of decoupling and make some comparison between the two approaches. This is related to recent joint work with Larry Guth and Hong Wang, as well as forthcoming joint work with Yuqiu Fu and Larry Guth.<br />
<br />
===Diogo Oliveira e Silva===<br />
<br />
Title: Global maximizers for spherical restriction<br />
<br />
Abstract: We prove that constant functions are the unique real-valued maximizers for all $L^2-L^{2n}$ adjoint Fourier restriction inequalities on the unit sphere $\mathbb{S}^{d-1}\subset\mathbb{R}^d$, $d\in\{3,4,5,6,7\}$, where $n\geq 3$ is an integer. The proof uses tools from probability theory, Lie theory, functional analysis, and the theory of special functions. It also relies on general solutions of the underlying Euler--Lagrange equation being smooth, a fact of independent interest which we discuss. We further show that complex-valued maximizers coincide with nonnegative maximizers multiplied by the character $e^{i\xi\cdot\omega}$, for some $\xi$, thereby extending previous work of Christ & Shao (2012) to arbitrary dimensions $d\geq 2$ and general even exponents. This talk is based on results obtained with René Quilodrán.<br />
<br />
===Oleg Safronov===<br />
<br />
Title: Relations between discrete and continuous spectra of differential operators<br />
<br />
Abstract: We will discuss relations between different parts of spectra of differential operators. In particular, we will see that negative and positive spectra of Schroedinger operators are related to each other. However, there is a stipulation: one needs to consider two operators one of which is obtained from the other<br />
by flipping the sign of the potential at each point x. If one knows only that the negative spectra of the two operators are discrete, then their positive spectra do not have gaps. If one knows more about the rate of accumulation of the discrete negative eigenvalues to zero, then one can say more about the absolutely continuous component of the positive spectrum.<br />
<br />
===Ziming Shi===<br />
<br />
Title: Sharp Sobolev $1/2$-estimate for $\bar\partial$ equations on strictly pseudoconvex domains with $C^2$ boundary <br />
<br />
Abstract: We give a solution operator for $\bar\partial$ equation that gains the sharp $1/2$-derivative in the Sobolev space $H^{s,p}$ on any strictly pseudoconvex domain with $C^2$-boundary, for all $1< p < \infty$ and $s>1/p$. <br />
We also show that the same solution operator gains a $1/2$-derivative in the H\"older-Zygmund space $\Lambda^s$ for any $s>0$, where previously it was known for $s>1$ by work of X. Gong. <br />
The main ingredients used in our proof are a Hardy-Littlewood lemma of Sobolev type and a new commutator estimate. <br />
Joint work with Liding Yao.<br />
<br />
===Xiumin Du===<br />
<br />
Title: Falconer's distance set problem<br />
<br />
Abstract: A classical question in geometric measure theory, introduced by Falconer in the 80s is, how large does the Hausdorff dimension of a compact subset in Euclidean space need to be to ensure that the Lebesgue measure of its set of pairwise Euclidean distances is positive. In this talk, I'll report some recent progress on this problem, which combines several ingredients including Orponen's radial projection theorem, Liu's L^2 identity obtained using a group action argument, and the refined decoupling theory. This is based on joint work with Alex Iosevich, Yumeng Ou, Hong Wang, and Ruixiang Zhang.<br />
<br />
===Etienne Le Masson===<br />
<br />
Title: Quantum ergodicity for Eisenstein series on large genus hyperbolic surfaces<br />
<br />
Abstract: We will present a delocalisation result for eigenfunctions of the Laplacian on finite area hyperbolic surfaces of large genus. This is a quantum ergodicity result analogous to a theorem of Zelditch showing that the mass of most L2 eigenfunctions and Eisenstein series (eigenfunctions associated with the continuous spectrum) equidistributes when the eigenvalues tend to infinity. Here we will fix a bounded spectral window and look at a similar equidistribution phenomenon when the area/genus goes to infinity (more precisely the surfaces Benjamini-Schramm converge to the plane). The conditions we require on the surfaces are satisfied with high probability in the Weil-Petersson model of random surfaces introduced by Mirzakhani. They also apply to congruence covers of the modular surface, where we recover a result of Nelson on the equidistribution of Maass forms (with weaker convergence rate). The proof is based on ergodic theory methods.<br />
Joint work with Tuomas Sahlsten.<br />
<br />
===Theresa Anderson===<br />
<br />
Title: Dyadic analysis (virtually) meets number theory<br />
<br />
Abstract: In this talk we discuss two ways in which dyadic analysis and number theory share a rich interaction. The first, which we will spend the most time motivating and discussing, involves a complete classification of "distinct dyadic systems". These are sets of grids which allow one to compare any Euclidean ball nicely with any dyadic cube, and allow for showing that a large number of continuous objects and operators can be "replaced" with their easier dyadic counterparts. If time remains, secondly, we define and make progress on showing the (failure) of a "Hasse principle" in harmonic analysis; specifically, we discuss the interplay between number theory and dyadic analysis that allows us to construct a measure that is "p-adic" doubling for any prime p (in a finite set of primes), yet not doubling overall.<br />
<br />
===Nathan Wagner===<br />
<br />
Title: Weighted Estimates for the Bergman and Szegő projections on strongly pseudoconvex domains with near minimal smoothness <br />
<br />
Abstract: The Bergman and Szegő projections are fundamental operators in complex analysis in one and several complex variables. Consequently, the mapping properties of these operators on L^p and other function spaces have been extensively studied. In this talk, we discuss some recent results for these operators on strongly pseudoconvex domains with near minimal smoothness. In particular, weighted L^p estimates are obtained, where the weight belongs to a suitable generalization of the Békollé-Bonami or Muckenhoupt class. For these domains with less boundary regularity, we use an operator-theoretic technique that goes back to Kerzman and Stein. We also obtain weighted estimates for the endpoint p=1, including weighted weak-type (1,1) estimates. Here we use a modified version of singular-integral theory and a generalization of the Riesz-Kolmogorov characterization of precompact subsets of Lebesgue spaces. This talk is based on joint work with Brett Wick and Cody Stockdale.<br />
<br />
===David Beltran===<br />
<br />
Title: Sobolev improving for averages over curves in $\mathbb{R}^4$ <br />
<br />
Abstract: Given a non-degenerate smooth space curve (that is, a curve whose n-1 curvature functions are non-vanishing), it is a classical question to study the smoothing properties of the averaging operators along a compact piece of such a curve. This question can be quantified, for example, by studying the $L^p$-Sobolev mapping properties of those operators. These are well understood in 2 and 3 dimensions, and in this talk, we present a new sharp result in 4 dimensions. We focus on the positive results; the non-trivial examples which show that our results are best possible were presented by Jonathan Hickman in December 1st. This is joint work with Shaoming Guo, Jonathan Hickman and Andreas Seeger.<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]<br />
<br />
<br />
Graduate Student Seminar:<br />
<br />
https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html</div>Beltranhttps://hilbert.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=21146Analysis Seminar2021-04-13T17:23:38Z<p>Beltran: /* Current Analysis Seminar Schedule */</p>
<hr />
<div><br />
The 2020-2021 Analysis Seminar will be organized by David Beltran and Andreas Seeger.<br />
It will be online for the entire academic year. The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar at different times, to accommodate speakers).<br />
<br />
Zoom links will be sent to those who have signed up for the Analysis Seminar List. If you would like to subscribe to the Analysis seminar list, send a blank email to analysis+join (at) g-groups (dot) wisc (dot) edu as well as an additional email to David and Andreas (dbeltran, seeger at math (dot) wisc (dot) edu) to notify the request.<br />
<br />
If you'd like to suggest speakers for the spring semester please contact David and Andreas.<br />
<br />
<br />
<br />
=[[Previous_Analysis_seminars]]=<br />
<br />
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars<br />
<br />
= Current Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 22<br />
|Alexei Poltoratski<br />
|UW Madison<br />
|[[#Alexei Poltoratski | Dirac inner functions ]]<br />
| <br />
|-<br />
|September 29<br />
|Joris Roos<br />
|University of Massachusetts - Lowell<br />
|[[#Polona Durcik and Joris Rooslinktoabstract | A triangular Hilbert transform with curvature, I ]]<br />
| <br />
|-<br />
|Wednesday September 30, 4 p.m.<br />
|Polona Durcik<br />
|Chapman University<br />
|[[#Polona Durcik and Joris Roos | A triangular Hilbert transform with curvature, II ]]<br />
| <br />
|-<br />
|October 6<br />
|Andrew Zimmer<br />
|UW Madison<br />
|[[#Andrew Zimmer | Complex analytic problems on domains with good intrinsic geometry ]]<br />
| <br />
|-<br />
|October 13<br />
|Hong Wang <br />
|Princeton/IAS<br />
|[[#Hong Wang | Improved decoupling for the parabola ]]<br />
| <br />
|-<br />
|October 20<br />
|Kevin Luli<br />
|UC Davis<br />
|[[#Kevin Luli | Smooth Nonnegative Interpolation ]]<br />
| <br />
|-<br />
|October 21, 4.00 p.m.<br />
|Niclas Technau<br />
|UW Madison<br />
|[[#Niclas Technau | Number theoretic applications of oscillatory integrals ]]<br />
| <br />
|-<br />
|October 27<br />
|Terence Harris<br />
| Cornell University<br />
|[[#Terence Harris | Low dimensional pinned distance sets via spherical averages ]]<br />
| <br />
|-<br />
|Monday, November 2, 4 p.m.<br />
|Yuval Wigderson<br />
|Stanford University<br />
|[[#Yuval Wigderson | New perspectives on the uncertainty principle ]]<br />
| <br />
|-<br />
|November 10, 10 a.m. <br />
|Óscar Domínguez<br />
| Universidad Complutense de Madrid<br />
|[[#Oscar Dominguez | New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions ]]<br />
| <br />
|-<br />
|November 17<br />
|Tamas Titkos<br />
|BBS U of Applied Sciences and Renyi Institute<br />
|[[#Tamas Titkos | Isometries of Wasserstein spaces ]]<br />
| <br />
|-<br />
|November 24<br />
|Shukun Wu<br />
|University of Illinois (Urbana-Champaign)<br />
||[[#Shukun Wu | On the Bochner-Riesz operator and the maximal Bochner-Riesz operator ]] <br />
| <br />
|-<br />
|December 1<br />
| Jonathan Hickman<br />
| The University of Edinburgh<br />
|[[#Jonathan Hickman | Sobolev improving for averages over space curves ]]<br />
| <br />
|-<br />
|February 2, 7:00 p.m.<br />
|Hanlong Fang<br />
|UW Madison<br />
|[[#Hanlong Fang | Canonical blow-ups of Grassmann manifolds ]]<br />
| <br />
|-<br />
|February 9<br />
|Bingyang Hu<br />
|Purdue University<br />
|[[#Bingyang Hu | Some structure theorems on general doubling measures ]]<br />
| <br />
|-<br />
|February 16<br />
|Krystal Taylor<br />
|The Ohio State University<br />
|[[#Krystal Taylor | Quantifications of the Besicovitch Projection theorem in a nonlinear setting ]]<br />
|<br />
|-<br />
|February 23<br />
|Dominique Maldague<br />
|MIT<br />
|[[#Dominique Maldague | A new proof of decoupling for the parabola ]]<br />
|<br />
|-<br />
|March 2<br />
|Diogo Oliveira e Silva<br />
|University of Birmingham<br />
|[[#Diogo Oliveira e Silva | Global maximizers for spherical restriction ]]<br />
|<br />
|-<br />
|March 9<br />
|Oleg Safronov <br />
|University of North Carolina Charlotte<br />
|[[#Oleg Safronov | Relations between discrete and continuous spectra of differential operators ]]<br />
|<br />
|-<br />
|March 16<br />
|Ziming Shi<br />
|Rutgers University<br />
|[[#Ziming Shi | Sharp Sobolev 1/2-estimate for dbar equations on strictly pseudoconvex domains with C^2 boundary ]]<br />
|<br />
|-<br />
|March 23<br />
|Xiumin Du<br />
|Northwestern University<br />
|[[#Xiumin Du | Falconer's distance set problem ]]<br />
|<br />
|-<br />
|March 30, 10:00 a.m.<br />
|Etienne Le Masson<br />
|Cergy Paris University<br />
|[[#Etienne Le Masson | Quantum ergodicity for Eisenstein series on large genus hyperbolic surfaces ]]<br />
|<br />
|-<br />
|April 6<br />
|Theresa Anderson <br />
|Purdue University<br />
|[[#Theresa Anderson | Dyadic analysis (virtually) meets number theory ]]<br />
|<br />
|-<br />
|April 13<br />
|Nathan Wagner<br />
|Washington University St. Louis<br />
|[[#Nathan Wagner | Weighted Estimates for the Bergman and Szegő projections on strongly pseudoconvex domains with near minimal smoothness ]]<br />
|<br />
|-<br />
|April 20<br />
|David Beltran<br />
| UW Madison<br />
|[[#David Beltran | Sobolev improving for averages over curves in $\mathbb{R}^4$]]<br />
|<br />
|-<br />
|April 27<br />
|Yumeng Ou<br />
|University of Pennsylvania<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===Alexei Poltoratski===<br />
<br />
Title: Dirac inner functions<br />
<br />
Abstract: My talk will focus on some new (and old) complex analytic objects arising from Dirac systems of differential equations.<br />
We will discuss connections between problems in complex function theory, spectral and scattering problems for differential<br />
operators and the non-linear Fourier transform.<br />
<br />
===Polona Durcik and Joris Roos===<br />
<br />
Title: A triangular Hilbert transform with curvature, I & II.<br />
<br />
Abstract: The triangular Hilbert is a two-dimensional bilinear singular<br />
originating in time-frequency analysis. No Lp bounds are currently<br />
known for this operator.<br />
In these two talks we discuss a recent joint work with Michael Christ<br />
on a variant of the triangular Hilbert transform involving curvature.<br />
This object is closely related to the bilinear Hilbert transform with<br />
curvature and a maximally modulated singular integral of Stein-Wainger<br />
type. As an application we also discuss a quantitative nonlinear Roth<br />
type theorem on patterns in the Euclidean plane.<br />
The second talk will focus on the proof of a key ingredient, a certain<br />
regularity estimate for a local operator.<br />
<br />
===Andrew Zimmer===<br />
<br />
Title: Complex analytic problems on domains with good intrinsic geometry<br />
<br />
Abstract: In this talk, I will describe a new class of domains in complex Euclidean space which is defined in terms of the existence of a Kaehler metric with good geometric properties. This class is invariant under biholomorphism and includes many well-studied classes of domains such as strongly pseudoconvex domains, finite type domains in dimension two, convex domains, homogeneous domains, and embeddings of Teichmuller spaces. Further, certain analytic problems are tractable for domains in this family even when the boundary is non-smooth. In particular, it is possible to characterize the domains in this family where the dbar-Neumann operator on (0, q)-forms is compact (which generalizes an old result of Fu-Straube for convex domains).<br />
<br />
===Hong Wang===<br />
<br />
Title: Improved decoupling for the parabola<br />
<br />
Abstract: In 2014, Bourgain and Demeter proved the $l^2$ decoupling estimates for the paraboloid with constant $R^{\epsilon}$. <br />
We prove an $(l^2, L^6)$ decoupling inequality for the parabola with constant $(\log R)^c$. This is joint work with Larry Guth and Dominique Maldague.<br />
<br />
===Kevin Luli===<br />
<br />
Title: Smooth Nonnegative Interpolation<br />
<br />
Abstract: Suppose E is an arbitrary subset of R^n. Let f: E \rightarrow [0, \infty). How can we decide if f extends to a nonnegative function C^m function F defined on all of R^n? Suppose E is finite. Can we compute a nonnegative C^m function F on R^n that agrees with f on E with the least possible C^m norm? How many computer operations does this take? In this talk, I will explain recent results on these problems. Non-negativity is one of the most important shape preserving properties for interpolants. In real life applications, the range of the interpolant is imposed by nature. For example, probability density, the amount of snow, rain, humidity, chemical concentration are all nonnegative quantities and are of interest in natural sciences. Even in one dimension, the existing techniques can only handle nonnegative interpolation under special assumptions on the data set. Our results work without any assumptions on the data sets. <br />
<br />
===Niclas Technau===<br />
<br />
Title: Number theoretic applications of oscillatory integrals<br />
<br />
Abstract: We discuss how the analysis of oscillatory integrals can be used to solve number theoretic problems. More specifically, the focus will be on understanding fine-scale statistics of sequences on the unit circle. Further, we shall briefly explain a connection to quantum chaos.<br />
<br />
===Terence Harris===<br />
<br />
Title: Low dimensional pinned distance sets via spherical averages<br />
<br />
Abstract: An inequality is derived for the average t-energy of weighted pinned distance measures, where 0 < t < 1, in terms of the L^2 spherical averages of Fourier transforms of measures. This generalises the result of Liu (originally for Lebesgue measure) to pinned distance sets of dimension smaller than 1, and strengthens Mattila's result from 1987, originally for the full distance set.<br />
<br />
===Yuval Wigderson===<br />
<br />
Title: New perspectives on the uncertainty principle<br />
<br />
Abstract: The phrase ``uncertainty principle'' refers to a wide array of results in several disparate fields of mathematics, all of which capture the notion that a function and its Fourier transform cannot both be ``very localized''. The measure of localization varies from one uncertainty principle to the next, and well-studied notions include the variance (and higher moments), the entropy, the support-size, and the rate of decay at infinity. Similarly, the proofs of the various uncertainty principles rely on a range of tools, from the elementary to the very deep. In this talk, I'll describe how many of the uncertainty principles all follow from a single, simple result, whose proof uses only a basic property of the Fourier transform: that it and its inverse are bounded as operators $L^1 \to L^\infty$. Using this result, one can also prove new variants of the uncertainty principle, which apply to new measures of localization and to operators other than the Fourier transform. This is joint work with Avi Wigderson.<br />
<br />
===Oscar Dominguez===<br />
<br />
Title: New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions<br />
<br />
Abstract: The celebrated Bourgain--Brezis--Mironescu formula enables us to recover Sobolev spaces in terms of limits of Gagliardo seminorms. Very recently, Brezis, Van Schaftingen and Yung have proposed an alternative methodology to approach Sobolev spaces via limits of weak-type Gagliardo functionals. The goal of this talk is twofold. Firstly, we will show that the BvSY result is a special case of a more general phenomenon based on maximal inequalities. In particular, we shall derive not only analogs of the BvSY theorem for different kinds of function spaces (Lebesgue, Calderon, higher-order Sobolev, …), but also applications to ergodic theory, Fourier series, etc. In the second part of the talk, we shall investigate the fractional setting in the BvSY theorem. Our approach is based on new Garsia-type inequalities and an application of the Caffarelli--Silvestre extension. This is joint work with Mario Milman.<br />
<br />
===Tamas Titkos===<br />
<br />
Title: Isometries of Wasserstein spaces<br />
<br />
Abstract: Due to its nice theoretical properties and an astonishing number of<br />
applications via optimal transport problems, probably the most<br />
intensively studied metric nowadays is the p-Wasserstein metric. Given<br />
a complete and separable metric space $X$ and a real number $p\geq1$,<br />
one defines the p-Wasserstein space $\mathcal{W}_p(X)$ as the collection<br />
of Borel probability measures with finite $p$-th moment, endowed with a<br />
distance which is calculated by means of transport plans \cite{5}.<br />
<br />
The main aim of our research project is to reveal the structure of the<br />
isometry group $\mathrm{Isom}(\mathcal{W}_p(X))$. Although<br />
$\mathrm{Isom}(X)$ embeds naturally into<br />
$\mathrm{Isom}(\mathcal{W}_p(X))$ by push-forward, and this embedding<br />
turned out to be surjective in many cases (see e.g. [1]), these two<br />
groups are not isomorphic in general. Kloeckner in [2] described<br />
the isometry group of the quadratic Wasserstein space<br />
$\mathcal{W}_2(\mathbb{R}^n)$, and it turned out that the case of $n=1$<br />
is special in the sense that $\mathrm{Isom}(\mathcal{W}_2(\mathbb{R})$<br />
is extremely rich. Namely, it contains a large subgroup of wild behaving<br />
isometries that distort the shape of measures. Following this line of<br />
investigation, in \cite{3} we described<br />
$\mathrm{Isom}(\mathcal{W}_p(\mathbb{R}))$ and<br />
$\mathrm{Isom}(\mathcal{W}_p([0,1])$ for all $p\geq 1$.<br />
<br />
In this talk I will survey first some of the earlier results in the<br />
subject, and then I will present the key results of [3]. If time<br />
permits, I will also report on our most recent manuscript [4] in<br />
which we extended Kloeckner's multidimensional results. Joint work with Gy\"orgy P\'al Geh\'er (University of Reading)<br />
and D\'aniel Virosztek (IST Austria).<br />
<br />
[1] J. Bertrand and B. Kloeckner, \emph{A geometric study of Wasserstein<br />
spaces: isometric rigidity in negative curvature}, International<br />
Mathematics Research Notices, 2016 (5), 1368--1386.<br />
<br />
[2] B. Kloeckner, \emph{A geometric study of Wasserstein spaces: Euclidean<br />
spaces}, Annali della Scuola Normale Superiore di Pisa - Classe di<br />
Scienze, Serie 5, Tome 9 (2010) no. 2, 297--323.<br />
<br />
[3] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{Isometric study of<br />
Wasserstein spaces – the real line}, Trans. Amer. Math. Soc., 373<br />
(2020), 5855--5883.<br />
<br />
[4] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{The isometry group of<br />
Wasserstein spaces: The Hilbertian case}, submitted manuscript.<br />
<br />
[5] C. Villani, \emph{Optimal Transport: Old and New,}<br />
(Grundlehren der mathematischen Wissenschaften)<br />
Springer, 2009.<br />
<br />
===Shukun Wu===<br />
<br />
Title: On the Bochner-Riesz operator and the maximal Bochner-Riesz operator<br />
<br />
Abstract: The Bochner-Riesz problem is one of the most important problems in the field of Fourier analysis. It has a strong connection to other famous problems, such as the restriction conjecture and the Kakeya conjecture. In this talk, I will present some recent improvements to the Bochner-Riesz conjecture and the maximal Bochner-Riesz conjecture. The main methods we used are polynomial partitioning and the Bourgain Demeter l^2 decoupling theorem. <br />
<br />
<br />
===Jonathan Hickman===<br />
<br />
Title: Sobolev improving for averages over space curves<br />
<br />
Abstract: Consider the averaging operator given by convolution with arclength measure on compact piece of a smooth curve in R^n. A simple question is to precisely quantify the gain in regularity induced by this averaging, for instance by studying the L^p-Sobolev mapping properties of the operator. This talk will report on ongoing developments towards understanding this problem. In particular, we will explore some non-trivial necessary conditions on the gain in regularity. Joint with D. Beltran, S. Guo and A. Seeger.<br />
<br />
===Hanlong Fang===<br />
<br />
Title: Canonical blow-ups of Grassmann manifolds<br />
<br />
Abstract: We introduce certain canonical blow-ups \mathcal T_{s,p,n}, as well as their distinct submanifolds \mathcal M_{s,p,n}, of Grassmann manifolds G(p,n) by partitioning the Plücker coordinates with respect to a parameter s. Various geometric aspects of \mathcal T_{s,p,n} and \mathcal M_{s,p,n} are studied, for instance, the smoothness, the holomorphic symmetries, the (semi-)positivity of the anti-canonical bundles, the existence of Kähler-Einstein metrics, the functoriality, etc. In particular, we introduce the notion of homeward compactification, of which \mathcal T_{s,p,n} are examples, as a generalization of the wonderful compactification. <br />
<br />
===Bingyang Hu===<br />
<br />
Title: Some structure theorems on general doubling measures.<br />
<br />
Abstract: In this talk, we will first several structure theorems about general doubling measures. Secondly, we will include some main idea to prove one of these results. More precisely, we will focus on the construction of an explicit family of measures that are p-adic doubling for any finite set of primes, however, not doubling. This part generalizes the work by Boylan, Mills and Ward in 2019 in a highly non-trivial way. As some application, we apply these results (that is, the same construction) to show analogous statements for Muckenhoupt Ap weights and reverse Holder weights. This is a joint work with Tess Anderson.<br />
<br />
===Krystal Taylor===<br />
<br />
Title: Quantifications of the Besicovitch Projection theorem in a nonlinear setting <br />
<br />
Abstract: There are several classical results relating the geometry, dimension, and measure of a set to the structure of its orthogonal projections. <br />
It turns out that many nonlinear projection-type operators also have special geometry that allows us to build similar relationships between a set and its "projections", just as in the linear setting. We will discuss a series of recent results from both geometric and probabilistic vantage points. In particular, we will see that the multi-scale analysis techniques of Tao, as well as the energy techniques of Mattila, can be strengthened and generalized to projection-type operators satisfying a transversality condition. As an application, we address the Buffon curve problem, which is to find upper and lower bounds for the rate of decay of the Favard curve length of the four-corner Cantor set.<br />
<br />
===Dominique Maldague===<br />
<br />
Title: A new proof of decoupling for the parabola<br />
<br />
Abstract: Decoupling has to do with measuring the size of functions with specialized Fourier support (in our case, in a neighborhood of the truncated parabola). Bourgain and Demeter resolved the l^2 decoupling conjecture in 2014, using ingredients like the multilinear Kakeya inequality, L^2 orthogonality, and induction-on-scales. I will present the ideas that go into a new proof of decoupling and make some comparison between the two approaches. This is related to recent joint work with Larry Guth and Hong Wang, as well as forthcoming joint work with Yuqiu Fu and Larry Guth.<br />
<br />
===Diogo Oliveira e Silva===<br />
<br />
Title: Global maximizers for spherical restriction<br />
<br />
Abstract: We prove that constant functions are the unique real-valued maximizers for all $L^2-L^{2n}$ adjoint Fourier restriction inequalities on the unit sphere $\mathbb{S}^{d-1}\subset\mathbb{R}^d$, $d\in\{3,4,5,6,7\}$, where $n\geq 3$ is an integer. The proof uses tools from probability theory, Lie theory, functional analysis, and the theory of special functions. It also relies on general solutions of the underlying Euler--Lagrange equation being smooth, a fact of independent interest which we discuss. We further show that complex-valued maximizers coincide with nonnegative maximizers multiplied by the character $e^{i\xi\cdot\omega}$, for some $\xi$, thereby extending previous work of Christ & Shao (2012) to arbitrary dimensions $d\geq 2$ and general even exponents. This talk is based on results obtained with René Quilodrán.<br />
<br />
===Oleg Safronov===<br />
<br />
Title: Relations between discrete and continuous spectra of differential operators<br />
<br />
Abstract: We will discuss relations between different parts of spectra of differential operators. In particular, we will see that negative and positive spectra of Schroedinger operators are related to each other. However, there is a stipulation: one needs to consider two operators one of which is obtained from the other<br />
by flipping the sign of the potential at each point x. If one knows only that the negative spectra of the two operators are discrete, then their positive spectra do not have gaps. If one knows more about the rate of accumulation of the discrete negative eigenvalues to zero, then one can say more about the absolutely continuous component of the positive spectrum.<br />
<br />
===Ziming Shi===<br />
<br />
Title: Sharp Sobolev $1/2$-estimate for $\bar\partial$ equations on strictly pseudoconvex domains with $C^2$ boundary <br />
<br />
Abstract: We give a solution operator for $\bar\partial$ equation that gains the sharp $1/2$-derivative in the Sobolev space $H^{s,p}$ on any strictly pseudoconvex domain with $C^2$-boundary, for all $1< p < \infty$ and $s>1/p$. <br />
We also show that the same solution operator gains a $1/2$-derivative in the H\"older-Zygmund space $\Lambda^s$ for any $s>0$, where previously it was known for $s>1$ by work of X. Gong. <br />
The main ingredients used in our proof are a Hardy-Littlewood lemma of Sobolev type and a new commutator estimate. <br />
Joint work with Liding Yao.<br />
<br />
===Xiumin Du===<br />
<br />
Title: Falconer's distance set problem<br />
<br />
Abstract: A classical question in geometric measure theory, introduced by Falconer in the 80s is, how large does the Hausdorff dimension of a compact subset in Euclidean space need to be to ensure that the Lebesgue measure of its set of pairwise Euclidean distances is positive. In this talk, I'll report some recent progress on this problem, which combines several ingredients including Orponen's radial projection theorem, Liu's L^2 identity obtained using a group action argument, and the refined decoupling theory. This is based on joint work with Alex Iosevich, Yumeng Ou, Hong Wang, and Ruixiang Zhang.<br />
<br />
===Etienne Le Masson===<br />
<br />
Title: Quantum ergodicity for Eisenstein series on large genus hyperbolic surfaces<br />
<br />
Abstract: We will present a delocalisation result for eigenfunctions of the Laplacian on finite area hyperbolic surfaces of large genus. This is a quantum ergodicity result analogous to a theorem of Zelditch showing that the mass of most L2 eigenfunctions and Eisenstein series (eigenfunctions associated with the continuous spectrum) equidistributes when the eigenvalues tend to infinity. Here we will fix a bounded spectral window and look at a similar equidistribution phenomenon when the area/genus goes to infinity (more precisely the surfaces Benjamini-Schramm converge to the plane). The conditions we require on the surfaces are satisfied with high probability in the Weil-Petersson model of random surfaces introduced by Mirzakhani. They also apply to congruence covers of the modular surface, where we recover a result of Nelson on the equidistribution of Maass forms (with weaker convergence rate). The proof is based on ergodic theory methods.<br />
Joint work with Tuomas Sahlsten.<br />
<br />
===Theresa Anderson===<br />
<br />
Title: Dyadic analysis (virtually) meets number theory<br />
<br />
Abstract: In this talk we discuss two ways in which dyadic analysis and number theory share a rich interaction. The first, which we will spend the most time motivating and discussing, involves a complete classification of "distinct dyadic systems". These are sets of grids which allow one to compare any Euclidean ball nicely with any dyadic cube, and allow for showing that a large number of continuous objects and operators can be "replaced" with their easier dyadic counterparts. If time remains, secondly, we define and make progress on showing the (failure) of a "Hasse principle" in harmonic analysis; specifically, we discuss the interplay between number theory and dyadic analysis that allows us to construct a measure that is "p-adic" doubling for any prime p (in a finite set of primes), yet not doubling overall.<br />
<br />
===Nathan Wagner===<br />
<br />
Title: Weighted Estimates for the Bergman and Szegő projections on strongly pseudoconvex domains with near minimal smoothness <br />
<br />
Abstract: The Bergman and Szegő projections are fundamental operators in complex analysis in one and several complex variables. Consequently, the mapping properties of these operators on L^p and other function spaces have been extensively studied. In this talk, we discuss some recent results for these operators on strongly pseudoconvex domains with near minimal smoothness. In particular, weighted L^p estimates are obtained, where the weight belongs to a suitable generalization of the Békollé-Bonami or Muckenhoupt class. For these domains with less boundary regularity, we use an operator-theoretic technique that goes back to Kerzman and Stein. We also obtain weighted estimates for the endpoint p=1, including weighted weak-type (1,1) estimates. Here we use a modified version of singular-integral theory and a generalization of the Riesz-Kolmogorov characterization of precompact subsets of Lebesgue spaces. This talk is based on joint work with Brett Wick and Cody Stockdale.<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]<br />
<br />
<br />
Graduate Student Seminar:<br />
<br />
https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html</div>Beltranhttps://hilbert.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=21001Analysis Seminar2021-03-15T16:09:41Z<p>Beltran: /* Abstracts */</p>
<hr />
<div><br />
The 2020-2021 Analysis Seminar will be organized by David Beltran and Andreas Seeger.<br />
It will be online for the entire academic year. The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar at different times, to accommodate speakers).<br />
<br />
Zoom links will be sent to those who have signed up for the Analysis Seminar List. If you would like to subscribe to the Analysis seminar list, send a blank email to analysis+join (at) g-groups (dot) wisc (dot) edu as well as an additional email to David and Andreas (dbeltran, seeger at math (dot) wisc (dot) edu) to notify the request.<br />
<br />
If you'd like to suggest speakers for the spring semester please contact David and Andreas.<br />
<br />
<br />
<br />
=[[Previous_Analysis_seminars]]=<br />
<br />
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars<br />
<br />
= Current Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 22<br />
|Alexei Poltoratski<br />
|UW Madison<br />
|[[#Alexei Poltoratski | Dirac inner functions ]]<br />
| <br />
|-<br />
|September 29<br />
|Joris Roos<br />
|University of Massachusetts - Lowell<br />
|[[#Polona Durcik and Joris Rooslinktoabstract | A triangular Hilbert transform with curvature, I ]]<br />
| <br />
|-<br />
|Wednesday September 30, 4 p.m.<br />
|Polona Durcik<br />
|Chapman University<br />
|[[#Polona Durcik and Joris Roos | A triangular Hilbert transform with curvature, II ]]<br />
| <br />
|-<br />
|October 6<br />
|Andrew Zimmer<br />
|UW Madison<br />
|[[#Andrew Zimmer | Complex analytic problems on domains with good intrinsic geometry ]]<br />
| <br />
|-<br />
|October 13<br />
|Hong Wang <br />
|Princeton/IAS<br />
|[[#Hong Wang | Improved decoupling for the parabola ]]<br />
| <br />
|-<br />
|October 20<br />
|Kevin Luli<br />
|UC Davis<br />
|[[#Kevin Luli | Smooth Nonnegative Interpolation ]]<br />
| <br />
|-<br />
|October 21, 4.00 p.m.<br />
|Niclas Technau<br />
|UW Madison<br />
|[[#Niclas Technau | Number theoretic applications of oscillatory integrals ]]<br />
| <br />
|-<br />
|October 27<br />
|Terence Harris<br />
| Cornell University<br />
|[[#Terence Harris | Low dimensional pinned distance sets via spherical averages ]]<br />
| <br />
|-<br />
|Monday, November 2, 4 p.m.<br />
|Yuval Wigderson<br />
|Stanford University<br />
|[[#Yuval Wigderson | New perspectives on the uncertainty principle ]]<br />
| <br />
|-<br />
|November 10, 10 a.m. <br />
|Óscar Domínguez<br />
| Universidad Complutense de Madrid<br />
|[[#Oscar Dominguez | New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions ]]<br />
| <br />
|-<br />
|November 17<br />
|Tamas Titkos<br />
|BBS U of Applied Sciences and Renyi Institute<br />
|[[#Tamas Titkos | Isometries of Wasserstein spaces ]]<br />
| <br />
|-<br />
|November 24<br />
|Shukun Wu<br />
|University of Illinois (Urbana-Champaign)<br />
||[[#Shukun Wu | On the Bochner-Riesz operator and the maximal Bochner-Riesz operator ]] <br />
| <br />
|-<br />
|December 1<br />
| Jonathan Hickman<br />
| The University of Edinburgh<br />
|[[#Jonathan Hickman | Sobolev improving for averages over space curves ]]<br />
| <br />
|-<br />
|February 2, 7:00 p.m.<br />
|Hanlong Fang<br />
|UW Madison<br />
|[[#Hanlong Fang | Canonical blow-ups of Grassmann manifolds ]]<br />
| <br />
|-<br />
|February 9<br />
|Bingyang Hu<br />
|Purdue University<br />
|[[#Bingyang Hu | Some structure theorems on general doubling measures ]]<br />
| <br />
|-<br />
|February 16<br />
|Krystal Taylor<br />
|The Ohio State University<br />
|[[#Krystal Taylor | Quantifications of the Besicovitch Projection theorem in a nonlinear setting ]]<br />
|<br />
|-<br />
|February 23<br />
|Dominique Maldague<br />
|MIT<br />
|[[#Dominique Maldague | A new proof of decoupling for the parabola ]]<br />
|<br />
|-<br />
|March 2<br />
|Diogo Oliveira e Silva<br />
|University of Birmingham<br />
|[[#Diogo Oliveira e Silva | Global maximizers for spherical restriction ]]<br />
|<br />
|-<br />
|March 9<br />
|Oleg Safronov <br />
|University of North Carolina Charlotte<br />
|[[#Oleg Safronov | Relations between discrete and continuous spectra of differential operators ]]<br />
|<br />
|-<br />
|March 16<br />
|Ziming Shi<br />
|Rutgers University<br />
|[[#Ziming Shi | Sharp Sobolev 1/2-estimate for dbar equations on strictly pseudoconvex domains with C^2 boundary ]]<br />
|<br />
|-<br />
|March 23<br />
|Xiumin Du<br />
|Northwestern University<br />
|[[#Xiumin Du | Falconer's distance set problem ]]<br />
|<br />
|-<br />
|March 30, 10:00 a.m.<br />
|Etienne Le Masson<br />
|Cergy Paris University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 6<br />
|Theresa Anderson <br />
|Purdue University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 13<br />
|Nathan Wagner<br />
|Washington University St. Louis<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 20<br />
|Jongchon Kim<br />
| University of British Columbia<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 27<br />
|Yumeng Ou<br />
|University of Pennsylvania<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|May 4<br />
|David Beltran<br />
|UW Madison<br />
|[[#linktoabstract | Title ]]<br />
|}<br />
<br />
=Abstracts=<br />
===Alexei Poltoratski===<br />
<br />
Title: Dirac inner functions<br />
<br />
Abstract: My talk will focus on some new (and old) complex analytic objects arising from Dirac systems of differential equations.<br />
We will discuss connections between problems in complex function theory, spectral and scattering problems for differential<br />
operators and the non-linear Fourier transform.<br />
<br />
===Polona Durcik and Joris Roos===<br />
<br />
Title: A triangular Hilbert transform with curvature, I & II.<br />
<br />
Abstract: The triangular Hilbert is a two-dimensional bilinear singular<br />
originating in time-frequency analysis. No Lp bounds are currently<br />
known for this operator.<br />
In these two talks we discuss a recent joint work with Michael Christ<br />
on a variant of the triangular Hilbert transform involving curvature.<br />
This object is closely related to the bilinear Hilbert transform with<br />
curvature and a maximally modulated singular integral of Stein-Wainger<br />
type. As an application we also discuss a quantitative nonlinear Roth<br />
type theorem on patterns in the Euclidean plane.<br />
The second talk will focus on the proof of a key ingredient, a certain<br />
regularity estimate for a local operator.<br />
<br />
===Andrew Zimmer===<br />
<br />
Title: Complex analytic problems on domains with good intrinsic geometry<br />
<br />
Abstract: In this talk, I will describe a new class of domains in complex Euclidean space which is defined in terms of the existence of a Kaehler metric with good geometric properties. This class is invariant under biholomorphism and includes many well-studied classes of domains such as strongly pseudoconvex domains, finite type domains in dimension two, convex domains, homogeneous domains, and embeddings of Teichmuller spaces. Further, certain analytic problems are tractable for domains in this family even when the boundary is non-smooth. In particular, it is possible to characterize the domains in this family where the dbar-Neumann operator on (0, q)-forms is compact (which generalizes an old result of Fu-Straube for convex domains).<br />
<br />
===Hong Wang===<br />
<br />
Title: Improved decoupling for the parabola<br />
<br />
Abstract: In 2014, Bourgain and Demeter proved the $l^2$ decoupling estimates for the paraboloid with constant $R^{\epsilon}$. <br />
We prove an $(l^2, L^6)$ decoupling inequality for the parabola with constant $(\log R)^c$. This is joint work with Larry Guth and Dominique Maldague.<br />
<br />
===Kevin Luli===<br />
<br />
Title: Smooth Nonnegative Interpolation<br />
<br />
Abstract: Suppose E is an arbitrary subset of R^n. Let f: E \rightarrow [0, \infty). How can we decide if f extends to a nonnegative function C^m function F defined on all of R^n? Suppose E is finite. Can we compute a nonnegative C^m function F on R^n that agrees with f on E with the least possible C^m norm? How many computer operations does this take? In this talk, I will explain recent results on these problems. Non-negativity is one of the most important shape preserving properties for interpolants. In real life applications, the range of the interpolant is imposed by nature. For example, probability density, the amount of snow, rain, humidity, chemical concentration are all nonnegative quantities and are of interest in natural sciences. Even in one dimension, the existing techniques can only handle nonnegative interpolation under special assumptions on the data set. Our results work without any assumptions on the data sets. <br />
<br />
===Niclas Technau===<br />
<br />
Title: Number theoretic applications of oscillatory integrals<br />
<br />
Abstract: We discuss how the analysis of oscillatory integrals can be used to solve number theoretic problems. More specifically, the focus will be on understanding fine-scale statistics of sequences on the unit circle. Further, we shall briefly explain a connection to quantum chaos.<br />
<br />
===Terence Harris===<br />
<br />
Title: Low dimensional pinned distance sets via spherical averages<br />
<br />
Abstract: An inequality is derived for the average t-energy of weighted pinned distance measures, where 0 < t < 1, in terms of the L^2 spherical averages of Fourier transforms of measures. This generalises the result of Liu (originally for Lebesgue measure) to pinned distance sets of dimension smaller than 1, and strengthens Mattila's result from 1987, originally for the full distance set.<br />
<br />
===Yuval Wigderson===<br />
<br />
Title: New perspectives on the uncertainty principle<br />
<br />
Abstract: The phrase ``uncertainty principle'' refers to a wide array of results in several disparate fields of mathematics, all of which capture the notion that a function and its Fourier transform cannot both be ``very localized''. The measure of localization varies from one uncertainty principle to the next, and well-studied notions include the variance (and higher moments), the entropy, the support-size, and the rate of decay at infinity. Similarly, the proofs of the various uncertainty principles rely on a range of tools, from the elementary to the very deep. In this talk, I'll describe how many of the uncertainty principles all follow from a single, simple result, whose proof uses only a basic property of the Fourier transform: that it and its inverse are bounded as operators $L^1 \to L^\infty$. Using this result, one can also prove new variants of the uncertainty principle, which apply to new measures of localization and to operators other than the Fourier transform. This is joint work with Avi Wigderson.<br />
<br />
===Oscar Dominguez===<br />
<br />
Title: New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions<br />
<br />
Abstract: The celebrated Bourgain--Brezis--Mironescu formula enables us to recover Sobolev spaces in terms of limits of Gagliardo seminorms. Very recently, Brezis, Van Schaftingen and Yung have proposed an alternative methodology to approach Sobolev spaces via limits of weak-type Gagliardo functionals. The goal of this talk is twofold. Firstly, we will show that the BvSY result is a special case of a more general phenomenon based on maximal inequalities. In particular, we shall derive not only analogs of the BvSY theorem for different kinds of function spaces (Lebesgue, Calderon, higher-order Sobolev, …), but also applications to ergodic theory, Fourier series, etc. In the second part of the talk, we shall investigate the fractional setting in the BvSY theorem. Our approach is based on new Garsia-type inequalities and an application of the Caffarelli--Silvestre extension. This is joint work with Mario Milman.<br />
<br />
===Tamas Titkos===<br />
<br />
Title: Isometries of Wasserstein spaces<br />
<br />
Abstract: Due to its nice theoretical properties and an astonishing number of<br />
applications via optimal transport problems, probably the most<br />
intensively studied metric nowadays is the p-Wasserstein metric. Given<br />
a complete and separable metric space $X$ and a real number $p\geq1$,<br />
one defines the p-Wasserstein space $\mathcal{W}_p(X)$ as the collection<br />
of Borel probability measures with finite $p$-th moment, endowed with a<br />
distance which is calculated by means of transport plans \cite{5}.<br />
<br />
The main aim of our research project is to reveal the structure of the<br />
isometry group $\mathrm{Isom}(\mathcal{W}_p(X))$. Although<br />
$\mathrm{Isom}(X)$ embeds naturally into<br />
$\mathrm{Isom}(\mathcal{W}_p(X))$ by push-forward, and this embedding<br />
turned out to be surjective in many cases (see e.g. [1]), these two<br />
groups are not isomorphic in general. Kloeckner in [2] described<br />
the isometry group of the quadratic Wasserstein space<br />
$\mathcal{W}_2(\mathbb{R}^n)$, and it turned out that the case of $n=1$<br />
is special in the sense that $\mathrm{Isom}(\mathcal{W}_2(\mathbb{R})$<br />
is extremely rich. Namely, it contains a large subgroup of wild behaving<br />
isometries that distort the shape of measures. Following this line of<br />
investigation, in \cite{3} we described<br />
$\mathrm{Isom}(\mathcal{W}_p(\mathbb{R}))$ and<br />
$\mathrm{Isom}(\mathcal{W}_p([0,1])$ for all $p\geq 1$.<br />
<br />
In this talk I will survey first some of the earlier results in the<br />
subject, and then I will present the key results of [3]. If time<br />
permits, I will also report on our most recent manuscript [4] in<br />
which we extended Kloeckner's multidimensional results. Joint work with Gy\"orgy P\'al Geh\'er (University of Reading)<br />
and D\'aniel Virosztek (IST Austria).<br />
<br />
[1] J. Bertrand and B. Kloeckner, \emph{A geometric study of Wasserstein<br />
spaces: isometric rigidity in negative curvature}, International<br />
Mathematics Research Notices, 2016 (5), 1368--1386.<br />
<br />
[2] B. Kloeckner, \emph{A geometric study of Wasserstein spaces: Euclidean<br />
spaces}, Annali della Scuola Normale Superiore di Pisa - Classe di<br />
Scienze, Serie 5, Tome 9 (2010) no. 2, 297--323.<br />
<br />
[3] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{Isometric study of<br />
Wasserstein spaces – the real line}, Trans. Amer. Math. Soc., 373<br />
(2020), 5855--5883.<br />
<br />
[4] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{The isometry group of<br />
Wasserstein spaces: The Hilbertian case}, submitted manuscript.<br />
<br />
[5] C. Villani, \emph{Optimal Transport: Old and New,}<br />
(Grundlehren der mathematischen Wissenschaften)<br />
Springer, 2009.<br />
<br />
===Shukun Wu===<br />
<br />
Title: On the Bochner-Riesz operator and the maximal Bochner-Riesz operator<br />
<br />
Abstract: The Bochner-Riesz problem is one of the most important problems in the field of Fourier analysis. It has a strong connection to other famous problems, such as the restriction conjecture and the Kakeya conjecture. In this talk, I will present some recent improvements to the Bochner-Riesz conjecture and the maximal Bochner-Riesz conjecture. The main methods we used are polynomial partitioning and the Bourgain Demeter l^2 decoupling theorem. <br />
<br />
<br />
===Jonathan Hickman===<br />
<br />
Title: Sobolev improving for averages over space curves<br />
<br />
Abstract: Consider the averaging operator given by convolution with arclength measure on compact piece of a smooth curve in R^n. A simple question is to precisely quantify the gain in regularity induced by this averaging, for instance by studying the L^p-Sobolev mapping properties of the operator. This talk will report on ongoing developments towards understanding this problem. In particular, we will explore some non-trivial necessary conditions on the gain in regularity. Joint with D. Beltran, S. Guo and A. Seeger.<br />
<br />
===Hanlong Fang===<br />
<br />
Title: Canonical blow-ups of Grassmann manifolds<br />
<br />
Abstract: We introduce certain canonical blow-ups \mathcal T_{s,p,n}, as well as their distinct submanifolds \mathcal M_{s,p,n}, of Grassmann manifolds G(p,n) by partitioning the Plücker coordinates with respect to a parameter s. Various geometric aspects of \mathcal T_{s,p,n} and \mathcal M_{s,p,n} are studied, for instance, the smoothness, the holomorphic symmetries, the (semi-)positivity of the anti-canonical bundles, the existence of Kähler-Einstein metrics, the functoriality, etc. In particular, we introduce the notion of homeward compactification, of which \mathcal T_{s,p,n} are examples, as a generalization of the wonderful compactification. <br />
<br />
===Bingyang Hu===<br />
<br />
Title: Some structure theorems on general doubling measures.<br />
<br />
Abstract: In this talk, we will first several structure theorems about general doubling measures. Secondly, we will include some main idea to prove one of these results. More precisely, we will focus on the construction of an explicit family of measures that are p-adic doubling for any finite set of primes, however, not doubling. This part generalizes the work by Boylan, Mills and Ward in 2019 in a highly non-trivial way. As some application, we apply these results (that is, the same construction) to show analogous statements for Muckenhoupt Ap weights and reverse Holder weights. This is a joint work with Tess Anderson.<br />
<br />
===Krystal Taylor===<br />
<br />
Title: Quantifications of the Besicovitch Projection theorem in a nonlinear setting <br />
<br />
Abstract: There are several classical results relating the geometry, dimension, and measure of a set to the structure of its orthogonal projections. <br />
It turns out that many nonlinear projection-type operators also have special geometry that allows us to build similar relationships between a set and its "projections", just as in the linear setting. We will discuss a series of recent results from both geometric and probabilistic vantage points. In particular, we will see that the multi-scale analysis techniques of Tao, as well as the energy techniques of Mattila, can be strengthened and generalized to projection-type operators satisfying a transversality condition. As an application, we address the Buffon curve problem, which is to find upper and lower bounds for the rate of decay of the Favard curve length of the four-corner Cantor set.<br />
<br />
===Dominique Maldague===<br />
<br />
Title: A new proof of decoupling for the parabola<br />
<br />
Abstract: Decoupling has to do with measuring the size of functions with specialized Fourier support (in our case, in a neighborhood of the truncated parabola). Bourgain and Demeter resolved the l^2 decoupling conjecture in 2014, using ingredients like the multilinear Kakeya inequality, L^2 orthogonality, and induction-on-scales. I will present the ideas that go into a new proof of decoupling and make some comparison between the two approaches. This is related to recent joint work with Larry Guth and Hong Wang, as well as forthcoming joint work with Yuqiu Fu and Larry Guth.<br />
<br />
===Diogo Oliveira e Silva===<br />
<br />
Title: Global maximizers for spherical restriction<br />
<br />
Abstract: We prove that constant functions are the unique real-valued maximizers for all $L^2-L^{2n}$ adjoint Fourier restriction inequalities on the unit sphere $\mathbb{S}^{d-1}\subset\mathbb{R}^d$, $d\in\{3,4,5,6,7\}$, where $n\geq 3$ is an integer. The proof uses tools from probability theory, Lie theory, functional analysis, and the theory of special functions. It also relies on general solutions of the underlying Euler--Lagrange equation being smooth, a fact of independent interest which we discuss. We further show that complex-valued maximizers coincide with nonnegative maximizers multiplied by the character $e^{i\xi\cdot\omega}$, for some $\xi$, thereby extending previous work of Christ & Shao (2012) to arbitrary dimensions $d\geq 2$ and general even exponents. This talk is based on results obtained with René Quilodrán.<br />
<br />
===Oleg Safronov===<br />
<br />
Title: Relations between discrete and continuous spectra of differential operators<br />
<br />
Abstract: We will discuss relations between different parts of spectra of differential operators. In particular, we will see that negative and positive spectra of Schroedinger operators are related to each other. However, there is a stipulation: one needs to consider two operators one of which is obtained from the other<br />
by flipping the sign of the potential at each point x. If one knows only that the negative spectra of the two operators are discrete, then their positive spectra do not have gaps. If one knows more about the rate of accumulation of the discrete negative eigenvalues to zero, then one can say more about the absolutely continuous component of the positive spectrum.<br />
<br />
===Ziming Shi===<br />
<br />
Title: Sharp Sobolev $1/2$-estimate for $\bar\partial$ equations on strictly pseudoconvex domains with $C^2$ boundary <br />
<br />
Abstract: We give a solution operator for $\bar\partial$ equation that gains the sharp $1/2$-derivative in the Sobolev space $H^{s,p}$ on any strictly pseudoconvex domain with $C^2$-boundary, for all $1< p < \infty$ and $s>1/p$. <br />
We also show that the same solution operator gains a $1/2$-derivative in the H\"older-Zygmund space $\Lambda^s$ for any $s>0$, where previously it was known for $s>1$ by work of X. Gong. <br />
The main ingredients used in our proof are a Hardy-Littlewood lemma of Sobolev type and a new commutator estimate. <br />
Joint work with Liding Yao.<br />
<br />
===Xiumin Du===<br />
<br />
Title: Falconer's distance set problem<br />
<br />
Abstract: A classical question in geometric measure theory, introduced by Falconer in the 80s is, how large does the Hausdorff dimension of a compact subset in Euclidean space need to be to ensure that the Lebesgue measure of its set of pairwise Euclidean distances is positive. In this talk, I'll report some recent progress on this problem, which combines several ingredients including Orponen's radial projection theorem, Liu's L^2 identity obtained using a group action argument, and the refined decoupling theory. This is based on joint work with Alex Iosevich, Yumeng Ou, Hong Wang, and Ruixiang Zhang.<br />
<br />
===Name===<br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
===Name===<br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]<br />
<br />
<br />
Graduate Student Seminar:<br />
<br />
https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html</div>Beltranhttps://hilbert.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=21000Analysis Seminar2021-03-15T16:09:13Z<p>Beltran: /* Current Analysis Seminar Schedule */</p>
<hr />
<div><br />
The 2020-2021 Analysis Seminar will be organized by David Beltran and Andreas Seeger.<br />
It will be online for the entire academic year. The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar at different times, to accommodate speakers).<br />
<br />
Zoom links will be sent to those who have signed up for the Analysis Seminar List. If you would like to subscribe to the Analysis seminar list, send a blank email to analysis+join (at) g-groups (dot) wisc (dot) edu as well as an additional email to David and Andreas (dbeltran, seeger at math (dot) wisc (dot) edu) to notify the request.<br />
<br />
If you'd like to suggest speakers for the spring semester please contact David and Andreas.<br />
<br />
<br />
<br />
=[[Previous_Analysis_seminars]]=<br />
<br />
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars<br />
<br />
= Current Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 22<br />
|Alexei Poltoratski<br />
|UW Madison<br />
|[[#Alexei Poltoratski | Dirac inner functions ]]<br />
| <br />
|-<br />
|September 29<br />
|Joris Roos<br />
|University of Massachusetts - Lowell<br />
|[[#Polona Durcik and Joris Rooslinktoabstract | A triangular Hilbert transform with curvature, I ]]<br />
| <br />
|-<br />
|Wednesday September 30, 4 p.m.<br />
|Polona Durcik<br />
|Chapman University<br />
|[[#Polona Durcik and Joris Roos | A triangular Hilbert transform with curvature, II ]]<br />
| <br />
|-<br />
|October 6<br />
|Andrew Zimmer<br />
|UW Madison<br />
|[[#Andrew Zimmer | Complex analytic problems on domains with good intrinsic geometry ]]<br />
| <br />
|-<br />
|October 13<br />
|Hong Wang <br />
|Princeton/IAS<br />
|[[#Hong Wang | Improved decoupling for the parabola ]]<br />
| <br />
|-<br />
|October 20<br />
|Kevin Luli<br />
|UC Davis<br />
|[[#Kevin Luli | Smooth Nonnegative Interpolation ]]<br />
| <br />
|-<br />
|October 21, 4.00 p.m.<br />
|Niclas Technau<br />
|UW Madison<br />
|[[#Niclas Technau | Number theoretic applications of oscillatory integrals ]]<br />
| <br />
|-<br />
|October 27<br />
|Terence Harris<br />
| Cornell University<br />
|[[#Terence Harris | Low dimensional pinned distance sets via spherical averages ]]<br />
| <br />
|-<br />
|Monday, November 2, 4 p.m.<br />
|Yuval Wigderson<br />
|Stanford University<br />
|[[#Yuval Wigderson | New perspectives on the uncertainty principle ]]<br />
| <br />
|-<br />
|November 10, 10 a.m. <br />
|Óscar Domínguez<br />
| Universidad Complutense de Madrid<br />
|[[#Oscar Dominguez | New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions ]]<br />
| <br />
|-<br />
|November 17<br />
|Tamas Titkos<br />
|BBS U of Applied Sciences and Renyi Institute<br />
|[[#Tamas Titkos | Isometries of Wasserstein spaces ]]<br />
| <br />
|-<br />
|November 24<br />
|Shukun Wu<br />
|University of Illinois (Urbana-Champaign)<br />
||[[#Shukun Wu | On the Bochner-Riesz operator and the maximal Bochner-Riesz operator ]] <br />
| <br />
|-<br />
|December 1<br />
| Jonathan Hickman<br />
| The University of Edinburgh<br />
|[[#Jonathan Hickman | Sobolev improving for averages over space curves ]]<br />
| <br />
|-<br />
|February 2, 7:00 p.m.<br />
|Hanlong Fang<br />
|UW Madison<br />
|[[#Hanlong Fang | Canonical blow-ups of Grassmann manifolds ]]<br />
| <br />
|-<br />
|February 9<br />
|Bingyang Hu<br />
|Purdue University<br />
|[[#Bingyang Hu | Some structure theorems on general doubling measures ]]<br />
| <br />
|-<br />
|February 16<br />
|Krystal Taylor<br />
|The Ohio State University<br />
|[[#Krystal Taylor | Quantifications of the Besicovitch Projection theorem in a nonlinear setting ]]<br />
|<br />
|-<br />
|February 23<br />
|Dominique Maldague<br />
|MIT<br />
|[[#Dominique Maldague | A new proof of decoupling for the parabola ]]<br />
|<br />
|-<br />
|March 2<br />
|Diogo Oliveira e Silva<br />
|University of Birmingham<br />
|[[#Diogo Oliveira e Silva | Global maximizers for spherical restriction ]]<br />
|<br />
|-<br />
|March 9<br />
|Oleg Safronov <br />
|University of North Carolina Charlotte<br />
|[[#Oleg Safronov | Relations between discrete and continuous spectra of differential operators ]]<br />
|<br />
|-<br />
|March 16<br />
|Ziming Shi<br />
|Rutgers University<br />
|[[#Ziming Shi | Sharp Sobolev 1/2-estimate for dbar equations on strictly pseudoconvex domains with C^2 boundary ]]<br />
|<br />
|-<br />
|March 23<br />
|Xiumin Du<br />
|Northwestern University<br />
|[[#Xiumin Du | Falconer's distance set problem ]]<br />
|<br />
|-<br />
|March 30, 10:00 a.m.<br />
|Etienne Le Masson<br />
|Cergy Paris University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 6<br />
|Theresa Anderson <br />
|Purdue University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 13<br />
|Nathan Wagner<br />
|Washington University St. Louis<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 20<br />
|Jongchon Kim<br />
| University of British Columbia<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 27<br />
|Yumeng Ou<br />
|University of Pennsylvania<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|May 4<br />
|David Beltran<br />
|UW Madison<br />
|[[#linktoabstract | Title ]]<br />
|}<br />
<br />
=Abstracts=<br />
===Alexei Poltoratski===<br />
<br />
Title: Dirac inner functions<br />
<br />
Abstract: My talk will focus on some new (and old) complex analytic objects arising from Dirac systems of differential equations.<br />
We will discuss connections between problems in complex function theory, spectral and scattering problems for differential<br />
operators and the non-linear Fourier transform.<br />
<br />
===Polona Durcik and Joris Roos===<br />
<br />
Title: A triangular Hilbert transform with curvature, I & II.<br />
<br />
Abstract: The triangular Hilbert is a two-dimensional bilinear singular<br />
originating in time-frequency analysis. No Lp bounds are currently<br />
known for this operator.<br />
In these two talks we discuss a recent joint work with Michael Christ<br />
on a variant of the triangular Hilbert transform involving curvature.<br />
This object is closely related to the bilinear Hilbert transform with<br />
curvature and a maximally modulated singular integral of Stein-Wainger<br />
type. As an application we also discuss a quantitative nonlinear Roth<br />
type theorem on patterns in the Euclidean plane.<br />
The second talk will focus on the proof of a key ingredient, a certain<br />
regularity estimate for a local operator.<br />
<br />
===Andrew Zimmer===<br />
<br />
Title: Complex analytic problems on domains with good intrinsic geometry<br />
<br />
Abstract: In this talk, I will describe a new class of domains in complex Euclidean space which is defined in terms of the existence of a Kaehler metric with good geometric properties. This class is invariant under biholomorphism and includes many well-studied classes of domains such as strongly pseudoconvex domains, finite type domains in dimension two, convex domains, homogeneous domains, and embeddings of Teichmuller spaces. Further, certain analytic problems are tractable for domains in this family even when the boundary is non-smooth. In particular, it is possible to characterize the domains in this family where the dbar-Neumann operator on (0, q)-forms is compact (which generalizes an old result of Fu-Straube for convex domains).<br />
<br />
===Hong Wang===<br />
<br />
Title: Improved decoupling for the parabola<br />
<br />
Abstract: In 2014, Bourgain and Demeter proved the $l^2$ decoupling estimates for the paraboloid with constant $R^{\epsilon}$. <br />
We prove an $(l^2, L^6)$ decoupling inequality for the parabola with constant $(\log R)^c$. This is joint work with Larry Guth and Dominique Maldague.<br />
<br />
===Kevin Luli===<br />
<br />
Title: Smooth Nonnegative Interpolation<br />
<br />
Abstract: Suppose E is an arbitrary subset of R^n. Let f: E \rightarrow [0, \infty). How can we decide if f extends to a nonnegative function C^m function F defined on all of R^n? Suppose E is finite. Can we compute a nonnegative C^m function F on R^n that agrees with f on E with the least possible C^m norm? How many computer operations does this take? In this talk, I will explain recent results on these problems. Non-negativity is one of the most important shape preserving properties for interpolants. In real life applications, the range of the interpolant is imposed by nature. For example, probability density, the amount of snow, rain, humidity, chemical concentration are all nonnegative quantities and are of interest in natural sciences. Even in one dimension, the existing techniques can only handle nonnegative interpolation under special assumptions on the data set. Our results work without any assumptions on the data sets. <br />
<br />
===Niclas Technau===<br />
<br />
Title: Number theoretic applications of oscillatory integrals<br />
<br />
Abstract: We discuss how the analysis of oscillatory integrals can be used to solve number theoretic problems. More specifically, the focus will be on understanding fine-scale statistics of sequences on the unit circle. Further, we shall briefly explain a connection to quantum chaos.<br />
<br />
===Terence Harris===<br />
<br />
Title: Low dimensional pinned distance sets via spherical averages<br />
<br />
Abstract: An inequality is derived for the average t-energy of weighted pinned distance measures, where 0 < t < 1, in terms of the L^2 spherical averages of Fourier transforms of measures. This generalises the result of Liu (originally for Lebesgue measure) to pinned distance sets of dimension smaller than 1, and strengthens Mattila's result from 1987, originally for the full distance set.<br />
<br />
===Yuval Wigderson===<br />
<br />
Title: New perspectives on the uncertainty principle<br />
<br />
Abstract: The phrase ``uncertainty principle'' refers to a wide array of results in several disparate fields of mathematics, all of which capture the notion that a function and its Fourier transform cannot both be ``very localized''. The measure of localization varies from one uncertainty principle to the next, and well-studied notions include the variance (and higher moments), the entropy, the support-size, and the rate of decay at infinity. Similarly, the proofs of the various uncertainty principles rely on a range of tools, from the elementary to the very deep. In this talk, I'll describe how many of the uncertainty principles all follow from a single, simple result, whose proof uses only a basic property of the Fourier transform: that it and its inverse are bounded as operators $L^1 \to L^\infty$. Using this result, one can also prove new variants of the uncertainty principle, which apply to new measures of localization and to operators other than the Fourier transform. This is joint work with Avi Wigderson.<br />
<br />
===Oscar Dominguez===<br />
<br />
Title: New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions<br />
<br />
Abstract: The celebrated Bourgain--Brezis--Mironescu formula enables us to recover Sobolev spaces in terms of limits of Gagliardo seminorms. Very recently, Brezis, Van Schaftingen and Yung have proposed an alternative methodology to approach Sobolev spaces via limits of weak-type Gagliardo functionals. The goal of this talk is twofold. Firstly, we will show that the BvSY result is a special case of a more general phenomenon based on maximal inequalities. In particular, we shall derive not only analogs of the BvSY theorem for different kinds of function spaces (Lebesgue, Calderon, higher-order Sobolev, …), but also applications to ergodic theory, Fourier series, etc. In the second part of the talk, we shall investigate the fractional setting in the BvSY theorem. Our approach is based on new Garsia-type inequalities and an application of the Caffarelli--Silvestre extension. This is joint work with Mario Milman.<br />
<br />
===Tamas Titkos===<br />
<br />
Title: Isometries of Wasserstein spaces<br />
<br />
Abstract: Due to its nice theoretical properties and an astonishing number of<br />
applications via optimal transport problems, probably the most<br />
intensively studied metric nowadays is the p-Wasserstein metric. Given<br />
a complete and separable metric space $X$ and a real number $p\geq1$,<br />
one defines the p-Wasserstein space $\mathcal{W}_p(X)$ as the collection<br />
of Borel probability measures with finite $p$-th moment, endowed with a<br />
distance which is calculated by means of transport plans \cite{5}.<br />
<br />
The main aim of our research project is to reveal the structure of the<br />
isometry group $\mathrm{Isom}(\mathcal{W}_p(X))$. Although<br />
$\mathrm{Isom}(X)$ embeds naturally into<br />
$\mathrm{Isom}(\mathcal{W}_p(X))$ by push-forward, and this embedding<br />
turned out to be surjective in many cases (see e.g. [1]), these two<br />
groups are not isomorphic in general. Kloeckner in [2] described<br />
the isometry group of the quadratic Wasserstein space<br />
$\mathcal{W}_2(\mathbb{R}^n)$, and it turned out that the case of $n=1$<br />
is special in the sense that $\mathrm{Isom}(\mathcal{W}_2(\mathbb{R})$<br />
is extremely rich. Namely, it contains a large subgroup of wild behaving<br />
isometries that distort the shape of measures. Following this line of<br />
investigation, in \cite{3} we described<br />
$\mathrm{Isom}(\mathcal{W}_p(\mathbb{R}))$ and<br />
$\mathrm{Isom}(\mathcal{W}_p([0,1])$ for all $p\geq 1$.<br />
<br />
In this talk I will survey first some of the earlier results in the<br />
subject, and then I will present the key results of [3]. If time<br />
permits, I will also report on our most recent manuscript [4] in<br />
which we extended Kloeckner's multidimensional results. Joint work with Gy\"orgy P\'al Geh\'er (University of Reading)<br />
and D\'aniel Virosztek (IST Austria).<br />
<br />
[1] J. Bertrand and B. Kloeckner, \emph{A geometric study of Wasserstein<br />
spaces: isometric rigidity in negative curvature}, International<br />
Mathematics Research Notices, 2016 (5), 1368--1386.<br />
<br />
[2] B. Kloeckner, \emph{A geometric study of Wasserstein spaces: Euclidean<br />
spaces}, Annali della Scuola Normale Superiore di Pisa - Classe di<br />
Scienze, Serie 5, Tome 9 (2010) no. 2, 297--323.<br />
<br />
[3] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{Isometric study of<br />
Wasserstein spaces – the real line}, Trans. Amer. Math. Soc., 373<br />
(2020), 5855--5883.<br />
<br />
[4] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{The isometry group of<br />
Wasserstein spaces: The Hilbertian case}, submitted manuscript.<br />
<br />
[5] C. Villani, \emph{Optimal Transport: Old and New,}<br />
(Grundlehren der mathematischen Wissenschaften)<br />
Springer, 2009.<br />
<br />
===Shukun Wu===<br />
<br />
Title: On the Bochner-Riesz operator and the maximal Bochner-Riesz operator<br />
<br />
Abstract: The Bochner-Riesz problem is one of the most important problems in the field of Fourier analysis. It has a strong connection to other famous problems, such as the restriction conjecture and the Kakeya conjecture. In this talk, I will present some recent improvements to the Bochner-Riesz conjecture and the maximal Bochner-Riesz conjecture. The main methods we used are polynomial partitioning and the Bourgain Demeter l^2 decoupling theorem. <br />
<br />
<br />
===Jonathan Hickman===<br />
<br />
Title: Sobolev improving for averages over space curves<br />
<br />
Abstract: Consider the averaging operator given by convolution with arclength measure on compact piece of a smooth curve in R^n. A simple question is to precisely quantify the gain in regularity induced by this averaging, for instance by studying the L^p-Sobolev mapping properties of the operator. This talk will report on ongoing developments towards understanding this problem. In particular, we will explore some non-trivial necessary conditions on the gain in regularity. Joint with D. Beltran, S. Guo and A. Seeger.<br />
<br />
===Hanlong Fang===<br />
<br />
Title: Canonical blow-ups of Grassmann manifolds<br />
<br />
Abstract: We introduce certain canonical blow-ups \mathcal T_{s,p,n}, as well as their distinct submanifolds \mathcal M_{s,p,n}, of Grassmann manifolds G(p,n) by partitioning the Plücker coordinates with respect to a parameter s. Various geometric aspects of \mathcal T_{s,p,n} and \mathcal M_{s,p,n} are studied, for instance, the smoothness, the holomorphic symmetries, the (semi-)positivity of the anti-canonical bundles, the existence of Kähler-Einstein metrics, the functoriality, etc. In particular, we introduce the notion of homeward compactification, of which \mathcal T_{s,p,n} are examples, as a generalization of the wonderful compactification. <br />
<br />
===Bingyang Hu===<br />
<br />
Title: Some structure theorems on general doubling measures.<br />
<br />
Abstract: In this talk, we will first several structure theorems about general doubling measures. Secondly, we will include some main idea to prove one of these results. More precisely, we will focus on the construction of an explicit family of measures that are p-adic doubling for any finite set of primes, however, not doubling. This part generalizes the work by Boylan, Mills and Ward in 2019 in a highly non-trivial way. As some application, we apply these results (that is, the same construction) to show analogous statements for Muckenhoupt Ap weights and reverse Holder weights. This is a joint work with Tess Anderson.<br />
<br />
===Krystal Taylor===<br />
<br />
Title: Quantifications of the Besicovitch Projection theorem in a nonlinear setting <br />
<br />
Abstract: There are several classical results relating the geometry, dimension, and measure of a set to the structure of its orthogonal projections. <br />
It turns out that many nonlinear projection-type operators also have special geometry that allows us to build similar relationships between a set and its "projections", just as in the linear setting. We will discuss a series of recent results from both geometric and probabilistic vantage points. In particular, we will see that the multi-scale analysis techniques of Tao, as well as the energy techniques of Mattila, can be strengthened and generalized to projection-type operators satisfying a transversality condition. As an application, we address the Buffon curve problem, which is to find upper and lower bounds for the rate of decay of the Favard curve length of the four-corner Cantor set.<br />
<br />
===Dominique Maldague===<br />
<br />
Title: A new proof of decoupling for the parabola<br />
<br />
Abstract: Decoupling has to do with measuring the size of functions with specialized Fourier support (in our case, in a neighborhood of the truncated parabola). Bourgain and Demeter resolved the l^2 decoupling conjecture in 2014, using ingredients like the multilinear Kakeya inequality, L^2 orthogonality, and induction-on-scales. I will present the ideas that go into a new proof of decoupling and make some comparison between the two approaches. This is related to recent joint work with Larry Guth and Hong Wang, as well as forthcoming joint work with Yuqiu Fu and Larry Guth.<br />
<br />
===Diogo Oliveira e Silva===<br />
<br />
Title: Global maximizers for spherical restriction<br />
<br />
Abstract: We prove that constant functions are the unique real-valued maximizers for all $L^2-L^{2n}$ adjoint Fourier restriction inequalities on the unit sphere $\mathbb{S}^{d-1}\subset\mathbb{R}^d$, $d\in\{3,4,5,6,7\}$, where $n\geq 3$ is an integer. The proof uses tools from probability theory, Lie theory, functional analysis, and the theory of special functions. It also relies on general solutions of the underlying Euler--Lagrange equation being smooth, a fact of independent interest which we discuss. We further show that complex-valued maximizers coincide with nonnegative maximizers multiplied by the character $e^{i\xi\cdot\omega}$, for some $\xi$, thereby extending previous work of Christ & Shao (2012) to arbitrary dimensions $d\geq 2$ and general even exponents. This talk is based on results obtained with René Quilodrán.<br />
<br />
===Oleg Safronov===<br />
<br />
Title: Relations between discrete and continuous spectra of differential operators<br />
<br />
Abstract: We will discuss relations between different parts of spectra of differential operators. In particular, we will see that negative and positive spectra of Schroedinger operators are related to each other. However, there is a stipulation: one needs to consider two operators one of which is obtained from the other<br />
by flipping the sign of the potential at each point x. If one knows only that the negative spectra of the two operators are discrete, then their positive spectra do not have gaps. If one knows more about the rate of accumulation of the discrete negative eigenvalues to zero, then one can say more about the absolutely continuous component of the positive spectrum.<br />
<br />
===Ziming Shi===<br />
<br />
Title: Sharp Sobolev $1/2$-estimate for $\bar\partial$ equations on strictly pseudoconvex domains with $C^2$ boundary <br />
<br />
Abstract: We give a solution operator for $\bar\partial$ equation that gains the sharp $1/2$-derivative in the Sobolev space $H^{s,p}$ on any strictly pseudoconvex domain with $C^2$-boundary, for all $1< p < \infty$ and $s>1/p$. <br />
We also show that the same solution operator gains a $1/2$-derivative in the H\"older-Zygmund space $\Lambda^s$ for any $s>0$, where previously it was known for $s>1$ by work of X. Gong. <br />
The main ingredients used in our proof are a Hardy-Littlewood lemma of Sobolev type and a new commutator estimate. <br />
Joint work with Liding Yao.<br />
<br />
===Name===<br />
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Title:<br />
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Abstract:<br />
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===Name===<br />
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Title:<br />
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===Name===<br />
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Title:<br />
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Abstract:<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]<br />
<br />
<br />
Graduate Student Seminar:<br />
<br />
https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html</div>Beltranhttps://hilbert.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=20871Analysis Seminar2021-02-22T19:51:00Z<p>Beltran: /* Abstracts */</p>
<hr />
<div><br />
The 2020-2021 Analysis Seminar will be organized by David Beltran and Andreas Seeger.<br />
It will be online for the entire academic year. The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar at different times, to accommodate speakers).<br />
<br />
Zoom links will be sent to those who have signed up for the Analysis Seminar List. If you would like to subscribe to the Analysis seminar list, send a blank email to analysis+join (at) g-groups (dot) wisc (dot) edu as well as an additional email to David and Andreas (dbeltran, seeger at math (dot) wisc (dot) edu) to notify the request.<br />
<br />
If you'd like to suggest speakers for the spring semester please contact David and Andreas.<br />
<br />
<br />
<br />
=[[Previous_Analysis_seminars]]=<br />
<br />
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars<br />
<br />
= Current Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 22<br />
|Alexei Poltoratski<br />
|UW Madison<br />
|[[#Alexei Poltoratski | Dirac inner functions ]]<br />
| <br />
|-<br />
|September 29<br />
|Joris Roos<br />
|University of Massachusetts - Lowell<br />
|[[#Polona Durcik and Joris Rooslinktoabstract | A triangular Hilbert transform with curvature, I ]]<br />
| <br />
|-<br />
|Wednesday September 30, 4 p.m.<br />
|Polona Durcik<br />
|Chapman University<br />
|[[#Polona Durcik and Joris Roos | A triangular Hilbert transform with curvature, II ]]<br />
| <br />
|-<br />
|October 6<br />
|Andrew Zimmer<br />
|UW Madison<br />
|[[#Andrew Zimmer | Complex analytic problems on domains with good intrinsic geometry ]]<br />
| <br />
|-<br />
|October 13<br />
|Hong Wang <br />
|Princeton/IAS<br />
|[[#Hong Wang | Improved decoupling for the parabola ]]<br />
| <br />
|-<br />
|October 20<br />
|Kevin Luli<br />
|UC Davis<br />
|[[#Kevin Luli | Smooth Nonnegative Interpolation ]]<br />
| <br />
|-<br />
|October 21, 4.00 p.m.<br />
|Niclas Technau<br />
|UW Madison<br />
|[[#Niclas Technau | Number theoretic applications of oscillatory integrals ]]<br />
| <br />
|-<br />
|October 27<br />
|Terence Harris<br />
| Cornell University<br />
|[[#Terence Harris | Low dimensional pinned distance sets via spherical averages ]]<br />
| <br />
|-<br />
|Monday, November 2, 4 p.m.<br />
|Yuval Wigderson<br />
|Stanford University<br />
|[[#Yuval Wigderson | New perspectives on the uncertainty principle ]]<br />
| <br />
|-<br />
|November 10, 10 a.m. <br />
|Óscar Domínguez<br />
| Universidad Complutense de Madrid<br />
|[[#Oscar Dominguez | New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions ]]<br />
| <br />
|-<br />
|November 17<br />
|Tamas Titkos<br />
|BBS U of Applied Sciences and Renyi Institute<br />
|[[#Tamas Titkos | Isometries of Wasserstein spaces ]]<br />
| <br />
|-<br />
|November 24<br />
|Shukun Wu<br />
|University of Illinois (Urbana-Champaign)<br />
||[[#Shukun Wu | On the Bochner-Riesz operator and the maximal Bochner-Riesz operator ]] <br />
| <br />
|-<br />
|December 1<br />
| Jonathan Hickman<br />
| The University of Edinburgh<br />
|[[#Jonathan Hickman | Sobolev improving for averages over space curves ]]<br />
| <br />
|-<br />
|February 2, 7:00 p.m.<br />
|Hanlong Fang<br />
|UW Madison<br />
|[[#Hanlong Fang | Canonical blow-ups of Grassmann manifolds ]]<br />
| <br />
|-<br />
|February 9<br />
|Bingyang Hu<br />
|Purdue University<br />
|[[#Bingyang Hu | Some structure theorems on general doubling measures ]]<br />
| <br />
|-<br />
|February 16<br />
|Krystal Taylor<br />
|The Ohio State University<br />
|[[#Krystal Taylor | Quantifications of the Besicovitch Projection theorem in a nonlinear setting ]]<br />
|<br />
|-<br />
|February 23<br />
|Dominique Maldague<br />
|MIT<br />
|[[#Dominique Maldague | A new proof of decoupling for the parabola ]]<br />
|<br />
|-<br />
|March 2<br />
|Diogo Oliveira e Silva<br />
|University of Birmingham<br />
|[[#Diogo Oliveira e Silva | Global maximizers for spherical restriction ]]<br />
|<br />
|-<br />
|March 9<br />
|Oleg Safronov <br />
|University of North Carolina Charlotte<br />
|[[#linktoabstract | Relations between discrete and continuous spectra of differential operators ]]<br />
|<br />
|-<br />
|March 16<br />
|Ziming Shi<br />
|Rutgers University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 23<br />
|Xiumin Du<br />
|Northwestern University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 30, 10:00 a.m.<br />
|Etienne Le Masson<br />
|Cergy Paris University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 6<br />
|Theresa Anderson <br />
|Purdue University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 13<br />
|Nathan Wagner<br />
|Washington University St. Louis<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 20<br />
|Jongchon Kim<br />
| University of British Columbia<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 27<br />
|Yumeng Ou<br />
|University of Pennsylvania<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|May 4<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|}<br />
<br />
=Abstracts=<br />
===Alexei Poltoratski===<br />
<br />
Title: Dirac inner functions<br />
<br />
Abstract: My talk will focus on some new (and old) complex analytic objects arising from Dirac systems of differential equations.<br />
We will discuss connections between problems in complex function theory, spectral and scattering problems for differential<br />
operators and the non-linear Fourier transform.<br />
<br />
===Polona Durcik and Joris Roos===<br />
<br />
Title: A triangular Hilbert transform with curvature, I & II.<br />
<br />
Abstract: The triangular Hilbert is a two-dimensional bilinear singular<br />
originating in time-frequency analysis. No Lp bounds are currently<br />
known for this operator.<br />
In these two talks we discuss a recent joint work with Michael Christ<br />
on a variant of the triangular Hilbert transform involving curvature.<br />
This object is closely related to the bilinear Hilbert transform with<br />
curvature and a maximally modulated singular integral of Stein-Wainger<br />
type. As an application we also discuss a quantitative nonlinear Roth<br />
type theorem on patterns in the Euclidean plane.<br />
The second talk will focus on the proof of a key ingredient, a certain<br />
regularity estimate for a local operator.<br />
<br />
===Andrew Zimmer===<br />
<br />
Title: Complex analytic problems on domains with good intrinsic geometry<br />
<br />
Abstract: In this talk, I will describe a new class of domains in complex Euclidean space which is defined in terms of the existence of a Kaehler metric with good geometric properties. This class is invariant under biholomorphism and includes many well-studied classes of domains such as strongly pseudoconvex domains, finite type domains in dimension two, convex domains, homogeneous domains, and embeddings of Teichmuller spaces. Further, certain analytic problems are tractable for domains in this family even when the boundary is non-smooth. In particular, it is possible to characterize the domains in this family where the dbar-Neumann operator on (0, q)-forms is compact (which generalizes an old result of Fu-Straube for convex domains).<br />
<br />
===Hong Wang===<br />
<br />
Title: Improved decoupling for the parabola<br />
<br />
Abstract: In 2014, Bourgain and Demeter proved the $l^2$ decoupling estimates for the paraboloid with constant $R^{\epsilon}$. <br />
We prove an $(l^2, L^6)$ decoupling inequality for the parabola with constant $(\log R)^c$. This is joint work with Larry Guth and Dominique Maldague.<br />
<br />
===Kevin Luli===<br />
<br />
Title: Smooth Nonnegative Interpolation<br />
<br />
Abstract: Suppose E is an arbitrary subset of R^n. Let f: E \rightarrow [0, \infty). How can we decide if f extends to a nonnegative function C^m function F defined on all of R^n? Suppose E is finite. Can we compute a nonnegative C^m function F on R^n that agrees with f on E with the least possible C^m norm? How many computer operations does this take? In this talk, I will explain recent results on these problems. Non-negativity is one of the most important shape preserving properties for interpolants. In real life applications, the range of the interpolant is imposed by nature. For example, probability density, the amount of snow, rain, humidity, chemical concentration are all nonnegative quantities and are of interest in natural sciences. Even in one dimension, the existing techniques can only handle nonnegative interpolation under special assumptions on the data set. Our results work without any assumptions on the data sets. <br />
<br />
===Niclas Technau===<br />
<br />
Title: Number theoretic applications of oscillatory integrals<br />
<br />
Abstract: We discuss how the analysis of oscillatory integrals can be used to solve number theoretic problems. More specifically, the focus will be on understanding fine-scale statistics of sequences on the unit circle. Further, we shall briefly explain a connection to quantum chaos.<br />
<br />
===Terence Harris===<br />
<br />
Title: Low dimensional pinned distance sets via spherical averages<br />
<br />
Abstract: An inequality is derived for the average t-energy of weighted pinned distance measures, where 0 < t < 1, in terms of the L^2 spherical averages of Fourier transforms of measures. This generalises the result of Liu (originally for Lebesgue measure) to pinned distance sets of dimension smaller than 1, and strengthens Mattila's result from 1987, originally for the full distance set.<br />
<br />
===Yuval Wigderson===<br />
<br />
Title: New perspectives on the uncertainty principle<br />
<br />
Abstract: The phrase ``uncertainty principle'' refers to a wide array of results in several disparate fields of mathematics, all of which capture the notion that a function and its Fourier transform cannot both be ``very localized''. The measure of localization varies from one uncertainty principle to the next, and well-studied notions include the variance (and higher moments), the entropy, the support-size, and the rate of decay at infinity. Similarly, the proofs of the various uncertainty principles rely on a range of tools, from the elementary to the very deep. In this talk, I'll describe how many of the uncertainty principles all follow from a single, simple result, whose proof uses only a basic property of the Fourier transform: that it and its inverse are bounded as operators $L^1 \to L^\infty$. Using this result, one can also prove new variants of the uncertainty principle, which apply to new measures of localization and to operators other than the Fourier transform. This is joint work with Avi Wigderson.<br />
<br />
===Oscar Dominguez===<br />
<br />
Title: New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions<br />
<br />
Abstract: The celebrated Bourgain--Brezis--Mironescu formula enables us to recover Sobolev spaces in terms of limits of Gagliardo seminorms. Very recently, Brezis, Van Schaftingen and Yung have proposed an alternative methodology to approach Sobolev spaces via limits of weak-type Gagliardo functionals. The goal of this talk is twofold. Firstly, we will show that the BvSY result is a special case of a more general phenomenon based on maximal inequalities. In particular, we shall derive not only analogs of the BvSY theorem for different kinds of function spaces (Lebesgue, Calderon, higher-order Sobolev, …), but also applications to ergodic theory, Fourier series, etc. In the second part of the talk, we shall investigate the fractional setting in the BvSY theorem. Our approach is based on new Garsia-type inequalities and an application of the Caffarelli--Silvestre extension. This is joint work with Mario Milman.<br />
<br />
===Tamas Titkos===<br />
<br />
Title: Isometries of Wasserstein spaces<br />
<br />
Abstract: Due to its nice theoretical properties and an astonishing number of<br />
applications via optimal transport problems, probably the most<br />
intensively studied metric nowadays is the p-Wasserstein metric. Given<br />
a complete and separable metric space $X$ and a real number $p\geq1$,<br />
one defines the p-Wasserstein space $\mathcal{W}_p(X)$ as the collection<br />
of Borel probability measures with finite $p$-th moment, endowed with a<br />
distance which is calculated by means of transport plans \cite{5}.<br />
<br />
The main aim of our research project is to reveal the structure of the<br />
isometry group $\mathrm{Isom}(\mathcal{W}_p(X))$. Although<br />
$\mathrm{Isom}(X)$ embeds naturally into<br />
$\mathrm{Isom}(\mathcal{W}_p(X))$ by push-forward, and this embedding<br />
turned out to be surjective in many cases (see e.g. [1]), these two<br />
groups are not isomorphic in general. Kloeckner in [2] described<br />
the isometry group of the quadratic Wasserstein space<br />
$\mathcal{W}_2(\mathbb{R}^n)$, and it turned out that the case of $n=1$<br />
is special in the sense that $\mathrm{Isom}(\mathcal{W}_2(\mathbb{R})$<br />
is extremely rich. Namely, it contains a large subgroup of wild behaving<br />
isometries that distort the shape of measures. Following this line of<br />
investigation, in \cite{3} we described<br />
$\mathrm{Isom}(\mathcal{W}_p(\mathbb{R}))$ and<br />
$\mathrm{Isom}(\mathcal{W}_p([0,1])$ for all $p\geq 1$.<br />
<br />
In this talk I will survey first some of the earlier results in the<br />
subject, and then I will present the key results of [3]. If time<br />
permits, I will also report on our most recent manuscript [4] in<br />
which we extended Kloeckner's multidimensional results. Joint work with Gy\"orgy P\'al Geh\'er (University of Reading)<br />
and D\'aniel Virosztek (IST Austria).<br />
<br />
[1] J. Bertrand and B. Kloeckner, \emph{A geometric study of Wasserstein<br />
spaces: isometric rigidity in negative curvature}, International<br />
Mathematics Research Notices, 2016 (5), 1368--1386.<br />
<br />
[2] B. Kloeckner, \emph{A geometric study of Wasserstein spaces: Euclidean<br />
spaces}, Annali della Scuola Normale Superiore di Pisa - Classe di<br />
Scienze, Serie 5, Tome 9 (2010) no. 2, 297--323.<br />
<br />
[3] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{Isometric study of<br />
Wasserstein spaces – the real line}, Trans. Amer. Math. Soc., 373<br />
(2020), 5855--5883.<br />
<br />
[4] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{The isometry group of<br />
Wasserstein spaces: The Hilbertian case}, submitted manuscript.<br />
<br />
[5] C. Villani, \emph{Optimal Transport: Old and New,}<br />
(Grundlehren der mathematischen Wissenschaften)<br />
Springer, 2009.<br />
<br />
===Shukun Wu===<br />
<br />
Title: On the Bochner-Riesz operator and the maximal Bochner-Riesz operator<br />
<br />
Abstract: The Bochner-Riesz problem is one of the most important problems in the field of Fourier analysis. It has a strong connection to other famous problems, such as the restriction conjecture and the Kakeya conjecture. In this talk, I will present some recent improvements to the Bochner-Riesz conjecture and the maximal Bochner-Riesz conjecture. The main methods we used are polynomial partitioning and the Bourgain Demeter l^2 decoupling theorem. <br />
<br />
<br />
===Jonathan Hickman===<br />
<br />
Title: Sobolev improving for averages over space curves<br />
<br />
Abstract: Consider the averaging operator given by convolution with arclength measure on compact piece of a smooth curve in R^n. A simple question is to precisely quantify the gain in regularity induced by this averaging, for instance by studying the L^p-Sobolev mapping properties of the operator. This talk will report on ongoing developments towards understanding this problem. In particular, we will explore some non-trivial necessary conditions on the gain in regularity. Joint with D. Beltran, S. Guo and A. Seeger.<br />
<br />
===Hanlong Fang===<br />
<br />
Title: Canonical blow-ups of Grassmann manifolds<br />
<br />
Abstract: We introduce certain canonical blow-ups \mathcal T_{s,p,n}, as well as their distinct submanifolds \mathcal M_{s,p,n}, of Grassmann manifolds G(p,n) by partitioning the Plücker coordinates with respect to a parameter s. Various geometric aspects of \mathcal T_{s,p,n} and \mathcal M_{s,p,n} are studied, for instance, the smoothness, the holomorphic symmetries, the (semi-)positivity of the anti-canonical bundles, the existence of Kähler-Einstein metrics, the functoriality, etc. In particular, we introduce the notion of homeward compactification, of which \mathcal T_{s,p,n} are examples, as a generalization of the wonderful compactification. <br />
<br />
===Bingyang Hu===<br />
<br />
Title: Some structure theorems on general doubling measures.<br />
<br />
Abstract: In this talk, we will first several structure theorems about general doubling measures. Secondly, we will include some main idea to prove one of these results. More precisely, we will focus on the construction of an explicit family of measures that are p-adic doubling for any finite set of primes, however, not doubling. This part generalizes the work by Boylan, Mills and Ward in 2019 in a highly non-trivial way. As some application, we apply these results (that is, the same construction) to show analogous statements for Muckenhoupt Ap weights and reverse Holder weights. This is a joint work with Tess Anderson.<br />
<br />
===Krystal Taylor===<br />
<br />
Title: Quantifications of the Besicovitch Projection theorem in a nonlinear setting <br />
<br />
Abstract: There are several classical results relating the geometry, dimension, and measure of a set to the structure of its orthogonal projections. <br />
It turns out that many nonlinear projection-type operators also have special geometry that allows us to build similar relationships between a set and its "projections", just as in the linear setting. We will discuss a series of recent results from both geometric and probabilistic vantage points. In particular, we will see that the multi-scale analysis techniques of Tao, as well as the energy techniques of Mattila, can be strengthened and generalized to projection-type operators satisfying a transversality condition. As an application, we address the Buffon curve problem, which is to find upper and lower bounds for the rate of decay of the Favard curve length of the four-corner Cantor set.<br />
<br />
===Dominique Maldague===<br />
<br />
Title: A new proof of decoupling for the parabola<br />
<br />
Abstract: Decoupling has to do with measuring the size of functions with specialized Fourier support (in our case, in a neighborhood of the truncated parabola). Bourgain and Demeter resolved the l^2 decoupling conjecture in 2014, using ingredients like the multilinear Kakeya inequality, L^2 orthogonality, and induction-on-scales. I will present the ideas that go into a new proof of decoupling and make some comparison between the two approaches. This is related to recent joint work with Larry Guth and Hong Wang, as well as forthcoming joint work with Yuqiu Fu and Larry Guth.<br />
<br />
===Diogo Oliveira e Silva===<br />
<br />
Title: Global maximizers for spherical restriction<br />
<br />
Abstract: We prove that constant functions are the unique real-valued maximizers for all $L^2-L^{2n}$ adjoint Fourier restriction inequalities on the unit sphere $\mathbb{S}^{d-1}\subset\mathbb{R}^d$, $d\in\{3,4,5,6,7\}$, where $n\geq 3$ is an integer. The proof uses tools from probability theory, Lie theory, functional analysis, and the theory of special functions. It also relies on general solutions of the underlying Euler--Lagrange equation being smooth, a fact of independent interest which we discuss. We further show that complex-valued maximizers coincide with nonnegative maximizers multiplied by the character $e^{i\xi\cdot\omega}$, for some $\xi$, thereby extending previous work of Christ & Shao (2012) to arbitrary dimensions $d\geq 2$ and general even exponents. This talk is based on results obtained with René Quilodrán.<br />
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=Extras=<br />
[[Blank Analysis Seminar Template]]<br />
<br />
<br />
Graduate Student Seminar:<br />
<br />
https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html</div>Beltranhttps://hilbert.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=20870Analysis Seminar2021-02-22T19:50:22Z<p>Beltran: /* Current Analysis Seminar Schedule */</p>
<hr />
<div><br />
The 2020-2021 Analysis Seminar will be organized by David Beltran and Andreas Seeger.<br />
It will be online for the entire academic year. The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar at different times, to accommodate speakers).<br />
<br />
Zoom links will be sent to those who have signed up for the Analysis Seminar List. If you would like to subscribe to the Analysis seminar list, send a blank email to analysis+join (at) g-groups (dot) wisc (dot) edu as well as an additional email to David and Andreas (dbeltran, seeger at math (dot) wisc (dot) edu) to notify the request.<br />
<br />
If you'd like to suggest speakers for the spring semester please contact David and Andreas.<br />
<br />
<br />
<br />
=[[Previous_Analysis_seminars]]=<br />
<br />
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars<br />
<br />
= Current Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 22<br />
|Alexei Poltoratski<br />
|UW Madison<br />
|[[#Alexei Poltoratski | Dirac inner functions ]]<br />
| <br />
|-<br />
|September 29<br />
|Joris Roos<br />
|University of Massachusetts - Lowell<br />
|[[#Polona Durcik and Joris Rooslinktoabstract | A triangular Hilbert transform with curvature, I ]]<br />
| <br />
|-<br />
|Wednesday September 30, 4 p.m.<br />
|Polona Durcik<br />
|Chapman University<br />
|[[#Polona Durcik and Joris Roos | A triangular Hilbert transform with curvature, II ]]<br />
| <br />
|-<br />
|October 6<br />
|Andrew Zimmer<br />
|UW Madison<br />
|[[#Andrew Zimmer | Complex analytic problems on domains with good intrinsic geometry ]]<br />
| <br />
|-<br />
|October 13<br />
|Hong Wang <br />
|Princeton/IAS<br />
|[[#Hong Wang | Improved decoupling for the parabola ]]<br />
| <br />
|-<br />
|October 20<br />
|Kevin Luli<br />
|UC Davis<br />
|[[#Kevin Luli | Smooth Nonnegative Interpolation ]]<br />
| <br />
|-<br />
|October 21, 4.00 p.m.<br />
|Niclas Technau<br />
|UW Madison<br />
|[[#Niclas Technau | Number theoretic applications of oscillatory integrals ]]<br />
| <br />
|-<br />
|October 27<br />
|Terence Harris<br />
| Cornell University<br />
|[[#Terence Harris | Low dimensional pinned distance sets via spherical averages ]]<br />
| <br />
|-<br />
|Monday, November 2, 4 p.m.<br />
|Yuval Wigderson<br />
|Stanford University<br />
|[[#Yuval Wigderson | New perspectives on the uncertainty principle ]]<br />
| <br />
|-<br />
|November 10, 10 a.m. <br />
|Óscar Domínguez<br />
| Universidad Complutense de Madrid<br />
|[[#Oscar Dominguez | New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions ]]<br />
| <br />
|-<br />
|November 17<br />
|Tamas Titkos<br />
|BBS U of Applied Sciences and Renyi Institute<br />
|[[#Tamas Titkos | Isometries of Wasserstein spaces ]]<br />
| <br />
|-<br />
|November 24<br />
|Shukun Wu<br />
|University of Illinois (Urbana-Champaign)<br />
||[[#Shukun Wu | On the Bochner-Riesz operator and the maximal Bochner-Riesz operator ]] <br />
| <br />
|-<br />
|December 1<br />
| Jonathan Hickman<br />
| The University of Edinburgh<br />
|[[#Jonathan Hickman | Sobolev improving for averages over space curves ]]<br />
| <br />
|-<br />
|February 2, 7:00 p.m.<br />
|Hanlong Fang<br />
|UW Madison<br />
|[[#Hanlong Fang | Canonical blow-ups of Grassmann manifolds ]]<br />
| <br />
|-<br />
|February 9<br />
|Bingyang Hu<br />
|Purdue University<br />
|[[#Bingyang Hu | Some structure theorems on general doubling measures ]]<br />
| <br />
|-<br />
|February 16<br />
|Krystal Taylor<br />
|The Ohio State University<br />
|[[#Krystal Taylor | Quantifications of the Besicovitch Projection theorem in a nonlinear setting ]]<br />
|<br />
|-<br />
|February 23<br />
|Dominique Maldague<br />
|MIT<br />
|[[#Dominique Maldague | A new proof of decoupling for the parabola ]]<br />
|<br />
|-<br />
|March 2<br />
|Diogo Oliveira e Silva<br />
|University of Birmingham<br />
|[[#Diogo Oliveira e Silva | Global maximizers for spherical restriction ]]<br />
|<br />
|-<br />
|March 9<br />
|Oleg Safronov <br />
|University of North Carolina Charlotte<br />
|[[#linktoabstract | Relations between discrete and continuous spectra of differential operators ]]<br />
|<br />
|-<br />
|March 16<br />
|Ziming Shi<br />
|Rutgers University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 23<br />
|Xiumin Du<br />
|Northwestern University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 30, 10:00 a.m.<br />
|Etienne Le Masson<br />
|Cergy Paris University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 6<br />
|Theresa Anderson <br />
|Purdue University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 13<br />
|Nathan Wagner<br />
|Washington University St. Louis<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 20<br />
|Jongchon Kim<br />
| University of British Columbia<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 27<br />
|Yumeng Ou<br />
|University of Pennsylvania<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|May 4<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|}<br />
<br />
=Abstracts=<br />
===Alexei Poltoratski===<br />
<br />
Title: Dirac inner functions<br />
<br />
Abstract: My talk will focus on some new (and old) complex analytic objects arising from Dirac systems of differential equations.<br />
We will discuss connections between problems in complex function theory, spectral and scattering problems for differential<br />
operators and the non-linear Fourier transform.<br />
<br />
===Polona Durcik and Joris Roos===<br />
<br />
Title: A triangular Hilbert transform with curvature, I & II.<br />
<br />
Abstract: The triangular Hilbert is a two-dimensional bilinear singular<br />
originating in time-frequency analysis. No Lp bounds are currently<br />
known for this operator.<br />
In these two talks we discuss a recent joint work with Michael Christ<br />
on a variant of the triangular Hilbert transform involving curvature.<br />
This object is closely related to the bilinear Hilbert transform with<br />
curvature and a maximally modulated singular integral of Stein-Wainger<br />
type. As an application we also discuss a quantitative nonlinear Roth<br />
type theorem on patterns in the Euclidean plane.<br />
The second talk will focus on the proof of a key ingredient, a certain<br />
regularity estimate for a local operator.<br />
<br />
===Andrew Zimmer===<br />
<br />
Title: Complex analytic problems on domains with good intrinsic geometry<br />
<br />
Abstract: In this talk, I will describe a new class of domains in complex Euclidean space which is defined in terms of the existence of a Kaehler metric with good geometric properties. This class is invariant under biholomorphism and includes many well-studied classes of domains such as strongly pseudoconvex domains, finite type domains in dimension two, convex domains, homogeneous domains, and embeddings of Teichmuller spaces. Further, certain analytic problems are tractable for domains in this family even when the boundary is non-smooth. In particular, it is possible to characterize the domains in this family where the dbar-Neumann operator on (0, q)-forms is compact (which generalizes an old result of Fu-Straube for convex domains).<br />
<br />
===Hong Wang===<br />
<br />
Title: Improved decoupling for the parabola<br />
<br />
Abstract: In 2014, Bourgain and Demeter proved the $l^2$ decoupling estimates for the paraboloid with constant $R^{\epsilon}$. <br />
We prove an $(l^2, L^6)$ decoupling inequality for the parabola with constant $(\log R)^c$. This is joint work with Larry Guth and Dominique Maldague.<br />
<br />
===Kevin Luli===<br />
<br />
Title: Smooth Nonnegative Interpolation<br />
<br />
Abstract: Suppose E is an arbitrary subset of R^n. Let f: E \rightarrow [0, \infty). How can we decide if f extends to a nonnegative function C^m function F defined on all of R^n? Suppose E is finite. Can we compute a nonnegative C^m function F on R^n that agrees with f on E with the least possible C^m norm? How many computer operations does this take? In this talk, I will explain recent results on these problems. Non-negativity is one of the most important shape preserving properties for interpolants. In real life applications, the range of the interpolant is imposed by nature. For example, probability density, the amount of snow, rain, humidity, chemical concentration are all nonnegative quantities and are of interest in natural sciences. Even in one dimension, the existing techniques can only handle nonnegative interpolation under special assumptions on the data set. Our results work without any assumptions on the data sets. <br />
<br />
===Niclas Technau===<br />
<br />
Title: Number theoretic applications of oscillatory integrals<br />
<br />
Abstract: We discuss how the analysis of oscillatory integrals can be used to solve number theoretic problems. More specifically, the focus will be on understanding fine-scale statistics of sequences on the unit circle. Further, we shall briefly explain a connection to quantum chaos.<br />
<br />
===Terence Harris===<br />
<br />
Title: Low dimensional pinned distance sets via spherical averages<br />
<br />
Abstract: An inequality is derived for the average t-energy of weighted pinned distance measures, where 0 < t < 1, in terms of the L^2 spherical averages of Fourier transforms of measures. This generalises the result of Liu (originally for Lebesgue measure) to pinned distance sets of dimension smaller than 1, and strengthens Mattila's result from 1987, originally for the full distance set.<br />
<br />
===Yuval Wigderson===<br />
<br />
Title: New perspectives on the uncertainty principle<br />
<br />
Abstract: The phrase ``uncertainty principle'' refers to a wide array of results in several disparate fields of mathematics, all of which capture the notion that a function and its Fourier transform cannot both be ``very localized''. The measure of localization varies from one uncertainty principle to the next, and well-studied notions include the variance (and higher moments), the entropy, the support-size, and the rate of decay at infinity. Similarly, the proofs of the various uncertainty principles rely on a range of tools, from the elementary to the very deep. In this talk, I'll describe how many of the uncertainty principles all follow from a single, simple result, whose proof uses only a basic property of the Fourier transform: that it and its inverse are bounded as operators $L^1 \to L^\infty$. Using this result, one can also prove new variants of the uncertainty principle, which apply to new measures of localization and to operators other than the Fourier transform. This is joint work with Avi Wigderson.<br />
<br />
===Oscar Dominguez===<br />
<br />
Title: New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions<br />
<br />
Abstract: The celebrated Bourgain--Brezis--Mironescu formula enables us to recover Sobolev spaces in terms of limits of Gagliardo seminorms. Very recently, Brezis, Van Schaftingen and Yung have proposed an alternative methodology to approach Sobolev spaces via limits of weak-type Gagliardo functionals. The goal of this talk is twofold. Firstly, we will show that the BvSY result is a special case of a more general phenomenon based on maximal inequalities. In particular, we shall derive not only analogs of the BvSY theorem for different kinds of function spaces (Lebesgue, Calderon, higher-order Sobolev, …), but also applications to ergodic theory, Fourier series, etc. In the second part of the talk, we shall investigate the fractional setting in the BvSY theorem. Our approach is based on new Garsia-type inequalities and an application of the Caffarelli--Silvestre extension. This is joint work with Mario Milman.<br />
<br />
===Tamas Titkos===<br />
<br />
Title: Isometries of Wasserstein spaces<br />
<br />
Abstract: Due to its nice theoretical properties and an astonishing number of<br />
applications via optimal transport problems, probably the most<br />
intensively studied metric nowadays is the p-Wasserstein metric. Given<br />
a complete and separable metric space $X$ and a real number $p\geq1$,<br />
one defines the p-Wasserstein space $\mathcal{W}_p(X)$ as the collection<br />
of Borel probability measures with finite $p$-th moment, endowed with a<br />
distance which is calculated by means of transport plans \cite{5}.<br />
<br />
The main aim of our research project is to reveal the structure of the<br />
isometry group $\mathrm{Isom}(\mathcal{W}_p(X))$. Although<br />
$\mathrm{Isom}(X)$ embeds naturally into<br />
$\mathrm{Isom}(\mathcal{W}_p(X))$ by push-forward, and this embedding<br />
turned out to be surjective in many cases (see e.g. [1]), these two<br />
groups are not isomorphic in general. Kloeckner in [2] described<br />
the isometry group of the quadratic Wasserstein space<br />
$\mathcal{W}_2(\mathbb{R}^n)$, and it turned out that the case of $n=1$<br />
is special in the sense that $\mathrm{Isom}(\mathcal{W}_2(\mathbb{R})$<br />
is extremely rich. Namely, it contains a large subgroup of wild behaving<br />
isometries that distort the shape of measures. Following this line of<br />
investigation, in \cite{3} we described<br />
$\mathrm{Isom}(\mathcal{W}_p(\mathbb{R}))$ and<br />
$\mathrm{Isom}(\mathcal{W}_p([0,1])$ for all $p\geq 1$.<br />
<br />
In this talk I will survey first some of the earlier results in the<br />
subject, and then I will present the key results of [3]. If time<br />
permits, I will also report on our most recent manuscript [4] in<br />
which we extended Kloeckner's multidimensional results. Joint work with Gy\"orgy P\'al Geh\'er (University of Reading)<br />
and D\'aniel Virosztek (IST Austria).<br />
<br />
[1] J. Bertrand and B. Kloeckner, \emph{A geometric study of Wasserstein<br />
spaces: isometric rigidity in negative curvature}, International<br />
Mathematics Research Notices, 2016 (5), 1368--1386.<br />
<br />
[2] B. Kloeckner, \emph{A geometric study of Wasserstein spaces: Euclidean<br />
spaces}, Annali della Scuola Normale Superiore di Pisa - Classe di<br />
Scienze, Serie 5, Tome 9 (2010) no. 2, 297--323.<br />
<br />
[3] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{Isometric study of<br />
Wasserstein spaces – the real line}, Trans. Amer. Math. Soc., 373<br />
(2020), 5855--5883.<br />
<br />
[4] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{The isometry group of<br />
Wasserstein spaces: The Hilbertian case}, submitted manuscript.<br />
<br />
[5] C. Villani, \emph{Optimal Transport: Old and New,}<br />
(Grundlehren der mathematischen Wissenschaften)<br />
Springer, 2009.<br />
<br />
===Shukun Wu===<br />
<br />
Title: On the Bochner-Riesz operator and the maximal Bochner-Riesz operator<br />
<br />
Abstract: The Bochner-Riesz problem is one of the most important problems in the field of Fourier analysis. It has a strong connection to other famous problems, such as the restriction conjecture and the Kakeya conjecture. In this talk, I will present some recent improvements to the Bochner-Riesz conjecture and the maximal Bochner-Riesz conjecture. The main methods we used are polynomial partitioning and the Bourgain Demeter l^2 decoupling theorem. <br />
<br />
<br />
===Jonathan Hickman===<br />
<br />
Title: Sobolev improving for averages over space curves<br />
<br />
Abstract: Consider the averaging operator given by convolution with arclength measure on compact piece of a smooth curve in R^n. A simple question is to precisely quantify the gain in regularity induced by this averaging, for instance by studying the L^p-Sobolev mapping properties of the operator. This talk will report on ongoing developments towards understanding this problem. In particular, we will explore some non-trivial necessary conditions on the gain in regularity. Joint with D. Beltran, S. Guo and A. Seeger.<br />
<br />
===Hanlong Fang===<br />
<br />
Title: Canonical blow-ups of Grassmann manifolds<br />
<br />
Abstract: We introduce certain canonical blow-ups \mathcal T_{s,p,n}, as well as their distinct submanifolds \mathcal M_{s,p,n}, of Grassmann manifolds G(p,n) by partitioning the Plücker coordinates with respect to a parameter s. Various geometric aspects of \mathcal T_{s,p,n} and \mathcal M_{s,p,n} are studied, for instance, the smoothness, the holomorphic symmetries, the (semi-)positivity of the anti-canonical bundles, the existence of Kähler-Einstein metrics, the functoriality, etc. In particular, we introduce the notion of homeward compactification, of which \mathcal T_{s,p,n} are examples, as a generalization of the wonderful compactification. <br />
<br />
===Bingyang Hu===<br />
<br />
Title: Some structure theorems on general doubling measures.<br />
<br />
Abstract: In this talk, we will first several structure theorems about general doubling measures. Secondly, we will include some main idea to prove one of these results. More precisely, we will focus on the construction of an explicit family of measures that are p-adic doubling for any finite set of primes, however, not doubling. This part generalizes the work by Boylan, Mills and Ward in 2019 in a highly non-trivial way. As some application, we apply these results (that is, the same construction) to show analogous statements for Muckenhoupt Ap weights and reverse Holder weights. This is a joint work with Tess Anderson.<br />
<br />
===Krystal Taylor===<br />
<br />
Title: Quantifications of the Besicovitch Projection theorem in a nonlinear setting <br />
<br />
Abstract: There are several classical results relating the geometry, dimension, and measure of a set to the structure of its orthogonal projections. <br />
It turns out that many nonlinear projection-type operators also have special geometry that allows us to build similar relationships between a set and its "projections", just as in the linear setting. We will discuss a series of recent results from both geometric and probabilistic vantage points. In particular, we will see that the multi-scale analysis techniques of Tao, as well as the energy techniques of Mattila, can be strengthened and generalized to projection-type operators satisfying a transversality condition. As an application, we address the Buffon curve problem, which is to find upper and lower bounds for the rate of decay of the Favard curve length of the four-corner Cantor set.<br />
<br />
===Dominique Maldague===<br />
<br />
Title: A new proof of decoupling for the parabola<br />
<br />
Abstract: Decoupling has to do with measuring the size of functions with specialized Fourier support (in our case, in a neighborhood of the truncated parabola). Bourgain and Demeter resolved the l^2 decoupling conjecture in 2014, using ingredients like the multilinear Kakeya inequality, L^2 orthogonality, and induction-on-scales. I will present the ideas that go into a new proof of decoupling and make some comparison between the two approaches. This is related to recent joint work with Larry Guth and Hong Wang, as well as forthcoming joint work with Yuqiu Fu and Larry Guth.<br />
<br />
===Name===<br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
===Name===<br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
===Name===<br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
===Name===<br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
===Name===<br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
===Name===<br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]<br />
<br />
<br />
Graduate Student Seminar:<br />
<br />
https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html</div>Beltranhttps://hilbert.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=20855Analysis Seminar2021-02-19T17:50:17Z<p>Beltran: </p>
<hr />
<div><br />
The 2020-2021 Analysis Seminar will be organized by David Beltran and Andreas Seeger.<br />
It will be online for the entire academic year. The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar at different times, to accommodate speakers).<br />
<br />
Zoom links will be sent to those who have signed up for the Analysis Seminar List. If you would like to subscribe to the Analysis seminar list, send a blank email to analysis+join (at) g-groups (dot) wisc (dot) edu as well as an additional email to David and Andreas (dbeltran, seeger at math (dot) wisc (dot) edu) to notify the request.<br />
<br />
If you'd like to suggest speakers for the spring semester please contact David and Andreas.<br />
<br />
<br />
<br />
=[[Previous_Analysis_seminars]]=<br />
<br />
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars<br />
<br />
= Current Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 22<br />
|Alexei Poltoratski<br />
|UW Madison<br />
|[[#Alexei Poltoratski | Dirac inner functions ]]<br />
| <br />
|-<br />
|September 29<br />
|Joris Roos<br />
|University of Massachusetts - Lowell<br />
|[[#Polona Durcik and Joris Rooslinktoabstract | A triangular Hilbert transform with curvature, I ]]<br />
| <br />
|-<br />
|Wednesday September 30, 4 p.m.<br />
|Polona Durcik<br />
|Chapman University<br />
|[[#Polona Durcik and Joris Roos | A triangular Hilbert transform with curvature, II ]]<br />
| <br />
|-<br />
|October 6<br />
|Andrew Zimmer<br />
|UW Madison<br />
|[[#Andrew Zimmer | Complex analytic problems on domains with good intrinsic geometry ]]<br />
| <br />
|-<br />
|October 13<br />
|Hong Wang <br />
|Princeton/IAS<br />
|[[#Hong Wang | Improved decoupling for the parabola ]]<br />
| <br />
|-<br />
|October 20<br />
|Kevin Luli<br />
|UC Davis<br />
|[[#Kevin Luli | Smooth Nonnegative Interpolation ]]<br />
| <br />
|-<br />
|October 21, 4.00 p.m.<br />
|Niclas Technau<br />
|UW Madison<br />
|[[#Niclas Technau | Number theoretic applications of oscillatory integrals ]]<br />
| <br />
|-<br />
|October 27<br />
|Terence Harris<br />
| Cornell University<br />
|[[#Terence Harris | Low dimensional pinned distance sets via spherical averages ]]<br />
| <br />
|-<br />
|Monday, November 2, 4 p.m.<br />
|Yuval Wigderson<br />
|Stanford University<br />
|[[#Yuval Wigderson | New perspectives on the uncertainty principle ]]<br />
| <br />
|-<br />
|November 10, 10 a.m. <br />
|Óscar Domínguez<br />
| Universidad Complutense de Madrid<br />
|[[#Oscar Dominguez | New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions ]]<br />
| <br />
|-<br />
|November 17<br />
|Tamas Titkos<br />
|BBS U of Applied Sciences and Renyi Institute<br />
|[[#Tamas Titkos | Isometries of Wasserstein spaces ]]<br />
| <br />
|-<br />
|November 24<br />
|Shukun Wu<br />
|University of Illinois (Urbana-Champaign)<br />
||[[#Shukun Wu | On the Bochner-Riesz operator and the maximal Bochner-Riesz operator ]] <br />
| <br />
|-<br />
|December 1<br />
| Jonathan Hickman<br />
| The University of Edinburgh<br />
|[[#Jonathan Hickman | Sobolev improving for averages over space curves ]]<br />
| <br />
|-<br />
|February 2, 7:00 p.m.<br />
|Hanlong Fang<br />
|UW Madison<br />
|[[#Hanlong Fang | Canonical blow-ups of Grassmann manifolds ]]<br />
| <br />
|-<br />
|February 9<br />
|Bingyang Hu<br />
|Purdue University<br />
|[[#Bingyang Hu | Some structure theorems on general doubling measures ]]<br />
| <br />
|-<br />
|February 16<br />
|Krystal Taylor<br />
|The Ohio State University<br />
|[[#Krystal Taylor | Quantifications of the Besicovitch Projection theorem in a nonlinear setting ]]<br />
|<br />
|-<br />
|February 23<br />
|Dominique Maldague<br />
|MIT<br />
|[[#Dominique Maldague | A new proof of decoupling for the parabola ]]<br />
|<br />
|-<br />
|March 2<br />
|Diogo Oliveira e Silva<br />
|University of Birmingham<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 9<br />
|Oleg Safronov <br />
|University of North Carolina Charlotte<br />
|[[#linktoabstract | Relations between discrete and continuous spectra of differential operators ]]<br />
|<br />
|-<br />
|March 16<br />
|Ziming Shi<br />
|Rutgers University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 23<br />
|Xiumin Du<br />
|Northwestern University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 30, 10:00 a.m.<br />
|Etienne Le Masson<br />
|Cergy Paris University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 6<br />
|Theresa Anderson <br />
|Purdue University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 13<br />
|Nathan Wagner<br />
|Washington University St. Louis<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 20<br />
|Jongchon Kim<br />
| University of British Columbia<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 27<br />
|Yumeng Ou<br />
|University of Pennsylvania<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|May 4<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|}<br />
<br />
=Abstracts=<br />
===Alexei Poltoratski===<br />
<br />
Title: Dirac inner functions<br />
<br />
Abstract: My talk will focus on some new (and old) complex analytic objects arising from Dirac systems of differential equations.<br />
We will discuss connections between problems in complex function theory, spectral and scattering problems for differential<br />
operators and the non-linear Fourier transform.<br />
<br />
===Polona Durcik and Joris Roos===<br />
<br />
Title: A triangular Hilbert transform with curvature, I & II.<br />
<br />
Abstract: The triangular Hilbert is a two-dimensional bilinear singular<br />
originating in time-frequency analysis. No Lp bounds are currently<br />
known for this operator.<br />
In these two talks we discuss a recent joint work with Michael Christ<br />
on a variant of the triangular Hilbert transform involving curvature.<br />
This object is closely related to the bilinear Hilbert transform with<br />
curvature and a maximally modulated singular integral of Stein-Wainger<br />
type. As an application we also discuss a quantitative nonlinear Roth<br />
type theorem on patterns in the Euclidean plane.<br />
The second talk will focus on the proof of a key ingredient, a certain<br />
regularity estimate for a local operator.<br />
<br />
===Andrew Zimmer===<br />
<br />
Title: Complex analytic problems on domains with good intrinsic geometry<br />
<br />
Abstract: In this talk, I will describe a new class of domains in complex Euclidean space which is defined in terms of the existence of a Kaehler metric with good geometric properties. This class is invariant under biholomorphism and includes many well-studied classes of domains such as strongly pseudoconvex domains, finite type domains in dimension two, convex domains, homogeneous domains, and embeddings of Teichmuller spaces. Further, certain analytic problems are tractable for domains in this family even when the boundary is non-smooth. In particular, it is possible to characterize the domains in this family where the dbar-Neumann operator on (0, q)-forms is compact (which generalizes an old result of Fu-Straube for convex domains).<br />
<br />
===Hong Wang===<br />
<br />
Title: Improved decoupling for the parabola<br />
<br />
Abstract: In 2014, Bourgain and Demeter proved the $l^2$ decoupling estimates for the paraboloid with constant $R^{\epsilon}$. <br />
We prove an $(l^2, L^6)$ decoupling inequality for the parabola with constant $(\log R)^c$. This is joint work with Larry Guth and Dominique Maldague.<br />
<br />
===Kevin Luli===<br />
<br />
Title: Smooth Nonnegative Interpolation<br />
<br />
Abstract: Suppose E is an arbitrary subset of R^n. Let f: E \rightarrow [0, \infty). How can we decide if f extends to a nonnegative function C^m function F defined on all of R^n? Suppose E is finite. Can we compute a nonnegative C^m function F on R^n that agrees with f on E with the least possible C^m norm? How many computer operations does this take? In this talk, I will explain recent results on these problems. Non-negativity is one of the most important shape preserving properties for interpolants. In real life applications, the range of the interpolant is imposed by nature. For example, probability density, the amount of snow, rain, humidity, chemical concentration are all nonnegative quantities and are of interest in natural sciences. Even in one dimension, the existing techniques can only handle nonnegative interpolation under special assumptions on the data set. Our results work without any assumptions on the data sets. <br />
<br />
===Niclas Technau===<br />
<br />
Title: Number theoretic applications of oscillatory integrals<br />
<br />
Abstract: We discuss how the analysis of oscillatory integrals can be used to solve number theoretic problems. More specifically, the focus will be on understanding fine-scale statistics of sequences on the unit circle. Further, we shall briefly explain a connection to quantum chaos.<br />
<br />
===Terence Harris===<br />
<br />
Title: Low dimensional pinned distance sets via spherical averages<br />
<br />
Abstract: An inequality is derived for the average t-energy of weighted pinned distance measures, where 0 < t < 1, in terms of the L^2 spherical averages of Fourier transforms of measures. This generalises the result of Liu (originally for Lebesgue measure) to pinned distance sets of dimension smaller than 1, and strengthens Mattila's result from 1987, originally for the full distance set.<br />
<br />
===Yuval Wigderson===<br />
<br />
Title: New perspectives on the uncertainty principle<br />
<br />
Abstract: The phrase ``uncertainty principle'' refers to a wide array of results in several disparate fields of mathematics, all of which capture the notion that a function and its Fourier transform cannot both be ``very localized''. The measure of localization varies from one uncertainty principle to the next, and well-studied notions include the variance (and higher moments), the entropy, the support-size, and the rate of decay at infinity. Similarly, the proofs of the various uncertainty principles rely on a range of tools, from the elementary to the very deep. In this talk, I'll describe how many of the uncertainty principles all follow from a single, simple result, whose proof uses only a basic property of the Fourier transform: that it and its inverse are bounded as operators $L^1 \to L^\infty$. Using this result, one can also prove new variants of the uncertainty principle, which apply to new measures of localization and to operators other than the Fourier transform. This is joint work with Avi Wigderson.<br />
<br />
===Oscar Dominguez===<br />
<br />
Title: New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions<br />
<br />
Abstract: The celebrated Bourgain--Brezis--Mironescu formula enables us to recover Sobolev spaces in terms of limits of Gagliardo seminorms. Very recently, Brezis, Van Schaftingen and Yung have proposed an alternative methodology to approach Sobolev spaces via limits of weak-type Gagliardo functionals. The goal of this talk is twofold. Firstly, we will show that the BvSY result is a special case of a more general phenomenon based on maximal inequalities. In particular, we shall derive not only analogs of the BvSY theorem for different kinds of function spaces (Lebesgue, Calderon, higher-order Sobolev, …), but also applications to ergodic theory, Fourier series, etc. In the second part of the talk, we shall investigate the fractional setting in the BvSY theorem. Our approach is based on new Garsia-type inequalities and an application of the Caffarelli--Silvestre extension. This is joint work with Mario Milman.<br />
<br />
===Tamas Titkos===<br />
<br />
Title: Isometries of Wasserstein spaces<br />
<br />
Abstract: Due to its nice theoretical properties and an astonishing number of<br />
applications via optimal transport problems, probably the most<br />
intensively studied metric nowadays is the p-Wasserstein metric. Given<br />
a complete and separable metric space $X$ and a real number $p\geq1$,<br />
one defines the p-Wasserstein space $\mathcal{W}_p(X)$ as the collection<br />
of Borel probability measures with finite $p$-th moment, endowed with a<br />
distance which is calculated by means of transport plans \cite{5}.<br />
<br />
The main aim of our research project is to reveal the structure of the<br />
isometry group $\mathrm{Isom}(\mathcal{W}_p(X))$. Although<br />
$\mathrm{Isom}(X)$ embeds naturally into<br />
$\mathrm{Isom}(\mathcal{W}_p(X))$ by push-forward, and this embedding<br />
turned out to be surjective in many cases (see e.g. [1]), these two<br />
groups are not isomorphic in general. Kloeckner in [2] described<br />
the isometry group of the quadratic Wasserstein space<br />
$\mathcal{W}_2(\mathbb{R}^n)$, and it turned out that the case of $n=1$<br />
is special in the sense that $\mathrm{Isom}(\mathcal{W}_2(\mathbb{R})$<br />
is extremely rich. Namely, it contains a large subgroup of wild behaving<br />
isometries that distort the shape of measures. Following this line of<br />
investigation, in \cite{3} we described<br />
$\mathrm{Isom}(\mathcal{W}_p(\mathbb{R}))$ and<br />
$\mathrm{Isom}(\mathcal{W}_p([0,1])$ for all $p\geq 1$.<br />
<br />
In this talk I will survey first some of the earlier results in the<br />
subject, and then I will present the key results of [3]. If time<br />
permits, I will also report on our most recent manuscript [4] in<br />
which we extended Kloeckner's multidimensional results. Joint work with Gy\"orgy P\'al Geh\'er (University of Reading)<br />
and D\'aniel Virosztek (IST Austria).<br />
<br />
[1] J. Bertrand and B. Kloeckner, \emph{A geometric study of Wasserstein<br />
spaces: isometric rigidity in negative curvature}, International<br />
Mathematics Research Notices, 2016 (5), 1368--1386.<br />
<br />
[2] B. Kloeckner, \emph{A geometric study of Wasserstein spaces: Euclidean<br />
spaces}, Annali della Scuola Normale Superiore di Pisa - Classe di<br />
Scienze, Serie 5, Tome 9 (2010) no. 2, 297--323.<br />
<br />
[3] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{Isometric study of<br />
Wasserstein spaces – the real line}, Trans. Amer. Math. Soc., 373<br />
(2020), 5855--5883.<br />
<br />
[4] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{The isometry group of<br />
Wasserstein spaces: The Hilbertian case}, submitted manuscript.<br />
<br />
[5] C. Villani, \emph{Optimal Transport: Old and New,}<br />
(Grundlehren der mathematischen Wissenschaften)<br />
Springer, 2009.<br />
<br />
===Shukun Wu===<br />
<br />
Title: On the Bochner-Riesz operator and the maximal Bochner-Riesz operator<br />
<br />
Abstract: The Bochner-Riesz problem is one of the most important problems in the field of Fourier analysis. It has a strong connection to other famous problems, such as the restriction conjecture and the Kakeya conjecture. In this talk, I will present some recent improvements to the Bochner-Riesz conjecture and the maximal Bochner-Riesz conjecture. The main methods we used are polynomial partitioning and the Bourgain Demeter l^2 decoupling theorem. <br />
<br />
<br />
===Jonathan Hickman===<br />
<br />
Title: Sobolev improving for averages over space curves<br />
<br />
Abstract: Consider the averaging operator given by convolution with arclength measure on compact piece of a smooth curve in R^n. A simple question is to precisely quantify the gain in regularity induced by this averaging, for instance by studying the L^p-Sobolev mapping properties of the operator. This talk will report on ongoing developments towards understanding this problem. In particular, we will explore some non-trivial necessary conditions on the gain in regularity. Joint with D. Beltran, S. Guo and A. Seeger.<br />
<br />
===Hanlong Fang===<br />
<br />
Title: Canonical blow-ups of Grassmann manifolds<br />
<br />
Abstract: We introduce certain canonical blow-ups \mathcal T_{s,p,n}, as well as their distinct submanifolds \mathcal M_{s,p,n}, of Grassmann manifolds G(p,n) by partitioning the Plücker coordinates with respect to a parameter s. Various geometric aspects of \mathcal T_{s,p,n} and \mathcal M_{s,p,n} are studied, for instance, the smoothness, the holomorphic symmetries, the (semi-)positivity of the anti-canonical bundles, the existence of Kähler-Einstein metrics, the functoriality, etc. In particular, we introduce the notion of homeward compactification, of which \mathcal T_{s,p,n} are examples, as a generalization of the wonderful compactification. <br />
<br />
===Bingyang Hu===<br />
<br />
Title: Some structure theorems on general doubling measures.<br />
<br />
Abstract: In this talk, we will first several structure theorems about general doubling measures. Secondly, we will include some main idea to prove one of these results. More precisely, we will focus on the construction of an explicit family of measures that are p-adic doubling for any finite set of primes, however, not doubling. This part generalizes the work by Boylan, Mills and Ward in 2019 in a highly non-trivial way. As some application, we apply these results (that is, the same construction) to show analogous statements for Muckenhoupt Ap weights and reverse Holder weights. This is a joint work with Tess Anderson.<br />
<br />
===Krystal Taylor===<br />
<br />
Title: Quantifications of the Besicovitch Projection theorem in a nonlinear setting <br />
<br />
Abstract: There are several classical results relating the geometry, dimension, and measure of a set to the structure of its orthogonal projections. <br />
It turns out that many nonlinear projection-type operators also have special geometry that allows us to build similar relationships between a set and its "projections", just as in the linear setting. We will discuss a series of recent results from both geometric and probabilistic vantage points. In particular, we will see that the multi-scale analysis techniques of Tao, as well as the energy techniques of Mattila, can be strengthened and generalized to projection-type operators satisfying a transversality condition. As an application, we address the Buffon curve problem, which is to find upper and lower bounds for the rate of decay of the Favard curve length of the four-corner Cantor set.<br />
<br />
===Dominique Maldague===<br />
<br />
Title: A new proof of decoupling for the parabola<br />
<br />
Abstract: Decoupling has to do with measuring the size of functions with specialized Fourier support (in our case, in a neighborhood of the truncated parabola). Bourgain and Demeter resolved the l^2 decoupling conjecture in 2014, using ingredients like the multilinear Kakeya inequality, L^2 orthogonality, and induction-on-scales. I will present the ideas that go into a new proof of decoupling and make some comparison between the two approaches. This is related to recent joint work with Larry Guth and Hong Wang, as well as forthcoming joint work with Yuqiu Fu and Larry Guth.<br />
<br />
===Name===<br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
===Name===<br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
===Name===<br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
===Name===<br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
===Name===<br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
===Name===<br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]<br />
<br />
<br />
Graduate Student Seminar:<br />
<br />
https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html</div>Beltranhttps://hilbert.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=20854Analysis Seminar2021-02-19T17:49:42Z<p>Beltran: </p>
<hr />
<div><br />
The 2020-2021 Analysis Seminar will be organized by David Beltran and Andreas Seeger.<br />
It will be online for the entire academic year. The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar at different times, to accommodate speakers).<br />
<br />
Zoom links will be sent to those who have signed up for the Analysis Seminar List. If you would like to subscribe to the Analysis seminar list, send a blank email to analysis+join (at) g-groups (dot) wisc (dot) edu as well as an additional email to David and Andreas (dbeltran, seeger at math (dot) wisc (dot) edu) to notify the request).<br />
<br />
If you'd like to suggest speakers for the spring semester please contact David and Andreas (dbeltran at math, seeger at math).<br />
<br />
<br />
<br />
=[[Previous_Analysis_seminars]]=<br />
<br />
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars<br />
<br />
= Current Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 22<br />
|Alexei Poltoratski<br />
|UW Madison<br />
|[[#Alexei Poltoratski | Dirac inner functions ]]<br />
| <br />
|-<br />
|September 29<br />
|Joris Roos<br />
|University of Massachusetts - Lowell<br />
|[[#Polona Durcik and Joris Rooslinktoabstract | A triangular Hilbert transform with curvature, I ]]<br />
| <br />
|-<br />
|Wednesday September 30, 4 p.m.<br />
|Polona Durcik<br />
|Chapman University<br />
|[[#Polona Durcik and Joris Roos | A triangular Hilbert transform with curvature, II ]]<br />
| <br />
|-<br />
|October 6<br />
|Andrew Zimmer<br />
|UW Madison<br />
|[[#Andrew Zimmer | Complex analytic problems on domains with good intrinsic geometry ]]<br />
| <br />
|-<br />
|October 13<br />
|Hong Wang <br />
|Princeton/IAS<br />
|[[#Hong Wang | Improved decoupling for the parabola ]]<br />
| <br />
|-<br />
|October 20<br />
|Kevin Luli<br />
|UC Davis<br />
|[[#Kevin Luli | Smooth Nonnegative Interpolation ]]<br />
| <br />
|-<br />
|October 21, 4.00 p.m.<br />
|Niclas Technau<br />
|UW Madison<br />
|[[#Niclas Technau | Number theoretic applications of oscillatory integrals ]]<br />
| <br />
|-<br />
|October 27<br />
|Terence Harris<br />
| Cornell University<br />
|[[#Terence Harris | Low dimensional pinned distance sets via spherical averages ]]<br />
| <br />
|-<br />
|Monday, November 2, 4 p.m.<br />
|Yuval Wigderson<br />
|Stanford University<br />
|[[#Yuval Wigderson | New perspectives on the uncertainty principle ]]<br />
| <br />
|-<br />
|November 10, 10 a.m. <br />
|Óscar Domínguez<br />
| Universidad Complutense de Madrid<br />
|[[#Oscar Dominguez | New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions ]]<br />
| <br />
|-<br />
|November 17<br />
|Tamas Titkos<br />
|BBS U of Applied Sciences and Renyi Institute<br />
|[[#Tamas Titkos | Isometries of Wasserstein spaces ]]<br />
| <br />
|-<br />
|November 24<br />
|Shukun Wu<br />
|University of Illinois (Urbana-Champaign)<br />
||[[#Shukun Wu | On the Bochner-Riesz operator and the maximal Bochner-Riesz operator ]] <br />
| <br />
|-<br />
|December 1<br />
| Jonathan Hickman<br />
| The University of Edinburgh<br />
|[[#Jonathan Hickman | Sobolev improving for averages over space curves ]]<br />
| <br />
|-<br />
|February 2, 7:00 p.m.<br />
|Hanlong Fang<br />
|UW Madison<br />
|[[#Hanlong Fang | Canonical blow-ups of Grassmann manifolds ]]<br />
| <br />
|-<br />
|February 9<br />
|Bingyang Hu<br />
|Purdue University<br />
|[[#Bingyang Hu | Some structure theorems on general doubling measures ]]<br />
| <br />
|-<br />
|February 16<br />
|Krystal Taylor<br />
|The Ohio State University<br />
|[[#Krystal Taylor | Quantifications of the Besicovitch Projection theorem in a nonlinear setting ]]<br />
|<br />
|-<br />
|February 23<br />
|Dominique Maldague<br />
|MIT<br />
|[[#Dominique Maldague | A new proof of decoupling for the parabola ]]<br />
|<br />
|-<br />
|March 2<br />
|Diogo Oliveira e Silva<br />
|University of Birmingham<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 9<br />
|Oleg Safronov <br />
|University of North Carolina Charlotte<br />
|[[#linktoabstract | Relations between discrete and continuous spectra of differential operators ]]<br />
|<br />
|-<br />
|March 16<br />
|Ziming Shi<br />
|Rutgers University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 23<br />
|Xiumin Du<br />
|Northwestern University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 30, 10:00 a.m.<br />
|Etienne Le Masson<br />
|Cergy Paris University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 6<br />
|Theresa Anderson <br />
|Purdue University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 13<br />
|Nathan Wagner<br />
|Washington University St. Louis<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 20<br />
|Jongchon Kim<br />
| University of British Columbia<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 27<br />
|Yumeng Ou<br />
|University of Pennsylvania<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|May 4<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|}<br />
<br />
=Abstracts=<br />
===Alexei Poltoratski===<br />
<br />
Title: Dirac inner functions<br />
<br />
Abstract: My talk will focus on some new (and old) complex analytic objects arising from Dirac systems of differential equations.<br />
We will discuss connections between problems in complex function theory, spectral and scattering problems for differential<br />
operators and the non-linear Fourier transform.<br />
<br />
===Polona Durcik and Joris Roos===<br />
<br />
Title: A triangular Hilbert transform with curvature, I & II.<br />
<br />
Abstract: The triangular Hilbert is a two-dimensional bilinear singular<br />
originating in time-frequency analysis. No Lp bounds are currently<br />
known for this operator.<br />
In these two talks we discuss a recent joint work with Michael Christ<br />
on a variant of the triangular Hilbert transform involving curvature.<br />
This object is closely related to the bilinear Hilbert transform with<br />
curvature and a maximally modulated singular integral of Stein-Wainger<br />
type. As an application we also discuss a quantitative nonlinear Roth<br />
type theorem on patterns in the Euclidean plane.<br />
The second talk will focus on the proof of a key ingredient, a certain<br />
regularity estimate for a local operator.<br />
<br />
===Andrew Zimmer===<br />
<br />
Title: Complex analytic problems on domains with good intrinsic geometry<br />
<br />
Abstract: In this talk, I will describe a new class of domains in complex Euclidean space which is defined in terms of the existence of a Kaehler metric with good geometric properties. This class is invariant under biholomorphism and includes many well-studied classes of domains such as strongly pseudoconvex domains, finite type domains in dimension two, convex domains, homogeneous domains, and embeddings of Teichmuller spaces. Further, certain analytic problems are tractable for domains in this family even when the boundary is non-smooth. In particular, it is possible to characterize the domains in this family where the dbar-Neumann operator on (0, q)-forms is compact (which generalizes an old result of Fu-Straube for convex domains).<br />
<br />
===Hong Wang===<br />
<br />
Title: Improved decoupling for the parabola<br />
<br />
Abstract: In 2014, Bourgain and Demeter proved the $l^2$ decoupling estimates for the paraboloid with constant $R^{\epsilon}$. <br />
We prove an $(l^2, L^6)$ decoupling inequality for the parabola with constant $(\log R)^c$. This is joint work with Larry Guth and Dominique Maldague.<br />
<br />
===Kevin Luli===<br />
<br />
Title: Smooth Nonnegative Interpolation<br />
<br />
Abstract: Suppose E is an arbitrary subset of R^n. Let f: E \rightarrow [0, \infty). How can we decide if f extends to a nonnegative function C^m function F defined on all of R^n? Suppose E is finite. Can we compute a nonnegative C^m function F on R^n that agrees with f on E with the least possible C^m norm? How many computer operations does this take? In this talk, I will explain recent results on these problems. Non-negativity is one of the most important shape preserving properties for interpolants. In real life applications, the range of the interpolant is imposed by nature. For example, probability density, the amount of snow, rain, humidity, chemical concentration are all nonnegative quantities and are of interest in natural sciences. Even in one dimension, the existing techniques can only handle nonnegative interpolation under special assumptions on the data set. Our results work without any assumptions on the data sets. <br />
<br />
===Niclas Technau===<br />
<br />
Title: Number theoretic applications of oscillatory integrals<br />
<br />
Abstract: We discuss how the analysis of oscillatory integrals can be used to solve number theoretic problems. More specifically, the focus will be on understanding fine-scale statistics of sequences on the unit circle. Further, we shall briefly explain a connection to quantum chaos.<br />
<br />
===Terence Harris===<br />
<br />
Title: Low dimensional pinned distance sets via spherical averages<br />
<br />
Abstract: An inequality is derived for the average t-energy of weighted pinned distance measures, where 0 < t < 1, in terms of the L^2 spherical averages of Fourier transforms of measures. This generalises the result of Liu (originally for Lebesgue measure) to pinned distance sets of dimension smaller than 1, and strengthens Mattila's result from 1987, originally for the full distance set.<br />
<br />
===Yuval Wigderson===<br />
<br />
Title: New perspectives on the uncertainty principle<br />
<br />
Abstract: The phrase ``uncertainty principle'' refers to a wide array of results in several disparate fields of mathematics, all of which capture the notion that a function and its Fourier transform cannot both be ``very localized''. The measure of localization varies from one uncertainty principle to the next, and well-studied notions include the variance (and higher moments), the entropy, the support-size, and the rate of decay at infinity. Similarly, the proofs of the various uncertainty principles rely on a range of tools, from the elementary to the very deep. In this talk, I'll describe how many of the uncertainty principles all follow from a single, simple result, whose proof uses only a basic property of the Fourier transform: that it and its inverse are bounded as operators $L^1 \to L^\infty$. Using this result, one can also prove new variants of the uncertainty principle, which apply to new measures of localization and to operators other than the Fourier transform. This is joint work with Avi Wigderson.<br />
<br />
===Oscar Dominguez===<br />
<br />
Title: New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions<br />
<br />
Abstract: The celebrated Bourgain--Brezis--Mironescu formula enables us to recover Sobolev spaces in terms of limits of Gagliardo seminorms. Very recently, Brezis, Van Schaftingen and Yung have proposed an alternative methodology to approach Sobolev spaces via limits of weak-type Gagliardo functionals. The goal of this talk is twofold. Firstly, we will show that the BvSY result is a special case of a more general phenomenon based on maximal inequalities. In particular, we shall derive not only analogs of the BvSY theorem for different kinds of function spaces (Lebesgue, Calderon, higher-order Sobolev, …), but also applications to ergodic theory, Fourier series, etc. In the second part of the talk, we shall investigate the fractional setting in the BvSY theorem. Our approach is based on new Garsia-type inequalities and an application of the Caffarelli--Silvestre extension. This is joint work with Mario Milman.<br />
<br />
===Tamas Titkos===<br />
<br />
Title: Isometries of Wasserstein spaces<br />
<br />
Abstract: Due to its nice theoretical properties and an astonishing number of<br />
applications via optimal transport problems, probably the most<br />
intensively studied metric nowadays is the p-Wasserstein metric. Given<br />
a complete and separable metric space $X$ and a real number $p\geq1$,<br />
one defines the p-Wasserstein space $\mathcal{W}_p(X)$ as the collection<br />
of Borel probability measures with finite $p$-th moment, endowed with a<br />
distance which is calculated by means of transport plans \cite{5}.<br />
<br />
The main aim of our research project is to reveal the structure of the<br />
isometry group $\mathrm{Isom}(\mathcal{W}_p(X))$. Although<br />
$\mathrm{Isom}(X)$ embeds naturally into<br />
$\mathrm{Isom}(\mathcal{W}_p(X))$ by push-forward, and this embedding<br />
turned out to be surjective in many cases (see e.g. [1]), these two<br />
groups are not isomorphic in general. Kloeckner in [2] described<br />
the isometry group of the quadratic Wasserstein space<br />
$\mathcal{W}_2(\mathbb{R}^n)$, and it turned out that the case of $n=1$<br />
is special in the sense that $\mathrm{Isom}(\mathcal{W}_2(\mathbb{R})$<br />
is extremely rich. Namely, it contains a large subgroup of wild behaving<br />
isometries that distort the shape of measures. Following this line of<br />
investigation, in \cite{3} we described<br />
$\mathrm{Isom}(\mathcal{W}_p(\mathbb{R}))$ and<br />
$\mathrm{Isom}(\mathcal{W}_p([0,1])$ for all $p\geq 1$.<br />
<br />
In this talk I will survey first some of the earlier results in the<br />
subject, and then I will present the key results of [3]. If time<br />
permits, I will also report on our most recent manuscript [4] in<br />
which we extended Kloeckner's multidimensional results. Joint work with Gy\"orgy P\'al Geh\'er (University of Reading)<br />
and D\'aniel Virosztek (IST Austria).<br />
<br />
[1] J. Bertrand and B. Kloeckner, \emph{A geometric study of Wasserstein<br />
spaces: isometric rigidity in negative curvature}, International<br />
Mathematics Research Notices, 2016 (5), 1368--1386.<br />
<br />
[2] B. Kloeckner, \emph{A geometric study of Wasserstein spaces: Euclidean<br />
spaces}, Annali della Scuola Normale Superiore di Pisa - Classe di<br />
Scienze, Serie 5, Tome 9 (2010) no. 2, 297--323.<br />
<br />
[3] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{Isometric study of<br />
Wasserstein spaces – the real line}, Trans. Amer. Math. Soc., 373<br />
(2020), 5855--5883.<br />
<br />
[4] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{The isometry group of<br />
Wasserstein spaces: The Hilbertian case}, submitted manuscript.<br />
<br />
[5] C. Villani, \emph{Optimal Transport: Old and New,}<br />
(Grundlehren der mathematischen Wissenschaften)<br />
Springer, 2009.<br />
<br />
===Shukun Wu===<br />
<br />
Title: On the Bochner-Riesz operator and the maximal Bochner-Riesz operator<br />
<br />
Abstract: The Bochner-Riesz problem is one of the most important problems in the field of Fourier analysis. It has a strong connection to other famous problems, such as the restriction conjecture and the Kakeya conjecture. In this talk, I will present some recent improvements to the Bochner-Riesz conjecture and the maximal Bochner-Riesz conjecture. The main methods we used are polynomial partitioning and the Bourgain Demeter l^2 decoupling theorem. <br />
<br />
<br />
===Jonathan Hickman===<br />
<br />
Title: Sobolev improving for averages over space curves<br />
<br />
Abstract: Consider the averaging operator given by convolution with arclength measure on compact piece of a smooth curve in R^n. A simple question is to precisely quantify the gain in regularity induced by this averaging, for instance by studying the L^p-Sobolev mapping properties of the operator. This talk will report on ongoing developments towards understanding this problem. In particular, we will explore some non-trivial necessary conditions on the gain in regularity. Joint with D. Beltran, S. Guo and A. Seeger.<br />
<br />
===Hanlong Fang===<br />
<br />
Title: Canonical blow-ups of Grassmann manifolds<br />
<br />
Abstract: We introduce certain canonical blow-ups \mathcal T_{s,p,n}, as well as their distinct submanifolds \mathcal M_{s,p,n}, of Grassmann manifolds G(p,n) by partitioning the Plücker coordinates with respect to a parameter s. Various geometric aspects of \mathcal T_{s,p,n} and \mathcal M_{s,p,n} are studied, for instance, the smoothness, the holomorphic symmetries, the (semi-)positivity of the anti-canonical bundles, the existence of Kähler-Einstein metrics, the functoriality, etc. In particular, we introduce the notion of homeward compactification, of which \mathcal T_{s,p,n} are examples, as a generalization of the wonderful compactification. <br />
<br />
===Bingyang Hu===<br />
<br />
Title: Some structure theorems on general doubling measures.<br />
<br />
Abstract: In this talk, we will first several structure theorems about general doubling measures. Secondly, we will include some main idea to prove one of these results. More precisely, we will focus on the construction of an explicit family of measures that are p-adic doubling for any finite set of primes, however, not doubling. This part generalizes the work by Boylan, Mills and Ward in 2019 in a highly non-trivial way. As some application, we apply these results (that is, the same construction) to show analogous statements for Muckenhoupt Ap weights and reverse Holder weights. This is a joint work with Tess Anderson.<br />
<br />
===Krystal Taylor===<br />
<br />
Title: Quantifications of the Besicovitch Projection theorem in a nonlinear setting <br />
<br />
Abstract: There are several classical results relating the geometry, dimension, and measure of a set to the structure of its orthogonal projections. <br />
It turns out that many nonlinear projection-type operators also have special geometry that allows us to build similar relationships between a set and its "projections", just as in the linear setting. We will discuss a series of recent results from both geometric and probabilistic vantage points. In particular, we will see that the multi-scale analysis techniques of Tao, as well as the energy techniques of Mattila, can be strengthened and generalized to projection-type operators satisfying a transversality condition. As an application, we address the Buffon curve problem, which is to find upper and lower bounds for the rate of decay of the Favard curve length of the four-corner Cantor set.<br />
<br />
===Dominique Maldague===<br />
<br />
Title: A new proof of decoupling for the parabola<br />
<br />
Abstract: Decoupling has to do with measuring the size of functions with specialized Fourier support (in our case, in a neighborhood of the truncated parabola). Bourgain and Demeter resolved the l^2 decoupling conjecture in 2014, using ingredients like the multilinear Kakeya inequality, L^2 orthogonality, and induction-on-scales. I will present the ideas that go into a new proof of decoupling and make some comparison between the two approaches. This is related to recent joint work with Larry Guth and Hong Wang, as well as forthcoming joint work with Yuqiu Fu and Larry Guth.<br />
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=Extras=<br />
[[Blank Analysis Seminar Template]]<br />
<br />
<br />
Graduate Student Seminar:<br />
<br />
https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html</div>Beltranhttps://hilbert.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=20853Analysis Seminar2021-02-19T17:44:05Z<p>Beltran: </p>
<hr />
<div><br />
The 2020-2021 Analysis Seminar will be organized by David Beltran and Andreas Seeger.<br />
It will be online for the entire academic year. The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar at different times, to accommodate speakers).<br />
<br />
Zoom links will be sent to those who have signed up for the Analysis Seminar List. If you would like to subscribe to the Analysis seminar list, send a blank email to analysis+join@g-groups.wisc.edu as well as an additional email to David and Andreas (dbeltran at math, seeger at math) to notify the request.<br />
<br />
If you'd like to suggest speakers for the spring semester please contact David and Andreas (dbeltran at math, seeger at math).<br />
<br />
<br />
<br />
=[[Previous_Analysis_seminars]]=<br />
<br />
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars<br />
<br />
= Current Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 22<br />
|Alexei Poltoratski<br />
|UW Madison<br />
|[[#Alexei Poltoratski | Dirac inner functions ]]<br />
| <br />
|-<br />
|September 29<br />
|Joris Roos<br />
|University of Massachusetts - Lowell<br />
|[[#Polona Durcik and Joris Rooslinktoabstract | A triangular Hilbert transform with curvature, I ]]<br />
| <br />
|-<br />
|Wednesday September 30, 4 p.m.<br />
|Polona Durcik<br />
|Chapman University<br />
|[[#Polona Durcik and Joris Roos | A triangular Hilbert transform with curvature, II ]]<br />
| <br />
|-<br />
|October 6<br />
|Andrew Zimmer<br />
|UW Madison<br />
|[[#Andrew Zimmer | Complex analytic problems on domains with good intrinsic geometry ]]<br />
| <br />
|-<br />
|October 13<br />
|Hong Wang <br />
|Princeton/IAS<br />
|[[#Hong Wang | Improved decoupling for the parabola ]]<br />
| <br />
|-<br />
|October 20<br />
|Kevin Luli<br />
|UC Davis<br />
|[[#Kevin Luli | Smooth Nonnegative Interpolation ]]<br />
| <br />
|-<br />
|October 21, 4.00 p.m.<br />
|Niclas Technau<br />
|UW Madison<br />
|[[#Niclas Technau | Number theoretic applications of oscillatory integrals ]]<br />
| <br />
|-<br />
|October 27<br />
|Terence Harris<br />
| Cornell University<br />
|[[#Terence Harris | Low dimensional pinned distance sets via spherical averages ]]<br />
| <br />
|-<br />
|Monday, November 2, 4 p.m.<br />
|Yuval Wigderson<br />
|Stanford University<br />
|[[#Yuval Wigderson | New perspectives on the uncertainty principle ]]<br />
| <br />
|-<br />
|November 10, 10 a.m. <br />
|Óscar Domínguez<br />
| Universidad Complutense de Madrid<br />
|[[#Oscar Dominguez | New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions ]]<br />
| <br />
|-<br />
|November 17<br />
|Tamas Titkos<br />
|BBS U of Applied Sciences and Renyi Institute<br />
|[[#Tamas Titkos | Isometries of Wasserstein spaces ]]<br />
| <br />
|-<br />
|November 24<br />
|Shukun Wu<br />
|University of Illinois (Urbana-Champaign)<br />
||[[#Shukun Wu | On the Bochner-Riesz operator and the maximal Bochner-Riesz operator ]] <br />
| <br />
|-<br />
|December 1<br />
| Jonathan Hickman<br />
| The University of Edinburgh<br />
|[[#Jonathan Hickman | Sobolev improving for averages over space curves ]]<br />
| <br />
|-<br />
|February 2, 7:00 p.m.<br />
|Hanlong Fang<br />
|UW Madison<br />
|[[#Hanlong Fang | Canonical blow-ups of Grassmann manifolds ]]<br />
| <br />
|-<br />
|February 9<br />
|Bingyang Hu<br />
|Purdue University<br />
|[[#Bingyang Hu | Some structure theorems on general doubling measures ]]<br />
| <br />
|-<br />
|February 16<br />
|Krystal Taylor<br />
|The Ohio State University<br />
|[[#Krystal Taylor | Quantifications of the Besicovitch Projection theorem in a nonlinear setting ]]<br />
|<br />
|-<br />
|February 23<br />
|Dominique Maldague<br />
|MIT<br />
|[[#Dominique Maldague | A new proof of decoupling for the parabola ]]<br />
|<br />
|-<br />
|March 2<br />
|Diogo Oliveira e Silva<br />
|University of Birmingham<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 9<br />
|Oleg Safronov <br />
|University of North Carolina Charlotte<br />
|[[#linktoabstract | Relations between discrete and continuous spectra of differential operators ]]<br />
|<br />
|-<br />
|March 16<br />
|Ziming Shi<br />
|Rutgers University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 23<br />
|Xiumin Du<br />
|Northwestern University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 30, 10:00 a.m.<br />
|Etienne Le Masson<br />
|Cergy Paris University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 6<br />
|Theresa Anderson <br />
|Purdue University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 13<br />
|Nathan Wagner<br />
|Washington University St. Louis<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 20<br />
|Jongchon Kim<br />
| University of British Columbia<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 27<br />
|Yumeng Ou<br />
|University of Pennsylvania<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|May 4<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|}<br />
<br />
=Abstracts=<br />
===Alexei Poltoratski===<br />
<br />
Title: Dirac inner functions<br />
<br />
Abstract: My talk will focus on some new (and old) complex analytic objects arising from Dirac systems of differential equations.<br />
We will discuss connections between problems in complex function theory, spectral and scattering problems for differential<br />
operators and the non-linear Fourier transform.<br />
<br />
===Polona Durcik and Joris Roos===<br />
<br />
Title: A triangular Hilbert transform with curvature, I & II.<br />
<br />
Abstract: The triangular Hilbert is a two-dimensional bilinear singular<br />
originating in time-frequency analysis. No Lp bounds are currently<br />
known for this operator.<br />
In these two talks we discuss a recent joint work with Michael Christ<br />
on a variant of the triangular Hilbert transform involving curvature.<br />
This object is closely related to the bilinear Hilbert transform with<br />
curvature and a maximally modulated singular integral of Stein-Wainger<br />
type. As an application we also discuss a quantitative nonlinear Roth<br />
type theorem on patterns in the Euclidean plane.<br />
The second talk will focus on the proof of a key ingredient, a certain<br />
regularity estimate for a local operator.<br />
<br />
===Andrew Zimmer===<br />
<br />
Title: Complex analytic problems on domains with good intrinsic geometry<br />
<br />
Abstract: In this talk, I will describe a new class of domains in complex Euclidean space which is defined in terms of the existence of a Kaehler metric with good geometric properties. This class is invariant under biholomorphism and includes many well-studied classes of domains such as strongly pseudoconvex domains, finite type domains in dimension two, convex domains, homogeneous domains, and embeddings of Teichmuller spaces. Further, certain analytic problems are tractable for domains in this family even when the boundary is non-smooth. In particular, it is possible to characterize the domains in this family where the dbar-Neumann operator on (0, q)-forms is compact (which generalizes an old result of Fu-Straube for convex domains).<br />
<br />
===Hong Wang===<br />
<br />
Title: Improved decoupling for the parabola<br />
<br />
Abstract: In 2014, Bourgain and Demeter proved the $l^2$ decoupling estimates for the paraboloid with constant $R^{\epsilon}$. <br />
We prove an $(l^2, L^6)$ decoupling inequality for the parabola with constant $(\log R)^c$. This is joint work with Larry Guth and Dominique Maldague.<br />
<br />
===Kevin Luli===<br />
<br />
Title: Smooth Nonnegative Interpolation<br />
<br />
Abstract: Suppose E is an arbitrary subset of R^n. Let f: E \rightarrow [0, \infty). How can we decide if f extends to a nonnegative function C^m function F defined on all of R^n? Suppose E is finite. Can we compute a nonnegative C^m function F on R^n that agrees with f on E with the least possible C^m norm? How many computer operations does this take? In this talk, I will explain recent results on these problems. Non-negativity is one of the most important shape preserving properties for interpolants. In real life applications, the range of the interpolant is imposed by nature. For example, probability density, the amount of snow, rain, humidity, chemical concentration are all nonnegative quantities and are of interest in natural sciences. Even in one dimension, the existing techniques can only handle nonnegative interpolation under special assumptions on the data set. Our results work without any assumptions on the data sets. <br />
<br />
===Niclas Technau===<br />
<br />
Title: Number theoretic applications of oscillatory integrals<br />
<br />
Abstract: We discuss how the analysis of oscillatory integrals can be used to solve number theoretic problems. More specifically, the focus will be on understanding fine-scale statistics of sequences on the unit circle. Further, we shall briefly explain a connection to quantum chaos.<br />
<br />
===Terence Harris===<br />
<br />
Title: Low dimensional pinned distance sets via spherical averages<br />
<br />
Abstract: An inequality is derived for the average t-energy of weighted pinned distance measures, where 0 < t < 1, in terms of the L^2 spherical averages of Fourier transforms of measures. This generalises the result of Liu (originally for Lebesgue measure) to pinned distance sets of dimension smaller than 1, and strengthens Mattila's result from 1987, originally for the full distance set.<br />
<br />
===Yuval Wigderson===<br />
<br />
Title: New perspectives on the uncertainty principle<br />
<br />
Abstract: The phrase ``uncertainty principle'' refers to a wide array of results in several disparate fields of mathematics, all of which capture the notion that a function and its Fourier transform cannot both be ``very localized''. The measure of localization varies from one uncertainty principle to the next, and well-studied notions include the variance (and higher moments), the entropy, the support-size, and the rate of decay at infinity. Similarly, the proofs of the various uncertainty principles rely on a range of tools, from the elementary to the very deep. In this talk, I'll describe how many of the uncertainty principles all follow from a single, simple result, whose proof uses only a basic property of the Fourier transform: that it and its inverse are bounded as operators $L^1 \to L^\infty$. Using this result, one can also prove new variants of the uncertainty principle, which apply to new measures of localization and to operators other than the Fourier transform. This is joint work with Avi Wigderson.<br />
<br />
===Oscar Dominguez===<br />
<br />
Title: New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions<br />
<br />
Abstract: The celebrated Bourgain--Brezis--Mironescu formula enables us to recover Sobolev spaces in terms of limits of Gagliardo seminorms. Very recently, Brezis, Van Schaftingen and Yung have proposed an alternative methodology to approach Sobolev spaces via limits of weak-type Gagliardo functionals. The goal of this talk is twofold. Firstly, we will show that the BvSY result is a special case of a more general phenomenon based on maximal inequalities. In particular, we shall derive not only analogs of the BvSY theorem for different kinds of function spaces (Lebesgue, Calderon, higher-order Sobolev, …), but also applications to ergodic theory, Fourier series, etc. In the second part of the talk, we shall investigate the fractional setting in the BvSY theorem. Our approach is based on new Garsia-type inequalities and an application of the Caffarelli--Silvestre extension. This is joint work with Mario Milman.<br />
<br />
===Tamas Titkos===<br />
<br />
Title: Isometries of Wasserstein spaces<br />
<br />
Abstract: Due to its nice theoretical properties and an astonishing number of<br />
applications via optimal transport problems, probably the most<br />
intensively studied metric nowadays is the p-Wasserstein metric. Given<br />
a complete and separable metric space $X$ and a real number $p\geq1$,<br />
one defines the p-Wasserstein space $\mathcal{W}_p(X)$ as the collection<br />
of Borel probability measures with finite $p$-th moment, endowed with a<br />
distance which is calculated by means of transport plans \cite{5}.<br />
<br />
The main aim of our research project is to reveal the structure of the<br />
isometry group $\mathrm{Isom}(\mathcal{W}_p(X))$. Although<br />
$\mathrm{Isom}(X)$ embeds naturally into<br />
$\mathrm{Isom}(\mathcal{W}_p(X))$ by push-forward, and this embedding<br />
turned out to be surjective in many cases (see e.g. [1]), these two<br />
groups are not isomorphic in general. Kloeckner in [2] described<br />
the isometry group of the quadratic Wasserstein space<br />
$\mathcal{W}_2(\mathbb{R}^n)$, and it turned out that the case of $n=1$<br />
is special in the sense that $\mathrm{Isom}(\mathcal{W}_2(\mathbb{R})$<br />
is extremely rich. Namely, it contains a large subgroup of wild behaving<br />
isometries that distort the shape of measures. Following this line of<br />
investigation, in \cite{3} we described<br />
$\mathrm{Isom}(\mathcal{W}_p(\mathbb{R}))$ and<br />
$\mathrm{Isom}(\mathcal{W}_p([0,1])$ for all $p\geq 1$.<br />
<br />
In this talk I will survey first some of the earlier results in the<br />
subject, and then I will present the key results of [3]. If time<br />
permits, I will also report on our most recent manuscript [4] in<br />
which we extended Kloeckner's multidimensional results. Joint work with Gy\"orgy P\'al Geh\'er (University of Reading)<br />
and D\'aniel Virosztek (IST Austria).<br />
<br />
[1] J. Bertrand and B. Kloeckner, \emph{A geometric study of Wasserstein<br />
spaces: isometric rigidity in negative curvature}, International<br />
Mathematics Research Notices, 2016 (5), 1368--1386.<br />
<br />
[2] B. Kloeckner, \emph{A geometric study of Wasserstein spaces: Euclidean<br />
spaces}, Annali della Scuola Normale Superiore di Pisa - Classe di<br />
Scienze, Serie 5, Tome 9 (2010) no. 2, 297--323.<br />
<br />
[3] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{Isometric study of<br />
Wasserstein spaces – the real line}, Trans. Amer. Math. Soc., 373<br />
(2020), 5855--5883.<br />
<br />
[4] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{The isometry group of<br />
Wasserstein spaces: The Hilbertian case}, submitted manuscript.<br />
<br />
[5] C. Villani, \emph{Optimal Transport: Old and New,}<br />
(Grundlehren der mathematischen Wissenschaften)<br />
Springer, 2009.<br />
<br />
===Shukun Wu===<br />
<br />
Title: On the Bochner-Riesz operator and the maximal Bochner-Riesz operator<br />
<br />
Abstract: The Bochner-Riesz problem is one of the most important problems in the field of Fourier analysis. It has a strong connection to other famous problems, such as the restriction conjecture and the Kakeya conjecture. In this talk, I will present some recent improvements to the Bochner-Riesz conjecture and the maximal Bochner-Riesz conjecture. The main methods we used are polynomial partitioning and the Bourgain Demeter l^2 decoupling theorem. <br />
<br />
<br />
===Jonathan Hickman===<br />
<br />
Title: Sobolev improving for averages over space curves<br />
<br />
Abstract: Consider the averaging operator given by convolution with arclength measure on compact piece of a smooth curve in R^n. A simple question is to precisely quantify the gain in regularity induced by this averaging, for instance by studying the L^p-Sobolev mapping properties of the operator. This talk will report on ongoing developments towards understanding this problem. In particular, we will explore some non-trivial necessary conditions on the gain in regularity. Joint with D. Beltran, S. Guo and A. Seeger.<br />
<br />
===Hanlong Fang===<br />
<br />
Title: Canonical blow-ups of Grassmann manifolds<br />
<br />
Abstract: We introduce certain canonical blow-ups \mathcal T_{s,p,n}, as well as their distinct submanifolds \mathcal M_{s,p,n}, of Grassmann manifolds G(p,n) by partitioning the Plücker coordinates with respect to a parameter s. Various geometric aspects of \mathcal T_{s,p,n} and \mathcal M_{s,p,n} are studied, for instance, the smoothness, the holomorphic symmetries, the (semi-)positivity of the anti-canonical bundles, the existence of Kähler-Einstein metrics, the functoriality, etc. In particular, we introduce the notion of homeward compactification, of which \mathcal T_{s,p,n} are examples, as a generalization of the wonderful compactification. <br />
<br />
===Bingyang Hu===<br />
<br />
Title: Some structure theorems on general doubling measures.<br />
<br />
Abstract: In this talk, we will first several structure theorems about general doubling measures. Secondly, we will include some main idea to prove one of these results. More precisely, we will focus on the construction of an explicit family of measures that are p-adic doubling for any finite set of primes, however, not doubling. This part generalizes the work by Boylan, Mills and Ward in 2019 in a highly non-trivial way. As some application, we apply these results (that is, the same construction) to show analogous statements for Muckenhoupt Ap weights and reverse Holder weights. This is a joint work with Tess Anderson.<br />
<br />
===Krystal Taylor===<br />
<br />
Title: Quantifications of the Besicovitch Projection theorem in a nonlinear setting <br />
<br />
Abstract: There are several classical results relating the geometry, dimension, and measure of a set to the structure of its orthogonal projections. <br />
It turns out that many nonlinear projection-type operators also have special geometry that allows us to build similar relationships between a set and its "projections", just as in the linear setting. We will discuss a series of recent results from both geometric and probabilistic vantage points. In particular, we will see that the multi-scale analysis techniques of Tao, as well as the energy techniques of Mattila, can be strengthened and generalized to projection-type operators satisfying a transversality condition. As an application, we address the Buffon curve problem, which is to find upper and lower bounds for the rate of decay of the Favard curve length of the four-corner Cantor set.<br />
<br />
===Dominique Maldague===<br />
<br />
Title: A new proof of decoupling for the parabola<br />
<br />
Abstract: Decoupling has to do with measuring the size of functions with specialized Fourier support (in our case, in a neighborhood of the truncated parabola). Bourgain and Demeter resolved the l^2 decoupling conjecture in 2014, using ingredients like the multilinear Kakeya inequality, L^2 orthogonality, and induction-on-scales. I will present the ideas that go into a new proof of decoupling and make some comparison between the two approaches. This is related to recent joint work with Larry Guth and Hong Wang, as well as forthcoming joint work with Yuqiu Fu and Larry Guth.<br />
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=Extras=<br />
[[Blank Analysis Seminar Template]]<br />
<br />
<br />
Graduate Student Seminar:<br />
<br />
https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html</div>Beltranhttps://hilbert.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=20846Analysis Seminar2021-02-16T00:33:13Z<p>Beltran: /* Abstracts */</p>
<hr />
<div><br />
The 2020-2021 Analysis Seminar will be organized by David Beltran and Andreas Seeger.<br />
It will be online for the entire academic year. The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar at different times, to accomodate speakers).<br />
<br />
Zoom links will be sent to those who have signed up for the Analysis Seminar List. For instructions how to sign up for seminar lists, see https://www.math.wisc.edu/node/230<br />
<br />
If you'd like to suggest speakers for the spring semester please contact David and Andreas (dbeltran at math, seeger at math).<br />
<br />
<br />
<br />
=[[Previous_Analysis_seminars]]=<br />
<br />
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars<br />
<br />
= Current Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 22<br />
|Alexei Poltoratski<br />
|UW Madison<br />
|[[#Alexei Poltoratski | Dirac inner functions ]]<br />
| <br />
|-<br />
|September 29<br />
|Joris Roos<br />
|University of Massachusetts - Lowell<br />
|[[#Polona Durcik and Joris Rooslinktoabstract | A triangular Hilbert transform with curvature, I ]]<br />
| <br />
|-<br />
|Wednesday September 30, 4 p.m.<br />
|Polona Durcik<br />
|Chapman University<br />
|[[#Polona Durcik and Joris Roos | A triangular Hilbert transform with curvature, II ]]<br />
| <br />
|-<br />
|October 6<br />
|Andrew Zimmer<br />
|UW Madison<br />
|[[#Andrew Zimmer | Complex analytic problems on domains with good intrinsic geometry ]]<br />
| <br />
|-<br />
|October 13<br />
|Hong Wang <br />
|Princeton/IAS<br />
|[[#Hong Wang | Improved decoupling for the parabola ]]<br />
| <br />
|-<br />
|October 20<br />
|Kevin Luli<br />
|UC Davis<br />
|[[#Kevin Luli | Smooth Nonnegative Interpolation ]]<br />
| <br />
|-<br />
|October 21, 4.00 p.m.<br />
|Niclas Technau<br />
|UW Madison<br />
|[[#Niclas Technau | Number theoretic applications of oscillatory integrals ]]<br />
| <br />
|-<br />
|October 27<br />
|Terence Harris<br />
| Cornell University<br />
|[[#Terence Harris | Low dimensional pinned distance sets via spherical averages ]]<br />
| <br />
|-<br />
|Monday, November 2, 4 p.m.<br />
|Yuval Wigderson<br />
|Stanford University<br />
|[[#Yuval Wigderson | New perspectives on the uncertainty principle ]]<br />
| <br />
|-<br />
|November 10, 10 a.m. <br />
|Óscar Domínguez<br />
| Universidad Complutense de Madrid<br />
|[[#Oscar Dominguez | New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions ]]<br />
| <br />
|-<br />
|November 17<br />
|Tamas Titkos<br />
|BBS U of Applied Sciences and Renyi Institute<br />
|[[#Tamas Titkos | Isometries of Wasserstein spaces ]]<br />
| <br />
|-<br />
|November 24<br />
|Shukun Wu<br />
|University of Illinois (Urbana-Champaign)<br />
||[[#Shukun Wu | On the Bochner-Riesz operator and the maximal Bochner-Riesz operator ]] <br />
| <br />
|-<br />
|December 1<br />
| Jonathan Hickman<br />
| The University of Edinburgh<br />
|[[#Jonathan Hickman | Sobolev improving for averages over space curves ]]<br />
| <br />
|-<br />
|February 2, 7:00 p.m.<br />
|Hanlong Fang<br />
|UW Madison<br />
|[[#Hanlong Fang | Canonical blow-ups of Grassmann manifolds ]]<br />
| <br />
|-<br />
|February 9<br />
|Bingyang Hu<br />
|Purdue University<br />
|[[#Bingyang Hu | Some structure theorems on general doubling measures ]]<br />
| <br />
|-<br />
|February 16<br />
|Krystal Taylor<br />
|The Ohio State University<br />
|[[#Krystal Taylor | Quantifications of the Besicovitch Projection theorem in a nonlinear setting ]]<br />
|<br />
|-<br />
|February 23<br />
|Dominique Maldague<br />
|MIT<br />
|[[#Dominique Maldague | A new proof of decoupling for the parabola ]]<br />
|<br />
|-<br />
|March 2<br />
|Diogo Oliveira e Silva<br />
|University of Birmingham<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 9<br />
|Oleg Safronov <br />
|University of North Carolina Charlotte<br />
|[[#linktoabstract | Relations between discrete and continuous spectra of differential operators ]]<br />
|<br />
|-<br />
|March 16<br />
|Ziming Shi<br />
|Rutgers University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 23<br />
|Xiumin Du<br />
|Northwestern University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 30, 10:00 a.m.<br />
|Etienne Le Masson<br />
|Cergy Paris University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 6<br />
|Theresa Anderson <br />
|Purdue University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 13<br />
|Nathan Wagner<br />
|Washington University St. Louis<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 20<br />
|Jongchon Kim<br />
| University of British Columbia<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 27<br />
|Yumeng Ou<br />
|University of Pennsylvania<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|May 4<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|}<br />
<br />
=Abstracts=<br />
===Alexei Poltoratski===<br />
<br />
Title: Dirac inner functions<br />
<br />
Abstract: My talk will focus on some new (and old) complex analytic objects arising from Dirac systems of differential equations.<br />
We will discuss connections between problems in complex function theory, spectral and scattering problems for differential<br />
operators and the non-linear Fourier transform.<br />
<br />
===Polona Durcik and Joris Roos===<br />
<br />
Title: A triangular Hilbert transform with curvature, I & II.<br />
<br />
Abstract: The triangular Hilbert is a two-dimensional bilinear singular<br />
originating in time-frequency analysis. No Lp bounds are currently<br />
known for this operator.<br />
In these two talks we discuss a recent joint work with Michael Christ<br />
on a variant of the triangular Hilbert transform involving curvature.<br />
This object is closely related to the bilinear Hilbert transform with<br />
curvature and a maximally modulated singular integral of Stein-Wainger<br />
type. As an application we also discuss a quantitative nonlinear Roth<br />
type theorem on patterns in the Euclidean plane.<br />
The second talk will focus on the proof of a key ingredient, a certain<br />
regularity estimate for a local operator.<br />
<br />
===Andrew Zimmer===<br />
<br />
Title: Complex analytic problems on domains with good intrinsic geometry<br />
<br />
Abstract: In this talk, I will describe a new class of domains in complex Euclidean space which is defined in terms of the existence of a Kaehler metric with good geometric properties. This class is invariant under biholomorphism and includes many well-studied classes of domains such as strongly pseudoconvex domains, finite type domains in dimension two, convex domains, homogeneous domains, and embeddings of Teichmuller spaces. Further, certain analytic problems are tractable for domains in this family even when the boundary is non-smooth. In particular, it is possible to characterize the domains in this family where the dbar-Neumann operator on (0, q)-forms is compact (which generalizes an old result of Fu-Straube for convex domains).<br />
<br />
===Hong Wang===<br />
<br />
Title: Improved decoupling for the parabola<br />
<br />
Abstract: In 2014, Bourgain and Demeter proved the $l^2$ decoupling estimates for the paraboloid with constant $R^{\epsilon}$. <br />
We prove an $(l^2, L^6)$ decoupling inequality for the parabola with constant $(\log R)^c$. This is joint work with Larry Guth and Dominique Maldague.<br />
<br />
===Kevin Luli===<br />
<br />
Title: Smooth Nonnegative Interpolation<br />
<br />
Abstract: Suppose E is an arbitrary subset of R^n. Let f: E \rightarrow [0, \infty). How can we decide if f extends to a nonnegative function C^m function F defined on all of R^n? Suppose E is finite. Can we compute a nonnegative C^m function F on R^n that agrees with f on E with the least possible C^m norm? How many computer operations does this take? In this talk, I will explain recent results on these problems. Non-negativity is one of the most important shape preserving properties for interpolants. In real life applications, the range of the interpolant is imposed by nature. For example, probability density, the amount of snow, rain, humidity, chemical concentration are all nonnegative quantities and are of interest in natural sciences. Even in one dimension, the existing techniques can only handle nonnegative interpolation under special assumptions on the data set. Our results work without any assumptions on the data sets. <br />
<br />
===Niclas Technau===<br />
<br />
Title: Number theoretic applications of oscillatory integrals<br />
<br />
Abstract: We discuss how the analysis of oscillatory integrals can be used to solve number theoretic problems. More specifically, the focus will be on understanding fine-scale statistics of sequences on the unit circle. Further, we shall briefly explain a connection to quantum chaos.<br />
<br />
===Terence Harris===<br />
<br />
Title: Low dimensional pinned distance sets via spherical averages<br />
<br />
Abstract: An inequality is derived for the average t-energy of weighted pinned distance measures, where 0 < t < 1, in terms of the L^2 spherical averages of Fourier transforms of measures. This generalises the result of Liu (originally for Lebesgue measure) to pinned distance sets of dimension smaller than 1, and strengthens Mattila's result from 1987, originally for the full distance set.<br />
<br />
===Yuval Wigderson===<br />
<br />
Title: New perspectives on the uncertainty principle<br />
<br />
Abstract: The phrase ``uncertainty principle'' refers to a wide array of results in several disparate fields of mathematics, all of which capture the notion that a function and its Fourier transform cannot both be ``very localized''. The measure of localization varies from one uncertainty principle to the next, and well-studied notions include the variance (and higher moments), the entropy, the support-size, and the rate of decay at infinity. Similarly, the proofs of the various uncertainty principles rely on a range of tools, from the elementary to the very deep. In this talk, I'll describe how many of the uncertainty principles all follow from a single, simple result, whose proof uses only a basic property of the Fourier transform: that it and its inverse are bounded as operators $L^1 \to L^\infty$. Using this result, one can also prove new variants of the uncertainty principle, which apply to new measures of localization and to operators other than the Fourier transform. This is joint work with Avi Wigderson.<br />
<br />
===Oscar Dominguez===<br />
<br />
Title: New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions<br />
<br />
Abstract: The celebrated Bourgain--Brezis--Mironescu formula enables us to recover Sobolev spaces in terms of limits of Gagliardo seminorms. Very recently, Brezis, Van Schaftingen and Yung have proposed an alternative methodology to approach Sobolev spaces via limits of weak-type Gagliardo functionals. The goal of this talk is twofold. Firstly, we will show that the BvSY result is a special case of a more general phenomenon based on maximal inequalities. In particular, we shall derive not only analogs of the BvSY theorem for different kinds of function spaces (Lebesgue, Calderon, higher-order Sobolev, …), but also applications to ergodic theory, Fourier series, etc. In the second part of the talk, we shall investigate the fractional setting in the BvSY theorem. Our approach is based on new Garsia-type inequalities and an application of the Caffarelli--Silvestre extension. This is joint work with Mario Milman.<br />
<br />
===Tamas Titkos===<br />
<br />
Title: Isometries of Wasserstein spaces<br />
<br />
Abstract: Due to its nice theoretical properties and an astonishing number of<br />
applications via optimal transport problems, probably the most<br />
intensively studied metric nowadays is the p-Wasserstein metric. Given<br />
a complete and separable metric space $X$ and a real number $p\geq1$,<br />
one defines the p-Wasserstein space $\mathcal{W}_p(X)$ as the collection<br />
of Borel probability measures with finite $p$-th moment, endowed with a<br />
distance which is calculated by means of transport plans \cite{5}.<br />
<br />
The main aim of our research project is to reveal the structure of the<br />
isometry group $\mathrm{Isom}(\mathcal{W}_p(X))$. Although<br />
$\mathrm{Isom}(X)$ embeds naturally into<br />
$\mathrm{Isom}(\mathcal{W}_p(X))$ by push-forward, and this embedding<br />
turned out to be surjective in many cases (see e.g. [1]), these two<br />
groups are not isomorphic in general. Kloeckner in [2] described<br />
the isometry group of the quadratic Wasserstein space<br />
$\mathcal{W}_2(\mathbb{R}^n)$, and it turned out that the case of $n=1$<br />
is special in the sense that $\mathrm{Isom}(\mathcal{W}_2(\mathbb{R})$<br />
is extremely rich. Namely, it contains a large subgroup of wild behaving<br />
isometries that distort the shape of measures. Following this line of<br />
investigation, in \cite{3} we described<br />
$\mathrm{Isom}(\mathcal{W}_p(\mathbb{R}))$ and<br />
$\mathrm{Isom}(\mathcal{W}_p([0,1])$ for all $p\geq 1$.<br />
<br />
In this talk I will survey first some of the earlier results in the<br />
subject, and then I will present the key results of [3]. If time<br />
permits, I will also report on our most recent manuscript [4] in<br />
which we extended Kloeckner's multidimensional results. Joint work with Gy\"orgy P\'al Geh\'er (University of Reading)<br />
and D\'aniel Virosztek (IST Austria).<br />
<br />
[1] J. Bertrand and B. Kloeckner, \emph{A geometric study of Wasserstein<br />
spaces: isometric rigidity in negative curvature}, International<br />
Mathematics Research Notices, 2016 (5), 1368--1386.<br />
<br />
[2] B. Kloeckner, \emph{A geometric study of Wasserstein spaces: Euclidean<br />
spaces}, Annali della Scuola Normale Superiore di Pisa - Classe di<br />
Scienze, Serie 5, Tome 9 (2010) no. 2, 297--323.<br />
<br />
[3] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{Isometric study of<br />
Wasserstein spaces – the real line}, Trans. Amer. Math. Soc., 373<br />
(2020), 5855--5883.<br />
<br />
[4] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{The isometry group of<br />
Wasserstein spaces: The Hilbertian case}, submitted manuscript.<br />
<br />
[5] C. Villani, \emph{Optimal Transport: Old and New,}<br />
(Grundlehren der mathematischen Wissenschaften)<br />
Springer, 2009.<br />
<br />
===Shukun Wu===<br />
<br />
Title: On the Bochner-Riesz operator and the maximal Bochner-Riesz operator<br />
<br />
Abstract: The Bochner-Riesz problem is one of the most important problems in the field of Fourier analysis. It has a strong connection to other famous problems, such as the restriction conjecture and the Kakeya conjecture. In this talk, I will present some recent improvements to the Bochner-Riesz conjecture and the maximal Bochner-Riesz conjecture. The main methods we used are polynomial partitioning and the Bourgain Demeter l^2 decoupling theorem. <br />
<br />
<br />
===Jonathan Hickman===<br />
<br />
Title: Sobolev improving for averages over space curves<br />
<br />
Abstract: Consider the averaging operator given by convolution with arclength measure on compact piece of a smooth curve in R^n. A simple question is to precisely quantify the gain in regularity induced by this averaging, for instance by studying the L^p-Sobolev mapping properties of the operator. This talk will report on ongoing developments towards understanding this problem. In particular, we will explore some non-trivial necessary conditions on the gain in regularity. Joint with D. Beltran, S. Guo and A. Seeger.<br />
<br />
===Hanlong Fang===<br />
<br />
Title: Canonical blow-ups of Grassmann manifolds<br />
<br />
Abstract: We introduce certain canonical blow-ups \mathcal T_{s,p,n}, as well as their distinct submanifolds \mathcal M_{s,p,n}, of Grassmann manifolds G(p,n) by partitioning the Plücker coordinates with respect to a parameter s. Various geometric aspects of \mathcal T_{s,p,n} and \mathcal M_{s,p,n} are studied, for instance, the smoothness, the holomorphic symmetries, the (semi-)positivity of the anti-canonical bundles, the existence of Kähler-Einstein metrics, the functoriality, etc. In particular, we introduce the notion of homeward compactification, of which \mathcal T_{s,p,n} are examples, as a generalization of the wonderful compactification. <br />
<br />
===Bingyang Hu===<br />
<br />
Title: Some structure theorems on general doubling measures.<br />
<br />
Abstract: In this talk, we will first several structure theorems about general doubling measures. Secondly, we will include some main idea to prove one of these results. More precisely, we will focus on the construction of an explicit family of measures that are p-adic doubling for any finite set of primes, however, not doubling. This part generalizes the work by Boylan, Mills and Ward in 2019 in a highly non-trivial way. As some application, we apply these results (that is, the same construction) to show analogous statements for Muckenhoupt Ap weights and reverse Holder weights. This is a joint work with Tess Anderson.<br />
<br />
===Krystal Taylor===<br />
<br />
Title: Quantifications of the Besicovitch Projection theorem in a nonlinear setting <br />
<br />
Abstract: There are several classical results relating the geometry, dimension, and measure of a set to the structure of its orthogonal projections. <br />
It turns out that many nonlinear projection-type operators also have special geometry that allows us to build similar relationships between a set and its "projections", just as in the linear setting. We will discuss a series of recent results from both geometric and probabilistic vantage points. In particular, we will see that the multi-scale analysis techniques of Tao, as well as the energy techniques of Mattila, can be strengthened and generalized to projection-type operators satisfying a transversality condition. As an application, we address the Buffon curve problem, which is to find upper and lower bounds for the rate of decay of the Favard curve length of the four-corner Cantor set.<br />
<br />
===Dominique Maldague===<br />
<br />
Title: A new proof of decoupling for the parabola<br />
<br />
Abstract: Decoupling has to do with measuring the size of functions with specialized Fourier support (in our case, in a neighborhood of the truncated parabola). Bourgain and Demeter resolved the l^2 decoupling conjecture in 2014, using ingredients like the multilinear Kakeya inequality, L^2 orthogonality, and induction-on-scales. I will present the ideas that go into a new proof of decoupling and make some comparison between the two approaches. This is related to recent joint work with Larry Guth and Hong Wang, as well as forthcoming joint work with Yuqiu Fu and Larry Guth.<br />
<br />
===Name===<br />
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===Name===<br />
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Title:<br />
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===Name===<br />
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Title:<br />
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===Name===<br />
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Title:<br />
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===Name===<br />
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===Name===<br />
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Title:<br />
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=Extras=<br />
[[Blank Analysis Seminar Template]]<br />
<br />
<br />
Graduate Student Seminar:<br />
<br />
https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html</div>Beltranhttps://hilbert.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=20845Analysis Seminar2021-02-16T00:32:27Z<p>Beltran: /* Current Analysis Seminar Schedule */</p>
<hr />
<div><br />
The 2020-2021 Analysis Seminar will be organized by David Beltran and Andreas Seeger.<br />
It will be online for the entire academic year. The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar at different times, to accomodate speakers).<br />
<br />
Zoom links will be sent to those who have signed up for the Analysis Seminar List. For instructions how to sign up for seminar lists, see https://www.math.wisc.edu/node/230<br />
<br />
If you'd like to suggest speakers for the spring semester please contact David and Andreas (dbeltran at math, seeger at math).<br />
<br />
<br />
<br />
=[[Previous_Analysis_seminars]]=<br />
<br />
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars<br />
<br />
= Current Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 22<br />
|Alexei Poltoratski<br />
|UW Madison<br />
|[[#Alexei Poltoratski | Dirac inner functions ]]<br />
| <br />
|-<br />
|September 29<br />
|Joris Roos<br />
|University of Massachusetts - Lowell<br />
|[[#Polona Durcik and Joris Rooslinktoabstract | A triangular Hilbert transform with curvature, I ]]<br />
| <br />
|-<br />
|Wednesday September 30, 4 p.m.<br />
|Polona Durcik<br />
|Chapman University<br />
|[[#Polona Durcik and Joris Roos | A triangular Hilbert transform with curvature, II ]]<br />
| <br />
|-<br />
|October 6<br />
|Andrew Zimmer<br />
|UW Madison<br />
|[[#Andrew Zimmer | Complex analytic problems on domains with good intrinsic geometry ]]<br />
| <br />
|-<br />
|October 13<br />
|Hong Wang <br />
|Princeton/IAS<br />
|[[#Hong Wang | Improved decoupling for the parabola ]]<br />
| <br />
|-<br />
|October 20<br />
|Kevin Luli<br />
|UC Davis<br />
|[[#Kevin Luli | Smooth Nonnegative Interpolation ]]<br />
| <br />
|-<br />
|October 21, 4.00 p.m.<br />
|Niclas Technau<br />
|UW Madison<br />
|[[#Niclas Technau | Number theoretic applications of oscillatory integrals ]]<br />
| <br />
|-<br />
|October 27<br />
|Terence Harris<br />
| Cornell University<br />
|[[#Terence Harris | Low dimensional pinned distance sets via spherical averages ]]<br />
| <br />
|-<br />
|Monday, November 2, 4 p.m.<br />
|Yuval Wigderson<br />
|Stanford University<br />
|[[#Yuval Wigderson | New perspectives on the uncertainty principle ]]<br />
| <br />
|-<br />
|November 10, 10 a.m. <br />
|Óscar Domínguez<br />
| Universidad Complutense de Madrid<br />
|[[#Oscar Dominguez | New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions ]]<br />
| <br />
|-<br />
|November 17<br />
|Tamas Titkos<br />
|BBS U of Applied Sciences and Renyi Institute<br />
|[[#Tamas Titkos | Isometries of Wasserstein spaces ]]<br />
| <br />
|-<br />
|November 24<br />
|Shukun Wu<br />
|University of Illinois (Urbana-Champaign)<br />
||[[#Shukun Wu | On the Bochner-Riesz operator and the maximal Bochner-Riesz operator ]] <br />
| <br />
|-<br />
|December 1<br />
| Jonathan Hickman<br />
| The University of Edinburgh<br />
|[[#Jonathan Hickman | Sobolev improving for averages over space curves ]]<br />
| <br />
|-<br />
|February 2, 7:00 p.m.<br />
|Hanlong Fang<br />
|UW Madison<br />
|[[#Hanlong Fang | Canonical blow-ups of Grassmann manifolds ]]<br />
| <br />
|-<br />
|February 9<br />
|Bingyang Hu<br />
|Purdue University<br />
|[[#Bingyang Hu | Some structure theorems on general doubling measures ]]<br />
| <br />
|-<br />
|February 16<br />
|Krystal Taylor<br />
|The Ohio State University<br />
|[[#Krystal Taylor | Quantifications of the Besicovitch Projection theorem in a nonlinear setting ]]<br />
|<br />
|-<br />
|February 23<br />
|Dominique Maldague<br />
|MIT<br />
|[[#Dominique Maldague | A new proof of decoupling for the parabola ]]<br />
|<br />
|-<br />
|March 2<br />
|Diogo Oliveira e Silva<br />
|University of Birmingham<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 9<br />
|Oleg Safronov <br />
|University of North Carolina Charlotte<br />
|[[#linktoabstract | Relations between discrete and continuous spectra of differential operators ]]<br />
|<br />
|-<br />
|March 16<br />
|Ziming Shi<br />
|Rutgers University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 23<br />
|Xiumin Du<br />
|Northwestern University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 30, 10:00 a.m.<br />
|Etienne Le Masson<br />
|Cergy Paris University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 6<br />
|Theresa Anderson <br />
|Purdue University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 13<br />
|Nathan Wagner<br />
|Washington University St. Louis<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 20<br />
|Jongchon Kim<br />
| University of British Columbia<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 27<br />
|Yumeng Ou<br />
|University of Pennsylvania<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|May 4<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|}<br />
<br />
=Abstracts=<br />
===Alexei Poltoratski===<br />
<br />
Title: Dirac inner functions<br />
<br />
Abstract: My talk will focus on some new (and old) complex analytic objects arising from Dirac systems of differential equations.<br />
We will discuss connections between problems in complex function theory, spectral and scattering problems for differential<br />
operators and the non-linear Fourier transform.<br />
<br />
===Polona Durcik and Joris Roos===<br />
<br />
Title: A triangular Hilbert transform with curvature, I & II.<br />
<br />
Abstract: The triangular Hilbert is a two-dimensional bilinear singular<br />
originating in time-frequency analysis. No Lp bounds are currently<br />
known for this operator.<br />
In these two talks we discuss a recent joint work with Michael Christ<br />
on a variant of the triangular Hilbert transform involving curvature.<br />
This object is closely related to the bilinear Hilbert transform with<br />
curvature and a maximally modulated singular integral of Stein-Wainger<br />
type. As an application we also discuss a quantitative nonlinear Roth<br />
type theorem on patterns in the Euclidean plane.<br />
The second talk will focus on the proof of a key ingredient, a certain<br />
regularity estimate for a local operator.<br />
<br />
===Andrew Zimmer===<br />
<br />
Title: Complex analytic problems on domains with good intrinsic geometry<br />
<br />
Abstract: In this talk, I will describe a new class of domains in complex Euclidean space which is defined in terms of the existence of a Kaehler metric with good geometric properties. This class is invariant under biholomorphism and includes many well-studied classes of domains such as strongly pseudoconvex domains, finite type domains in dimension two, convex domains, homogeneous domains, and embeddings of Teichmuller spaces. Further, certain analytic problems are tractable for domains in this family even when the boundary is non-smooth. In particular, it is possible to characterize the domains in this family where the dbar-Neumann operator on (0, q)-forms is compact (which generalizes an old result of Fu-Straube for convex domains).<br />
<br />
===Hong Wang===<br />
<br />
Title: Improved decoupling for the parabola<br />
<br />
Abstract: In 2014, Bourgain and Demeter proved the $l^2$ decoupling estimates for the paraboloid with constant $R^{\epsilon}$. <br />
We prove an $(l^2, L^6)$ decoupling inequality for the parabola with constant $(\log R)^c$. This is joint work with Larry Guth and Dominique Maldague.<br />
<br />
===Kevin Luli===<br />
<br />
Title: Smooth Nonnegative Interpolation<br />
<br />
Abstract: Suppose E is an arbitrary subset of R^n. Let f: E \rightarrow [0, \infty). How can we decide if f extends to a nonnegative function C^m function F defined on all of R^n? Suppose E is finite. Can we compute a nonnegative C^m function F on R^n that agrees with f on E with the least possible C^m norm? How many computer operations does this take? In this talk, I will explain recent results on these problems. Non-negativity is one of the most important shape preserving properties for interpolants. In real life applications, the range of the interpolant is imposed by nature. For example, probability density, the amount of snow, rain, humidity, chemical concentration are all nonnegative quantities and are of interest in natural sciences. Even in one dimension, the existing techniques can only handle nonnegative interpolation under special assumptions on the data set. Our results work without any assumptions on the data sets. <br />
<br />
===Niclas Technau===<br />
<br />
Title: Number theoretic applications of oscillatory integrals<br />
<br />
Abstract: We discuss how the analysis of oscillatory integrals can be used to solve number theoretic problems. More specifically, the focus will be on understanding fine-scale statistics of sequences on the unit circle. Further, we shall briefly explain a connection to quantum chaos.<br />
<br />
===Terence Harris===<br />
<br />
Title: Low dimensional pinned distance sets via spherical averages<br />
<br />
Abstract: An inequality is derived for the average t-energy of weighted pinned distance measures, where 0 < t < 1, in terms of the L^2 spherical averages of Fourier transforms of measures. This generalises the result of Liu (originally for Lebesgue measure) to pinned distance sets of dimension smaller than 1, and strengthens Mattila's result from 1987, originally for the full distance set.<br />
<br />
===Yuval Wigderson===<br />
<br />
Title: New perspectives on the uncertainty principle<br />
<br />
Abstract: The phrase ``uncertainty principle'' refers to a wide array of results in several disparate fields of mathematics, all of which capture the notion that a function and its Fourier transform cannot both be ``very localized''. The measure of localization varies from one uncertainty principle to the next, and well-studied notions include the variance (and higher moments), the entropy, the support-size, and the rate of decay at infinity. Similarly, the proofs of the various uncertainty principles rely on a range of tools, from the elementary to the very deep. In this talk, I'll describe how many of the uncertainty principles all follow from a single, simple result, whose proof uses only a basic property of the Fourier transform: that it and its inverse are bounded as operators $L^1 \to L^\infty$. Using this result, one can also prove new variants of the uncertainty principle, which apply to new measures of localization and to operators other than the Fourier transform. This is joint work with Avi Wigderson.<br />
<br />
===Oscar Dominguez===<br />
<br />
Title: New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions<br />
<br />
Abstract: The celebrated Bourgain--Brezis--Mironescu formula enables us to recover Sobolev spaces in terms of limits of Gagliardo seminorms. Very recently, Brezis, Van Schaftingen and Yung have proposed an alternative methodology to approach Sobolev spaces via limits of weak-type Gagliardo functionals. The goal of this talk is twofold. Firstly, we will show that the BvSY result is a special case of a more general phenomenon based on maximal inequalities. In particular, we shall derive not only analogs of the BvSY theorem for different kinds of function spaces (Lebesgue, Calderon, higher-order Sobolev, …), but also applications to ergodic theory, Fourier series, etc. In the second part of the talk, we shall investigate the fractional setting in the BvSY theorem. Our approach is based on new Garsia-type inequalities and an application of the Caffarelli--Silvestre extension. This is joint work with Mario Milman.<br />
<br />
===Tamas Titkos===<br />
<br />
Title: Isometries of Wasserstein spaces<br />
<br />
Abstract: Due to its nice theoretical properties and an astonishing number of<br />
applications via optimal transport problems, probably the most<br />
intensively studied metric nowadays is the p-Wasserstein metric. Given<br />
a complete and separable metric space $X$ and a real number $p\geq1$,<br />
one defines the p-Wasserstein space $\mathcal{W}_p(X)$ as the collection<br />
of Borel probability measures with finite $p$-th moment, endowed with a<br />
distance which is calculated by means of transport plans \cite{5}.<br />
<br />
The main aim of our research project is to reveal the structure of the<br />
isometry group $\mathrm{Isom}(\mathcal{W}_p(X))$. Although<br />
$\mathrm{Isom}(X)$ embeds naturally into<br />
$\mathrm{Isom}(\mathcal{W}_p(X))$ by push-forward, and this embedding<br />
turned out to be surjective in many cases (see e.g. [1]), these two<br />
groups are not isomorphic in general. Kloeckner in [2] described<br />
the isometry group of the quadratic Wasserstein space<br />
$\mathcal{W}_2(\mathbb{R}^n)$, and it turned out that the case of $n=1$<br />
is special in the sense that $\mathrm{Isom}(\mathcal{W}_2(\mathbb{R})$<br />
is extremely rich. Namely, it contains a large subgroup of wild behaving<br />
isometries that distort the shape of measures. Following this line of<br />
investigation, in \cite{3} we described<br />
$\mathrm{Isom}(\mathcal{W}_p(\mathbb{R}))$ and<br />
$\mathrm{Isom}(\mathcal{W}_p([0,1])$ for all $p\geq 1$.<br />
<br />
In this talk I will survey first some of the earlier results in the<br />
subject, and then I will present the key results of [3]. If time<br />
permits, I will also report on our most recent manuscript [4] in<br />
which we extended Kloeckner's multidimensional results. Joint work with Gy\"orgy P\'al Geh\'er (University of Reading)<br />
and D\'aniel Virosztek (IST Austria).<br />
<br />
[1] J. Bertrand and B. Kloeckner, \emph{A geometric study of Wasserstein<br />
spaces: isometric rigidity in negative curvature}, International<br />
Mathematics Research Notices, 2016 (5), 1368--1386.<br />
<br />
[2] B. Kloeckner, \emph{A geometric study of Wasserstein spaces: Euclidean<br />
spaces}, Annali della Scuola Normale Superiore di Pisa - Classe di<br />
Scienze, Serie 5, Tome 9 (2010) no. 2, 297--323.<br />
<br />
[3] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{Isometric study of<br />
Wasserstein spaces – the real line}, Trans. Amer. Math. Soc., 373<br />
(2020), 5855--5883.<br />
<br />
[4] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{The isometry group of<br />
Wasserstein spaces: The Hilbertian case}, submitted manuscript.<br />
<br />
[5] C. Villani, \emph{Optimal Transport: Old and New,}<br />
(Grundlehren der mathematischen Wissenschaften)<br />
Springer, 2009.<br />
<br />
===Shukun Wu===<br />
<br />
Title: On the Bochner-Riesz operator and the maximal Bochner-Riesz operator<br />
<br />
Abstract: The Bochner-Riesz problem is one of the most important problems in the field of Fourier analysis. It has a strong connection to other famous problems, such as the restriction conjecture and the Kakeya conjecture. In this talk, I will present some recent improvements to the Bochner-Riesz conjecture and the maximal Bochner-Riesz conjecture. The main methods we used are polynomial partitioning and the Bourgain Demeter l^2 decoupling theorem. <br />
<br />
<br />
===Jonathan Hickman===<br />
<br />
Title: Sobolev improving for averages over space curves<br />
<br />
Abstract: Consider the averaging operator given by convolution with arclength measure on compact piece of a smooth curve in R^n. A simple question is to precisely quantify the gain in regularity induced by this averaging, for instance by studying the L^p-Sobolev mapping properties of the operator. This talk will report on ongoing developments towards understanding this problem. In particular, we will explore some non-trivial necessary conditions on the gain in regularity. Joint with D. Beltran, S. Guo and A. Seeger.<br />
<br />
===Hanlong Fang===<br />
<br />
Title: Canonical blow-ups of Grassmann manifolds<br />
<br />
Abstract: We introduce certain canonical blow-ups \mathcal T_{s,p,n}, as well as their distinct submanifolds \mathcal M_{s,p,n}, of Grassmann manifolds G(p,n) by partitioning the Plücker coordinates with respect to a parameter s. Various geometric aspects of \mathcal T_{s,p,n} and \mathcal M_{s,p,n} are studied, for instance, the smoothness, the holomorphic symmetries, the (semi-)positivity of the anti-canonical bundles, the existence of Kähler-Einstein metrics, the functoriality, etc. In particular, we introduce the notion of homeward compactification, of which \mathcal T_{s,p,n} are examples, as a generalization of the wonderful compactification. <br />
<br />
===Bingyang Hu===<br />
<br />
Title: Some structure theorems on general doubling measures.<br />
<br />
Abstract: In this talk, we will first several structure theorems about general doubling measures. Secondly, we will include some main idea to prove one of these results. More precisely, we will focus on the construction of an explicit family of measures that are p-adic doubling for any finite set of primes, however, not doubling. This part generalizes the work by Boylan, Mills and Ward in 2019 in a highly non-trivial way. As some application, we apply these results (that is, the same construction) to show analogous statements for Muckenhoupt Ap weights and reverse Holder weights. This is a joint work with Tess Anderson.<br />
<br />
===Krystal Taylor===<br />
<br />
Title: Quantifications of the Besicovitch Projection theorem in a nonlinear setting <br />
<br />
Abstract: There are several classical results relating the geometry, dimension, and measure of a set to the structure of its orthogonal projections. <br />
It turns out that many nonlinear projection-type operators also have special geometry that allows us to build similar relationships between a set and its "projections", just as in the linear setting. We will discuss a series of recent results from both geometric and probabilistic vantage points. In particular, we will see that the multi-scale analysis techniques of Tao, as well as the energy techniques of Mattila, can be strengthened and generalized to projection-type operators satisfying a transversality condition. As an application, we address the Buffon curve problem, which is to find upper and lower bounds for the rate of decay of the Favard curve length of the four-corner Cantor set.<br />
<br />
===Name===<br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
===Name===<br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
===Name===<br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
===Name===<br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
===Name===<br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
===Name===<br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
===Name===<br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]<br />
<br />
<br />
Graduate Student Seminar:<br />
<br />
https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html</div>Beltranhttps://hilbert.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=20802Analysis Seminar2021-02-08T22:28:05Z<p>Beltran: /* Krystal Taylor */</p>
<hr />
<div><br />
The 2020-2021 Analysis Seminar will be organized by David Beltran and Andreas Seeger.<br />
It will be online for the entire academic year. The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar at different times, to accomodate speakers).<br />
<br />
Zoom links will be sent to those who have signed up for the Analysis Seminar List. For instructions how to sign up for seminar lists, see https://www.math.wisc.edu/node/230<br />
<br />
If you'd like to suggest speakers for the spring semester please contact David and Andreas (dbeltran at math, seeger at math).<br />
<br />
<br />
<br />
=[[Previous_Analysis_seminars]]=<br />
<br />
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars<br />
<br />
= Current Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 22<br />
|Alexei Poltoratski<br />
|UW Madison<br />
|[[#Alexei Poltoratski | Dirac inner functions ]]<br />
| <br />
|-<br />
|September 29<br />
|Joris Roos<br />
|University of Massachusetts - Lowell<br />
|[[#Polona Durcik and Joris Rooslinktoabstract | A triangular Hilbert transform with curvature, I ]]<br />
| <br />
|-<br />
|Wednesday September 30, 4 p.m.<br />
|Polona Durcik<br />
|Chapman University<br />
|[[#Polona Durcik and Joris Roos | A triangular Hilbert transform with curvature, II ]]<br />
| <br />
|-<br />
|October 6<br />
|Andrew Zimmer<br />
|UW Madison<br />
|[[#Andrew Zimmer | Complex analytic problems on domains with good intrinsic geometry ]]<br />
| <br />
|-<br />
|October 13<br />
|Hong Wang <br />
|Princeton/IAS<br />
|[[#Hong Wang | Improved decoupling for the parabola ]]<br />
| <br />
|-<br />
|October 20<br />
|Kevin Luli<br />
|UC Davis<br />
|[[#Kevin Luli | Smooth Nonnegative Interpolation ]]<br />
| <br />
|-<br />
|October 21, 4.00 p.m.<br />
|Niclas Technau<br />
|UW Madison<br />
|[[#Niclas Technau | Number theoretic applications of oscillatory integrals ]]<br />
| <br />
|-<br />
|October 27<br />
|Terence Harris<br />
| Cornell University<br />
|[[#Terence Harris | Low dimensional pinned distance sets via spherical averages ]]<br />
| <br />
|-<br />
|Monday, November 2, 4 p.m.<br />
|Yuval Wigderson<br />
|Stanford University<br />
|[[#Yuval Wigderson | New perspectives on the uncertainty principle ]]<br />
| <br />
|-<br />
|November 10, 10 a.m. <br />
|Óscar Domínguez<br />
| Universidad Complutense de Madrid<br />
|[[#Oscar Dominguez | New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions ]]<br />
| <br />
|-<br />
|November 17<br />
|Tamas Titkos<br />
|BBS U of Applied Sciences and Renyi Institute<br />
|[[#Tamas Titkos | Isometries of Wasserstein spaces ]]<br />
| <br />
|-<br />
|November 24<br />
|Shukun Wu<br />
|University of Illinois (Urbana-Champaign)<br />
||[[#Shukun Wu | On the Bochner-Riesz operator and the maximal Bochner-Riesz operator ]] <br />
| <br />
|-<br />
|December 1<br />
| Jonathan Hickman<br />
| The University of Edinburgh<br />
|[[#Jonathan Hickman | Sobolev improving for averages over space curves ]]<br />
| <br />
|-<br />
|February 2, 7:00 p.m.<br />
|Hanlong Fang<br />
|UW Madison<br />
|[[#Hanlong Fang | Canonical blow-ups of Grassmann manifolds ]]<br />
| <br />
|-<br />
|February 9<br />
|Bingyang Hu<br />
|Purdue University<br />
|[[#Bingyang Hu | Some structure theorems on general doubling measures ]]<br />
| <br />
|-<br />
|February 16<br />
|Krystal Taylor<br />
|The Ohio State University<br />
|[[#Krystal Taylor | Quantifications of the Besicovitch Projection theorem in a nonlinear setting ]]<br />
|<br />
|-<br />
|February 23<br />
|Dominique Maldague<br />
|MIT<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 2<br />
|Diogo Oliveira e Silva<br />
|University of Birmingham<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 9<br />
|Oleg Safronov <br />
|University of North Carolina Charlotte<br />
|[[#linktoabstract | Relations between discrete and continuous spectra of differential operators ]]<br />
|<br />
|-<br />
|March 16<br />
|Ziming Shi<br />
|Rutgers University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 23<br />
|Xiumin Du<br />
|Northwestern University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 30, 10:00 a.m.<br />
|Etienne Le Masson<br />
|Cergy Paris University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 6<br />
|Theresa Anderson <br />
|Purdue University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 13<br />
|Nathan Wagner<br />
|Washington University St. Louis<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 20<br />
|Jongchon Kim<br />
| University of British Columbia<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 27<br />
|Yumeng Ou<br />
|University of Pennsylvania<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|May 4<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|}<br />
<br />
=Abstracts=<br />
===Alexei Poltoratski===<br />
<br />
Title: Dirac inner functions<br />
<br />
Abstract: My talk will focus on some new (and old) complex analytic objects arising from Dirac systems of differential equations.<br />
We will discuss connections between problems in complex function theory, spectral and scattering problems for differential<br />
operators and the non-linear Fourier transform.<br />
<br />
===Polona Durcik and Joris Roos===<br />
<br />
Title: A triangular Hilbert transform with curvature, I & II.<br />
<br />
Abstract: The triangular Hilbert is a two-dimensional bilinear singular<br />
originating in time-frequency analysis. No Lp bounds are currently<br />
known for this operator.<br />
In these two talks we discuss a recent joint work with Michael Christ<br />
on a variant of the triangular Hilbert transform involving curvature.<br />
This object is closely related to the bilinear Hilbert transform with<br />
curvature and a maximally modulated singular integral of Stein-Wainger<br />
type. As an application we also discuss a quantitative nonlinear Roth<br />
type theorem on patterns in the Euclidean plane.<br />
The second talk will focus on the proof of a key ingredient, a certain<br />
regularity estimate for a local operator.<br />
<br />
===Andrew Zimmer===<br />
<br />
Title: Complex analytic problems on domains with good intrinsic geometry<br />
<br />
Abstract: In this talk, I will describe a new class of domains in complex Euclidean space which is defined in terms of the existence of a Kaehler metric with good geometric properties. This class is invariant under biholomorphism and includes many well-studied classes of domains such as strongly pseudoconvex domains, finite type domains in dimension two, convex domains, homogeneous domains, and embeddings of Teichmuller spaces. Further, certain analytic problems are tractable for domains in this family even when the boundary is non-smooth. In particular, it is possible to characterize the domains in this family where the dbar-Neumann operator on (0, q)-forms is compact (which generalizes an old result of Fu-Straube for convex domains).<br />
<br />
===Hong Wang===<br />
<br />
Title: Improved decoupling for the parabola<br />
<br />
Abstract: In 2014, Bourgain and Demeter proved the $l^2$ decoupling estimates for the paraboloid with constant $R^{\epsilon}$. <br />
We prove an $(l^2, L^6)$ decoupling inequality for the parabola with constant $(\log R)^c$. This is joint work with Larry Guth and Dominique Maldague.<br />
<br />
===Kevin Luli===<br />
<br />
Title: Smooth Nonnegative Interpolation<br />
<br />
Abstract: Suppose E is an arbitrary subset of R^n. Let f: E \rightarrow [0, \infty). How can we decide if f extends to a nonnegative function C^m function F defined on all of R^n? Suppose E is finite. Can we compute a nonnegative C^m function F on R^n that agrees with f on E with the least possible C^m norm? How many computer operations does this take? In this talk, I will explain recent results on these problems. Non-negativity is one of the most important shape preserving properties for interpolants. In real life applications, the range of the interpolant is imposed by nature. For example, probability density, the amount of snow, rain, humidity, chemical concentration are all nonnegative quantities and are of interest in natural sciences. Even in one dimension, the existing techniques can only handle nonnegative interpolation under special assumptions on the data set. Our results work without any assumptions on the data sets. <br />
<br />
===Niclas Technau===<br />
<br />
Title: Number theoretic applications of oscillatory integrals<br />
<br />
Abstract: We discuss how the analysis of oscillatory integrals can be used to solve number theoretic problems. More specifically, the focus will be on understanding fine-scale statistics of sequences on the unit circle. Further, we shall briefly explain a connection to quantum chaos.<br />
<br />
===Terence Harris===<br />
<br />
Title: Low dimensional pinned distance sets via spherical averages<br />
<br />
Abstract: An inequality is derived for the average t-energy of weighted pinned distance measures, where 0 < t < 1, in terms of the L^2 spherical averages of Fourier transforms of measures. This generalises the result of Liu (originally for Lebesgue measure) to pinned distance sets of dimension smaller than 1, and strengthens Mattila's result from 1987, originally for the full distance set.<br />
<br />
===Yuval Wigderson===<br />
<br />
Title: New perspectives on the uncertainty principle<br />
<br />
Abstract: The phrase ``uncertainty principle'' refers to a wide array of results in several disparate fields of mathematics, all of which capture the notion that a function and its Fourier transform cannot both be ``very localized''. The measure of localization varies from one uncertainty principle to the next, and well-studied notions include the variance (and higher moments), the entropy, the support-size, and the rate of decay at infinity. Similarly, the proofs of the various uncertainty principles rely on a range of tools, from the elementary to the very deep. In this talk, I'll describe how many of the uncertainty principles all follow from a single, simple result, whose proof uses only a basic property of the Fourier transform: that it and its inverse are bounded as operators $L^1 \to L^\infty$. Using this result, one can also prove new variants of the uncertainty principle, which apply to new measures of localization and to operators other than the Fourier transform. This is joint work with Avi Wigderson.<br />
<br />
===Oscar Dominguez===<br />
<br />
Title: New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions<br />
<br />
Abstract: The celebrated Bourgain--Brezis--Mironescu formula enables us to recover Sobolev spaces in terms of limits of Gagliardo seminorms. Very recently, Brezis, Van Schaftingen and Yung have proposed an alternative methodology to approach Sobolev spaces via limits of weak-type Gagliardo functionals. The goal of this talk is twofold. Firstly, we will show that the BvSY result is a special case of a more general phenomenon based on maximal inequalities. In particular, we shall derive not only analogs of the BvSY theorem for different kinds of function spaces (Lebesgue, Calderon, higher-order Sobolev, …), but also applications to ergodic theory, Fourier series, etc. In the second part of the talk, we shall investigate the fractional setting in the BvSY theorem. Our approach is based on new Garsia-type inequalities and an application of the Caffarelli--Silvestre extension. This is joint work with Mario Milman.<br />
<br />
===Tamas Titkos===<br />
<br />
Title: Isometries of Wasserstein spaces<br />
<br />
Abstract: Due to its nice theoretical properties and an astonishing number of<br />
applications via optimal transport problems, probably the most<br />
intensively studied metric nowadays is the p-Wasserstein metric. Given<br />
a complete and separable metric space $X$ and a real number $p\geq1$,<br />
one defines the p-Wasserstein space $\mathcal{W}_p(X)$ as the collection<br />
of Borel probability measures with finite $p$-th moment, endowed with a<br />
distance which is calculated by means of transport plans \cite{5}.<br />
<br />
The main aim of our research project is to reveal the structure of the<br />
isometry group $\mathrm{Isom}(\mathcal{W}_p(X))$. Although<br />
$\mathrm{Isom}(X)$ embeds naturally into<br />
$\mathrm{Isom}(\mathcal{W}_p(X))$ by push-forward, and this embedding<br />
turned out to be surjective in many cases (see e.g. [1]), these two<br />
groups are not isomorphic in general. Kloeckner in [2] described<br />
the isometry group of the quadratic Wasserstein space<br />
$\mathcal{W}_2(\mathbb{R}^n)$, and it turned out that the case of $n=1$<br />
is special in the sense that $\mathrm{Isom}(\mathcal{W}_2(\mathbb{R})$<br />
is extremely rich. Namely, it contains a large subgroup of wild behaving<br />
isometries that distort the shape of measures. Following this line of<br />
investigation, in \cite{3} we described<br />
$\mathrm{Isom}(\mathcal{W}_p(\mathbb{R}))$ and<br />
$\mathrm{Isom}(\mathcal{W}_p([0,1])$ for all $p\geq 1$.<br />
<br />
In this talk I will survey first some of the earlier results in the<br />
subject, and then I will present the key results of [3]. If time<br />
permits, I will also report on our most recent manuscript [4] in<br />
which we extended Kloeckner's multidimensional results. Joint work with Gy\"orgy P\'al Geh\'er (University of Reading)<br />
and D\'aniel Virosztek (IST Austria).<br />
<br />
[1] J. Bertrand and B. Kloeckner, \emph{A geometric study of Wasserstein<br />
spaces: isometric rigidity in negative curvature}, International<br />
Mathematics Research Notices, 2016 (5), 1368--1386.<br />
<br />
[2] B. Kloeckner, \emph{A geometric study of Wasserstein spaces: Euclidean<br />
spaces}, Annali della Scuola Normale Superiore di Pisa - Classe di<br />
Scienze, Serie 5, Tome 9 (2010) no. 2, 297--323.<br />
<br />
[3] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{Isometric study of<br />
Wasserstein spaces – the real line}, Trans. Amer. Math. Soc., 373<br />
(2020), 5855--5883.<br />
<br />
[4] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{The isometry group of<br />
Wasserstein spaces: The Hilbertian case}, submitted manuscript.<br />
<br />
[5] C. Villani, \emph{Optimal Transport: Old and New,}<br />
(Grundlehren der mathematischen Wissenschaften)<br />
Springer, 2009.<br />
<br />
===Shukun Wu===<br />
<br />
Title: On the Bochner-Riesz operator and the maximal Bochner-Riesz operator<br />
<br />
Abstract: The Bochner-Riesz problem is one of the most important problems in the field of Fourier analysis. It has a strong connection to other famous problems, such as the restriction conjecture and the Kakeya conjecture. In this talk, I will present some recent improvements to the Bochner-Riesz conjecture and the maximal Bochner-Riesz conjecture. The main methods we used are polynomial partitioning and the Bourgain Demeter l^2 decoupling theorem. <br />
<br />
<br />
===Jonathan Hickman===<br />
<br />
Title: Sobolev improving for averages over space curves<br />
<br />
Abstract: Consider the averaging operator given by convolution with arclength measure on compact piece of a smooth curve in R^n. A simple question is to precisely quantify the gain in regularity induced by this averaging, for instance by studying the L^p-Sobolev mapping properties of the operator. This talk will report on ongoing developments towards understanding this problem. In particular, we will explore some non-trivial necessary conditions on the gain in regularity. Joint with D. Beltran, S. Guo and A. Seeger.<br />
<br />
===Hanlong Fang===<br />
<br />
Title: Canonical blow-ups of Grassmann manifolds<br />
<br />
Abstract: We introduce certain canonical blow-ups \mathcal T_{s,p,n}, as well as their distinct submanifolds \mathcal M_{s,p,n}, of Grassmann manifolds G(p,n) by partitioning the Plücker coordinates with respect to a parameter s. Various geometric aspects of \mathcal T_{s,p,n} and \mathcal M_{s,p,n} are studied, for instance, the smoothness, the holomorphic symmetries, the (semi-)positivity of the anti-canonical bundles, the existence of Kähler-Einstein metrics, the functoriality, etc. In particular, we introduce the notion of homeward compactification, of which \mathcal T_{s,p,n} are examples, as a generalization of the wonderful compactification. <br />
<br />
===Bingyang Hu===<br />
<br />
Title: Some structure theorems on general doubling measures.<br />
<br />
Abstract: In this talk, we will first several structure theorems about general doubling measures. Secondly, we will include some main idea to prove one of these results. More precisely, we will focus on the construction of an explicit family of measures that are p-adic doubling for any finite set of primes, however, not doubling. This part generalizes the work by Boylan, Mills and Ward in 2019 in a highly non-trivial way. As some application, we apply these results (that is, the same construction) to show analogous statements for Mukenhoupt Ap weights and reverse Holder weights. This is a joint walk with Tess Anderson.<br />
<br />
===Krystal Taylor===<br />
<br />
Title: Quantifications of the Besicovitch Projection theorem in a nonlinear setting <br />
<br />
Abstract: There are several classical results relating the geometry, dimension, and measure of a set to the structure of its orthogonal projections. <br />
It turns out that many nonlinear projection-type operators also have special geometry that allows us to build similar relationships between a set and its "projections", just as in the linear setting. We will discuss a series of recent results from both geometric and probabilistic vantage points. In particular, we will see that the multi-scale analysis techniques of Tao, as well as the energy techniques of Mattila, can be strengthened and generalized to projection-type operators satisfying a transversality condition. As an application, we address the Buffon curve problem, which is to find upper and lower bounds for the rate of decay of the Favard curve length of the four-corner Cantor set.<br />
<br />
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=Extras=<br />
[[Blank Analysis Seminar Template]]<br />
<br />
<br />
Graduate Student Seminar:<br />
<br />
https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html</div>Beltranhttps://hilbert.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=20801Analysis Seminar2021-02-08T22:27:41Z<p>Beltran: /* Abstracts */</p>
<hr />
<div><br />
The 2020-2021 Analysis Seminar will be organized by David Beltran and Andreas Seeger.<br />
It will be online for the entire academic year. The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar at different times, to accomodate speakers).<br />
<br />
Zoom links will be sent to those who have signed up for the Analysis Seminar List. For instructions how to sign up for seminar lists, see https://www.math.wisc.edu/node/230<br />
<br />
If you'd like to suggest speakers for the spring semester please contact David and Andreas (dbeltran at math, seeger at math).<br />
<br />
<br />
<br />
=[[Previous_Analysis_seminars]]=<br />
<br />
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars<br />
<br />
= Current Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 22<br />
|Alexei Poltoratski<br />
|UW Madison<br />
|[[#Alexei Poltoratski | Dirac inner functions ]]<br />
| <br />
|-<br />
|September 29<br />
|Joris Roos<br />
|University of Massachusetts - Lowell<br />
|[[#Polona Durcik and Joris Rooslinktoabstract | A triangular Hilbert transform with curvature, I ]]<br />
| <br />
|-<br />
|Wednesday September 30, 4 p.m.<br />
|Polona Durcik<br />
|Chapman University<br />
|[[#Polona Durcik and Joris Roos | A triangular Hilbert transform with curvature, II ]]<br />
| <br />
|-<br />
|October 6<br />
|Andrew Zimmer<br />
|UW Madison<br />
|[[#Andrew Zimmer | Complex analytic problems on domains with good intrinsic geometry ]]<br />
| <br />
|-<br />
|October 13<br />
|Hong Wang <br />
|Princeton/IAS<br />
|[[#Hong Wang | Improved decoupling for the parabola ]]<br />
| <br />
|-<br />
|October 20<br />
|Kevin Luli<br />
|UC Davis<br />
|[[#Kevin Luli | Smooth Nonnegative Interpolation ]]<br />
| <br />
|-<br />
|October 21, 4.00 p.m.<br />
|Niclas Technau<br />
|UW Madison<br />
|[[#Niclas Technau | Number theoretic applications of oscillatory integrals ]]<br />
| <br />
|-<br />
|October 27<br />
|Terence Harris<br />
| Cornell University<br />
|[[#Terence Harris | Low dimensional pinned distance sets via spherical averages ]]<br />
| <br />
|-<br />
|Monday, November 2, 4 p.m.<br />
|Yuval Wigderson<br />
|Stanford University<br />
|[[#Yuval Wigderson | New perspectives on the uncertainty principle ]]<br />
| <br />
|-<br />
|November 10, 10 a.m. <br />
|Óscar Domínguez<br />
| Universidad Complutense de Madrid<br />
|[[#Oscar Dominguez | New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions ]]<br />
| <br />
|-<br />
|November 17<br />
|Tamas Titkos<br />
|BBS U of Applied Sciences and Renyi Institute<br />
|[[#Tamas Titkos | Isometries of Wasserstein spaces ]]<br />
| <br />
|-<br />
|November 24<br />
|Shukun Wu<br />
|University of Illinois (Urbana-Champaign)<br />
||[[#Shukun Wu | On the Bochner-Riesz operator and the maximal Bochner-Riesz operator ]] <br />
| <br />
|-<br />
|December 1<br />
| Jonathan Hickman<br />
| The University of Edinburgh<br />
|[[#Jonathan Hickman | Sobolev improving for averages over space curves ]]<br />
| <br />
|-<br />
|February 2, 7:00 p.m.<br />
|Hanlong Fang<br />
|UW Madison<br />
|[[#Hanlong Fang | Canonical blow-ups of Grassmann manifolds ]]<br />
| <br />
|-<br />
|February 9<br />
|Bingyang Hu<br />
|Purdue University<br />
|[[#Bingyang Hu | Some structure theorems on general doubling measures ]]<br />
| <br />
|-<br />
|February 16<br />
|Krystal Taylor<br />
|The Ohio State University<br />
|[[#Krystal Taylor | Quantifications of the Besicovitch Projection theorem in a nonlinear setting ]]<br />
|<br />
|-<br />
|February 23<br />
|Dominique Maldague<br />
|MIT<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 2<br />
|Diogo Oliveira e Silva<br />
|University of Birmingham<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 9<br />
|Oleg Safronov <br />
|University of North Carolina Charlotte<br />
|[[#linktoabstract | Relations between discrete and continuous spectra of differential operators ]]<br />
|<br />
|-<br />
|March 16<br />
|Ziming Shi<br />
|Rutgers University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 23<br />
|Xiumin Du<br />
|Northwestern University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 30, 10:00 a.m.<br />
|Etienne Le Masson<br />
|Cergy Paris University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 6<br />
|Theresa Anderson <br />
|Purdue University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 13<br />
|Nathan Wagner<br />
|Washington University St. Louis<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 20<br />
|Jongchon Kim<br />
| University of British Columbia<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 27<br />
|Yumeng Ou<br />
|University of Pennsylvania<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|May 4<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|}<br />
<br />
=Abstracts=<br />
===Alexei Poltoratski===<br />
<br />
Title: Dirac inner functions<br />
<br />
Abstract: My talk will focus on some new (and old) complex analytic objects arising from Dirac systems of differential equations.<br />
We will discuss connections between problems in complex function theory, spectral and scattering problems for differential<br />
operators and the non-linear Fourier transform.<br />
<br />
===Polona Durcik and Joris Roos===<br />
<br />
Title: A triangular Hilbert transform with curvature, I & II.<br />
<br />
Abstract: The triangular Hilbert is a two-dimensional bilinear singular<br />
originating in time-frequency analysis. No Lp bounds are currently<br />
known for this operator.<br />
In these two talks we discuss a recent joint work with Michael Christ<br />
on a variant of the triangular Hilbert transform involving curvature.<br />
This object is closely related to the bilinear Hilbert transform with<br />
curvature and a maximally modulated singular integral of Stein-Wainger<br />
type. As an application we also discuss a quantitative nonlinear Roth<br />
type theorem on patterns in the Euclidean plane.<br />
The second talk will focus on the proof of a key ingredient, a certain<br />
regularity estimate for a local operator.<br />
<br />
===Andrew Zimmer===<br />
<br />
Title: Complex analytic problems on domains with good intrinsic geometry<br />
<br />
Abstract: In this talk, I will describe a new class of domains in complex Euclidean space which is defined in terms of the existence of a Kaehler metric with good geometric properties. This class is invariant under biholomorphism and includes many well-studied classes of domains such as strongly pseudoconvex domains, finite type domains in dimension two, convex domains, homogeneous domains, and embeddings of Teichmuller spaces. Further, certain analytic problems are tractable for domains in this family even when the boundary is non-smooth. In particular, it is possible to characterize the domains in this family where the dbar-Neumann operator on (0, q)-forms is compact (which generalizes an old result of Fu-Straube for convex domains).<br />
<br />
===Hong Wang===<br />
<br />
Title: Improved decoupling for the parabola<br />
<br />
Abstract: In 2014, Bourgain and Demeter proved the $l^2$ decoupling estimates for the paraboloid with constant $R^{\epsilon}$. <br />
We prove an $(l^2, L^6)$ decoupling inequality for the parabola with constant $(\log R)^c$. This is joint work with Larry Guth and Dominique Maldague.<br />
<br />
===Kevin Luli===<br />
<br />
Title: Smooth Nonnegative Interpolation<br />
<br />
Abstract: Suppose E is an arbitrary subset of R^n. Let f: E \rightarrow [0, \infty). How can we decide if f extends to a nonnegative function C^m function F defined on all of R^n? Suppose E is finite. Can we compute a nonnegative C^m function F on R^n that agrees with f on E with the least possible C^m norm? How many computer operations does this take? In this talk, I will explain recent results on these problems. Non-negativity is one of the most important shape preserving properties for interpolants. In real life applications, the range of the interpolant is imposed by nature. For example, probability density, the amount of snow, rain, humidity, chemical concentration are all nonnegative quantities and are of interest in natural sciences. Even in one dimension, the existing techniques can only handle nonnegative interpolation under special assumptions on the data set. Our results work without any assumptions on the data sets. <br />
<br />
===Niclas Technau===<br />
<br />
Title: Number theoretic applications of oscillatory integrals<br />
<br />
Abstract: We discuss how the analysis of oscillatory integrals can be used to solve number theoretic problems. More specifically, the focus will be on understanding fine-scale statistics of sequences on the unit circle. Further, we shall briefly explain a connection to quantum chaos.<br />
<br />
===Terence Harris===<br />
<br />
Title: Low dimensional pinned distance sets via spherical averages<br />
<br />
Abstract: An inequality is derived for the average t-energy of weighted pinned distance measures, where 0 < t < 1, in terms of the L^2 spherical averages of Fourier transforms of measures. This generalises the result of Liu (originally for Lebesgue measure) to pinned distance sets of dimension smaller than 1, and strengthens Mattila's result from 1987, originally for the full distance set.<br />
<br />
===Yuval Wigderson===<br />
<br />
Title: New perspectives on the uncertainty principle<br />
<br />
Abstract: The phrase ``uncertainty principle'' refers to a wide array of results in several disparate fields of mathematics, all of which capture the notion that a function and its Fourier transform cannot both be ``very localized''. The measure of localization varies from one uncertainty principle to the next, and well-studied notions include the variance (and higher moments), the entropy, the support-size, and the rate of decay at infinity. Similarly, the proofs of the various uncertainty principles rely on a range of tools, from the elementary to the very deep. In this talk, I'll describe how many of the uncertainty principles all follow from a single, simple result, whose proof uses only a basic property of the Fourier transform: that it and its inverse are bounded as operators $L^1 \to L^\infty$. Using this result, one can also prove new variants of the uncertainty principle, which apply to new measures of localization and to operators other than the Fourier transform. This is joint work with Avi Wigderson.<br />
<br />
===Oscar Dominguez===<br />
<br />
Title: New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions<br />
<br />
Abstract: The celebrated Bourgain--Brezis--Mironescu formula enables us to recover Sobolev spaces in terms of limits of Gagliardo seminorms. Very recently, Brezis, Van Schaftingen and Yung have proposed an alternative methodology to approach Sobolev spaces via limits of weak-type Gagliardo functionals. The goal of this talk is twofold. Firstly, we will show that the BvSY result is a special case of a more general phenomenon based on maximal inequalities. In particular, we shall derive not only analogs of the BvSY theorem for different kinds of function spaces (Lebesgue, Calderon, higher-order Sobolev, …), but also applications to ergodic theory, Fourier series, etc. In the second part of the talk, we shall investigate the fractional setting in the BvSY theorem. Our approach is based on new Garsia-type inequalities and an application of the Caffarelli--Silvestre extension. This is joint work with Mario Milman.<br />
<br />
===Tamas Titkos===<br />
<br />
Title: Isometries of Wasserstein spaces<br />
<br />
Abstract: Due to its nice theoretical properties and an astonishing number of<br />
applications via optimal transport problems, probably the most<br />
intensively studied metric nowadays is the p-Wasserstein metric. Given<br />
a complete and separable metric space $X$ and a real number $p\geq1$,<br />
one defines the p-Wasserstein space $\mathcal{W}_p(X)$ as the collection<br />
of Borel probability measures with finite $p$-th moment, endowed with a<br />
distance which is calculated by means of transport plans \cite{5}.<br />
<br />
The main aim of our research project is to reveal the structure of the<br />
isometry group $\mathrm{Isom}(\mathcal{W}_p(X))$. Although<br />
$\mathrm{Isom}(X)$ embeds naturally into<br />
$\mathrm{Isom}(\mathcal{W}_p(X))$ by push-forward, and this embedding<br />
turned out to be surjective in many cases (see e.g. [1]), these two<br />
groups are not isomorphic in general. Kloeckner in [2] described<br />
the isometry group of the quadratic Wasserstein space<br />
$\mathcal{W}_2(\mathbb{R}^n)$, and it turned out that the case of $n=1$<br />
is special in the sense that $\mathrm{Isom}(\mathcal{W}_2(\mathbb{R})$<br />
is extremely rich. Namely, it contains a large subgroup of wild behaving<br />
isometries that distort the shape of measures. Following this line of<br />
investigation, in \cite{3} we described<br />
$\mathrm{Isom}(\mathcal{W}_p(\mathbb{R}))$ and<br />
$\mathrm{Isom}(\mathcal{W}_p([0,1])$ for all $p\geq 1$.<br />
<br />
In this talk I will survey first some of the earlier results in the<br />
subject, and then I will present the key results of [3]. If time<br />
permits, I will also report on our most recent manuscript [4] in<br />
which we extended Kloeckner's multidimensional results. Joint work with Gy\"orgy P\'al Geh\'er (University of Reading)<br />
and D\'aniel Virosztek (IST Austria).<br />
<br />
[1] J. Bertrand and B. Kloeckner, \emph{A geometric study of Wasserstein<br />
spaces: isometric rigidity in negative curvature}, International<br />
Mathematics Research Notices, 2016 (5), 1368--1386.<br />
<br />
[2] B. Kloeckner, \emph{A geometric study of Wasserstein spaces: Euclidean<br />
spaces}, Annali della Scuola Normale Superiore di Pisa - Classe di<br />
Scienze, Serie 5, Tome 9 (2010) no. 2, 297--323.<br />
<br />
[3] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{Isometric study of<br />
Wasserstein spaces – the real line}, Trans. Amer. Math. Soc., 373<br />
(2020), 5855--5883.<br />
<br />
[4] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{The isometry group of<br />
Wasserstein spaces: The Hilbertian case}, submitted manuscript.<br />
<br />
[5] C. Villani, \emph{Optimal Transport: Old and New,}<br />
(Grundlehren der mathematischen Wissenschaften)<br />
Springer, 2009.<br />
<br />
===Shukun Wu===<br />
<br />
Title: On the Bochner-Riesz operator and the maximal Bochner-Riesz operator<br />
<br />
Abstract: The Bochner-Riesz problem is one of the most important problems in the field of Fourier analysis. It has a strong connection to other famous problems, such as the restriction conjecture and the Kakeya conjecture. In this talk, I will present some recent improvements to the Bochner-Riesz conjecture and the maximal Bochner-Riesz conjecture. The main methods we used are polynomial partitioning and the Bourgain Demeter l^2 decoupling theorem. <br />
<br />
<br />
===Jonathan Hickman===<br />
<br />
Title: Sobolev improving for averages over space curves<br />
<br />
Abstract: Consider the averaging operator given by convolution with arclength measure on compact piece of a smooth curve in R^n. A simple question is to precisely quantify the gain in regularity induced by this averaging, for instance by studying the L^p-Sobolev mapping properties of the operator. This talk will report on ongoing developments towards understanding this problem. In particular, we will explore some non-trivial necessary conditions on the gain in regularity. Joint with D. Beltran, S. Guo and A. Seeger.<br />
<br />
===Hanlong Fang===<br />
<br />
Title: Canonical blow-ups of Grassmann manifolds<br />
<br />
Abstract: We introduce certain canonical blow-ups \mathcal T_{s,p,n}, as well as their distinct submanifolds \mathcal M_{s,p,n}, of Grassmann manifolds G(p,n) by partitioning the Plücker coordinates with respect to a parameter s. Various geometric aspects of \mathcal T_{s,p,n} and \mathcal M_{s,p,n} are studied, for instance, the smoothness, the holomorphic symmetries, the (semi-)positivity of the anti-canonical bundles, the existence of Kähler-Einstein metrics, the functoriality, etc. In particular, we introduce the notion of homeward compactification, of which \mathcal T_{s,p,n} are examples, as a generalization of the wonderful compactification. <br />
<br />
===Bingyang Hu===<br />
<br />
Title: Some structure theorems on general doubling measures.<br />
<br />
Abstract: In this talk, we will first several structure theorems about general doubling measures. Secondly, we will include some main idea to prove one of these results. More precisely, we will focus on the construction of an explicit family of measures that are p-adic doubling for any finite set of primes, however, not doubling. This part generalizes the work by Boylan, Mills and Ward in 2019 in a highly non-trivial way. As some application, we apply these results (that is, the same construction) to show analogous statements for Mukenhoupt Ap weights and reverse Holder weights. This is a joint walk with Tess Anderson.<br />
<br />
===Krystal Taylor===<br />
<br />
Title: Quantifications of the Besicovitch Projection theorem in a nonlinear setting <br />
<br />
Abstract: There are several classical results relating the geometry, dimension, and measure of a set to the structure of its orthogonal projections. <br />
It turns out that many nonlinear projection-type operators also have special geometry that allows us to build similar relationships between a set and its ``projections'', just as in the linear setting. We will discuss a series of recent results from both geometric and probabilistic vantage points. In particular, we will see that the multi-scale analysis techniques of Tao, as well as the energy techniques of Mattila, can be strengthened and generalized to projection-type operators satisfying a transversality condition. As an application, we address the Buffon curve problem, which is to find upper and lower bounds for the rate of decay of the Favard curve length of the four-corner Cantor set.<br />
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=Extras=<br />
[[Blank Analysis Seminar Template]]<br />
<br />
<br />
Graduate Student Seminar:<br />
<br />
https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html</div>Beltranhttps://hilbert.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=20800Analysis Seminar2021-02-08T22:26:43Z<p>Beltran: /* Current Analysis Seminar Schedule */</p>
<hr />
<div><br />
The 2020-2021 Analysis Seminar will be organized by David Beltran and Andreas Seeger.<br />
It will be online for the entire academic year. The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar at different times, to accomodate speakers).<br />
<br />
Zoom links will be sent to those who have signed up for the Analysis Seminar List. For instructions how to sign up for seminar lists, see https://www.math.wisc.edu/node/230<br />
<br />
If you'd like to suggest speakers for the spring semester please contact David and Andreas (dbeltran at math, seeger at math).<br />
<br />
<br />
<br />
=[[Previous_Analysis_seminars]]=<br />
<br />
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars<br />
<br />
= Current Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 22<br />
|Alexei Poltoratski<br />
|UW Madison<br />
|[[#Alexei Poltoratski | Dirac inner functions ]]<br />
| <br />
|-<br />
|September 29<br />
|Joris Roos<br />
|University of Massachusetts - Lowell<br />
|[[#Polona Durcik and Joris Rooslinktoabstract | A triangular Hilbert transform with curvature, I ]]<br />
| <br />
|-<br />
|Wednesday September 30, 4 p.m.<br />
|Polona Durcik<br />
|Chapman University<br />
|[[#Polona Durcik and Joris Roos | A triangular Hilbert transform with curvature, II ]]<br />
| <br />
|-<br />
|October 6<br />
|Andrew Zimmer<br />
|UW Madison<br />
|[[#Andrew Zimmer | Complex analytic problems on domains with good intrinsic geometry ]]<br />
| <br />
|-<br />
|October 13<br />
|Hong Wang <br />
|Princeton/IAS<br />
|[[#Hong Wang | Improved decoupling for the parabola ]]<br />
| <br />
|-<br />
|October 20<br />
|Kevin Luli<br />
|UC Davis<br />
|[[#Kevin Luli | Smooth Nonnegative Interpolation ]]<br />
| <br />
|-<br />
|October 21, 4.00 p.m.<br />
|Niclas Technau<br />
|UW Madison<br />
|[[#Niclas Technau | Number theoretic applications of oscillatory integrals ]]<br />
| <br />
|-<br />
|October 27<br />
|Terence Harris<br />
| Cornell University<br />
|[[#Terence Harris | Low dimensional pinned distance sets via spherical averages ]]<br />
| <br />
|-<br />
|Monday, November 2, 4 p.m.<br />
|Yuval Wigderson<br />
|Stanford University<br />
|[[#Yuval Wigderson | New perspectives on the uncertainty principle ]]<br />
| <br />
|-<br />
|November 10, 10 a.m. <br />
|Óscar Domínguez<br />
| Universidad Complutense de Madrid<br />
|[[#Oscar Dominguez | New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions ]]<br />
| <br />
|-<br />
|November 17<br />
|Tamas Titkos<br />
|BBS U of Applied Sciences and Renyi Institute<br />
|[[#Tamas Titkos | Isometries of Wasserstein spaces ]]<br />
| <br />
|-<br />
|November 24<br />
|Shukun Wu<br />
|University of Illinois (Urbana-Champaign)<br />
||[[#Shukun Wu | On the Bochner-Riesz operator and the maximal Bochner-Riesz operator ]] <br />
| <br />
|-<br />
|December 1<br />
| Jonathan Hickman<br />
| The University of Edinburgh<br />
|[[#Jonathan Hickman | Sobolev improving for averages over space curves ]]<br />
| <br />
|-<br />
|February 2, 7:00 p.m.<br />
|Hanlong Fang<br />
|UW Madison<br />
|[[#Hanlong Fang | Canonical blow-ups of Grassmann manifolds ]]<br />
| <br />
|-<br />
|February 9<br />
|Bingyang Hu<br />
|Purdue University<br />
|[[#Bingyang Hu | Some structure theorems on general doubling measures ]]<br />
| <br />
|-<br />
|February 16<br />
|Krystal Taylor<br />
|The Ohio State University<br />
|[[#Krystal Taylor | Quantifications of the Besicovitch Projection theorem in a nonlinear setting ]]<br />
|<br />
|-<br />
|February 23<br />
|Dominique Maldague<br />
|MIT<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 2<br />
|Diogo Oliveira e Silva<br />
|University of Birmingham<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 9<br />
|Oleg Safronov <br />
|University of North Carolina Charlotte<br />
|[[#linktoabstract | Relations between discrete and continuous spectra of differential operators ]]<br />
|<br />
|-<br />
|March 16<br />
|Ziming Shi<br />
|Rutgers University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 23<br />
|Xiumin Du<br />
|Northwestern University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 30, 10:00 a.m.<br />
|Etienne Le Masson<br />
|Cergy Paris University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 6<br />
|Theresa Anderson <br />
|Purdue University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 13<br />
|Nathan Wagner<br />
|Washington University St. Louis<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 20<br />
|Jongchon Kim<br />
| University of British Columbia<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 27<br />
|Yumeng Ou<br />
|University of Pennsylvania<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|May 4<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|}<br />
<br />
=Abstracts=<br />
===Alexei Poltoratski===<br />
<br />
Title: Dirac inner functions<br />
<br />
Abstract: My talk will focus on some new (and old) complex analytic objects arising from Dirac systems of differential equations.<br />
We will discuss connections between problems in complex function theory, spectral and scattering problems for differential<br />
operators and the non-linear Fourier transform.<br />
<br />
===Polona Durcik and Joris Roos===<br />
<br />
Title: A triangular Hilbert transform with curvature, I & II.<br />
<br />
Abstract: The triangular Hilbert is a two-dimensional bilinear singular<br />
originating in time-frequency analysis. No Lp bounds are currently<br />
known for this operator.<br />
In these two talks we discuss a recent joint work with Michael Christ<br />
on a variant of the triangular Hilbert transform involving curvature.<br />
This object is closely related to the bilinear Hilbert transform with<br />
curvature and a maximally modulated singular integral of Stein-Wainger<br />
type. As an application we also discuss a quantitative nonlinear Roth<br />
type theorem on patterns in the Euclidean plane.<br />
The second talk will focus on the proof of a key ingredient, a certain<br />
regularity estimate for a local operator.<br />
<br />
===Andrew Zimmer===<br />
<br />
Title: Complex analytic problems on domains with good intrinsic geometry<br />
<br />
Abstract: In this talk, I will describe a new class of domains in complex Euclidean space which is defined in terms of the existence of a Kaehler metric with good geometric properties. This class is invariant under biholomorphism and includes many well-studied classes of domains such as strongly pseudoconvex domains, finite type domains in dimension two, convex domains, homogeneous domains, and embeddings of Teichmuller spaces. Further, certain analytic problems are tractable for domains in this family even when the boundary is non-smooth. In particular, it is possible to characterize the domains in this family where the dbar-Neumann operator on (0, q)-forms is compact (which generalizes an old result of Fu-Straube for convex domains).<br />
<br />
===Hong Wang===<br />
<br />
Title: Improved decoupling for the parabola<br />
<br />
Abstract: In 2014, Bourgain and Demeter proved the $l^2$ decoupling estimates for the paraboloid with constant $R^{\epsilon}$. <br />
We prove an $(l^2, L^6)$ decoupling inequality for the parabola with constant $(\log R)^c$. This is joint work with Larry Guth and Dominique Maldague.<br />
<br />
===Kevin Luli===<br />
<br />
Title: Smooth Nonnegative Interpolation<br />
<br />
Abstract: Suppose E is an arbitrary subset of R^n. Let f: E \rightarrow [0, \infty). How can we decide if f extends to a nonnegative function C^m function F defined on all of R^n? Suppose E is finite. Can we compute a nonnegative C^m function F on R^n that agrees with f on E with the least possible C^m norm? How many computer operations does this take? In this talk, I will explain recent results on these problems. Non-negativity is one of the most important shape preserving properties for interpolants. In real life applications, the range of the interpolant is imposed by nature. For example, probability density, the amount of snow, rain, humidity, chemical concentration are all nonnegative quantities and are of interest in natural sciences. Even in one dimension, the existing techniques can only handle nonnegative interpolation under special assumptions on the data set. Our results work without any assumptions on the data sets. <br />
<br />
===Niclas Technau===<br />
<br />
Title: Number theoretic applications of oscillatory integrals<br />
<br />
Abstract: We discuss how the analysis of oscillatory integrals can be used to solve number theoretic problems. More specifically, the focus will be on understanding fine-scale statistics of sequences on the unit circle. Further, we shall briefly explain a connection to quantum chaos.<br />
<br />
===Terence Harris===<br />
<br />
Title: Low dimensional pinned distance sets via spherical averages<br />
<br />
Abstract: An inequality is derived for the average t-energy of weighted pinned distance measures, where 0 < t < 1, in terms of the L^2 spherical averages of Fourier transforms of measures. This generalises the result of Liu (originally for Lebesgue measure) to pinned distance sets of dimension smaller than 1, and strengthens Mattila's result from 1987, originally for the full distance set.<br />
<br />
===Yuval Wigderson===<br />
<br />
Title: New perspectives on the uncertainty principle<br />
<br />
Abstract: The phrase ``uncertainty principle'' refers to a wide array of results in several disparate fields of mathematics, all of which capture the notion that a function and its Fourier transform cannot both be ``very localized''. The measure of localization varies from one uncertainty principle to the next, and well-studied notions include the variance (and higher moments), the entropy, the support-size, and the rate of decay at infinity. Similarly, the proofs of the various uncertainty principles rely on a range of tools, from the elementary to the very deep. In this talk, I'll describe how many of the uncertainty principles all follow from a single, simple result, whose proof uses only a basic property of the Fourier transform: that it and its inverse are bounded as operators $L^1 \to L^\infty$. Using this result, one can also prove new variants of the uncertainty principle, which apply to new measures of localization and to operators other than the Fourier transform. This is joint work with Avi Wigderson.<br />
<br />
===Oscar Dominguez===<br />
<br />
Title: New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions<br />
<br />
Abstract: The celebrated Bourgain--Brezis--Mironescu formula enables us to recover Sobolev spaces in terms of limits of Gagliardo seminorms. Very recently, Brezis, Van Schaftingen and Yung have proposed an alternative methodology to approach Sobolev spaces via limits of weak-type Gagliardo functionals. The goal of this talk is twofold. Firstly, we will show that the BvSY result is a special case of a more general phenomenon based on maximal inequalities. In particular, we shall derive not only analogs of the BvSY theorem for different kinds of function spaces (Lebesgue, Calderon, higher-order Sobolev, …), but also applications to ergodic theory, Fourier series, etc. In the second part of the talk, we shall investigate the fractional setting in the BvSY theorem. Our approach is based on new Garsia-type inequalities and an application of the Caffarelli--Silvestre extension. This is joint work with Mario Milman.<br />
<br />
===Tamas Titkos===<br />
<br />
Title: Isometries of Wasserstein spaces<br />
<br />
Abstract: Due to its nice theoretical properties and an astonishing number of<br />
applications via optimal transport problems, probably the most<br />
intensively studied metric nowadays is the p-Wasserstein metric. Given<br />
a complete and separable metric space $X$ and a real number $p\geq1$,<br />
one defines the p-Wasserstein space $\mathcal{W}_p(X)$ as the collection<br />
of Borel probability measures with finite $p$-th moment, endowed with a<br />
distance which is calculated by means of transport plans \cite{5}.<br />
<br />
The main aim of our research project is to reveal the structure of the<br />
isometry group $\mathrm{Isom}(\mathcal{W}_p(X))$. Although<br />
$\mathrm{Isom}(X)$ embeds naturally into<br />
$\mathrm{Isom}(\mathcal{W}_p(X))$ by push-forward, and this embedding<br />
turned out to be surjective in many cases (see e.g. [1]), these two<br />
groups are not isomorphic in general. Kloeckner in [2] described<br />
the isometry group of the quadratic Wasserstein space<br />
$\mathcal{W}_2(\mathbb{R}^n)$, and it turned out that the case of $n=1$<br />
is special in the sense that $\mathrm{Isom}(\mathcal{W}_2(\mathbb{R})$<br />
is extremely rich. Namely, it contains a large subgroup of wild behaving<br />
isometries that distort the shape of measures. Following this line of<br />
investigation, in \cite{3} we described<br />
$\mathrm{Isom}(\mathcal{W}_p(\mathbb{R}))$ and<br />
$\mathrm{Isom}(\mathcal{W}_p([0,1])$ for all $p\geq 1$.<br />
<br />
In this talk I will survey first some of the earlier results in the<br />
subject, and then I will present the key results of [3]. If time<br />
permits, I will also report on our most recent manuscript [4] in<br />
which we extended Kloeckner's multidimensional results. Joint work with Gy\"orgy P\'al Geh\'er (University of Reading)<br />
and D\'aniel Virosztek (IST Austria).<br />
<br />
[1] J. Bertrand and B. Kloeckner, \emph{A geometric study of Wasserstein<br />
spaces: isometric rigidity in negative curvature}, International<br />
Mathematics Research Notices, 2016 (5), 1368--1386.<br />
<br />
[2] B. Kloeckner, \emph{A geometric study of Wasserstein spaces: Euclidean<br />
spaces}, Annali della Scuola Normale Superiore di Pisa - Classe di<br />
Scienze, Serie 5, Tome 9 (2010) no. 2, 297--323.<br />
<br />
[3] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{Isometric study of<br />
Wasserstein spaces – the real line}, Trans. Amer. Math. Soc., 373<br />
(2020), 5855--5883.<br />
<br />
[4] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{The isometry group of<br />
Wasserstein spaces: The Hilbertian case}, submitted manuscript.<br />
<br />
[5] C. Villani, \emph{Optimal Transport: Old and New,}<br />
(Grundlehren der mathematischen Wissenschaften)<br />
Springer, 2009.<br />
<br />
===Shukun Wu===<br />
<br />
Title: On the Bochner-Riesz operator and the maximal Bochner-Riesz operator<br />
<br />
Abstract: The Bochner-Riesz problem is one of the most important problems in the field of Fourier analysis. It has a strong connection to other famous problems, such as the restriction conjecture and the Kakeya conjecture. In this talk, I will present some recent improvements to the Bochner-Riesz conjecture and the maximal Bochner-Riesz conjecture. The main methods we used are polynomial partitioning and the Bourgain Demeter l^2 decoupling theorem. <br />
<br />
<br />
===Jonathan Hickman===<br />
<br />
Title: Sobolev improving for averages over space curves<br />
<br />
Abstract: Consider the averaging operator given by convolution with arclength measure on compact piece of a smooth curve in R^n. A simple question is to precisely quantify the gain in regularity induced by this averaging, for instance by studying the L^p-Sobolev mapping properties of the operator. This talk will report on ongoing developments towards understanding this problem. In particular, we will explore some non-trivial necessary conditions on the gain in regularity. Joint with D. Beltran, S. Guo and A. Seeger.<br />
<br />
===Hanlong Fang===<br />
<br />
Title: Canonical blow-ups of Grassmann manifolds<br />
<br />
Abstract: We introduce certain canonical blow-ups \mathcal T_{s,p,n}, as well as their distinct submanifolds \mathcal M_{s,p,n}, of Grassmann manifolds G(p,n) by partitioning the Plücker coordinates with respect to a parameter s. Various geometric aspects of \mathcal T_{s,p,n} and \mathcal M_{s,p,n} are studied, for instance, the smoothness, the holomorphic symmetries, the (semi-)positivity of the anti-canonical bundles, the existence of Kähler-Einstein metrics, the functoriality, etc. In particular, we introduce the notion of homeward compactification, of which \mathcal T_{s,p,n} are examples, as a generalization of the wonderful compactification. <br />
<br />
===Bingyang Hu===<br />
<br />
Title: Some structure theorems on general doubling measures.<br />
<br />
Abstract: In this talk, we will first several structure theorems about general doubling measures. Secondly, we will include some main idea to prove one of these results. More precisely, we will focus on the construction of an explicit family of measures that are p-adic doubling for any finite set of primes, however, not doubling. This part generalizes the work by Boylan, Mills and Ward in 2019 in a highly non-trivial way. As some application, we apply these results (that is, the same construction) to show analogous statements for Mukenhoupt Ap weights and reverse Holder weights. This is a joint walk with Tess Anderson.<br />
<br />
===Name===<br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
===Name===<br />
<br />
Title:<br />
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Abstract:<br />
<br />
===Name===<br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
===Name===<br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
===Name===<br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
===Name===<br />
<br />
Title:<br />
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Abstract:<br />
<br />
===Name===<br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
===Name===<br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]<br />
<br />
<br />
Graduate Student Seminar:<br />
<br />
https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html</div>Beltranhttps://hilbert.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=20653Analysis Seminar2021-01-27T00:31:23Z<p>Beltran: /* Current Analysis Seminar Schedule */</p>
<hr />
<div><br />
The 2020-2021 Analysis Seminar will be organized by David Beltran and Andreas Seeger.<br />
It will be online for the entire academic year. The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar at different times, to accomodate speakers).<br />
<br />
Zoom links will be sent to those who have signed up for the Analysis Seminar List. For instructions how to sign up for seminar lists, see https://www.math.wisc.edu/node/230<br />
<br />
If you'd like to suggest speakers for the spring semester please contact David and Andreas (dbeltran at math, seeger at math).<br />
<br />
<br />
<br />
=[[Previous_Analysis_seminars]]=<br />
<br />
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars<br />
<br />
= Current Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 22<br />
|Alexei Poltoratski<br />
|UW Madison<br />
|[[#Alexei Poltoratski | Dirac inner functions ]]<br />
| <br />
|-<br />
|September 29<br />
|Joris Roos<br />
|University of Massachusetts - Lowell<br />
|[[#Polona Durcik and Joris Rooslinktoabstract | A triangular Hilbert transform with curvature, I ]]<br />
| <br />
|-<br />
|Wednesday September 30, 4 p.m.<br />
|Polona Durcik<br />
|Chapman University<br />
|[[#Polona Durcik and Joris Roos | A triangular Hilbert transform with curvature, II ]]<br />
| <br />
|-<br />
|October 6<br />
|Andrew Zimmer<br />
|UW Madison<br />
|[[#Andrew Zimmer | Complex analytic problems on domains with good intrinsic geometry ]]<br />
| <br />
|-<br />
|October 13<br />
|Hong Wang <br />
|Princeton/IAS<br />
|[[#Hong Wang | Improved decoupling for the parabola ]]<br />
| <br />
|-<br />
|October 20<br />
|Kevin Luli<br />
|UC Davis<br />
|[[#Kevin Luli | Smooth Nonnegative Interpolation ]]<br />
| <br />
|-<br />
|October 21, 4.00 p.m.<br />
|Niclas Technau<br />
|UW Madison<br />
|[[#Niclas Technau | Number theoretic applications of oscillatory integrals ]]<br />
| <br />
|-<br />
|October 27<br />
|Terence Harris<br />
| Cornell University<br />
|[[#Terence Harris | Low dimensional pinned distance sets via spherical averages ]]<br />
| <br />
|-<br />
|Monday, November 2, 4 p.m.<br />
|Yuval Wigderson<br />
|Stanford University<br />
|[[#Yuval Wigderson | New perspectives on the uncertainty principle ]]<br />
| <br />
|-<br />
|November 10, 10 a.m. <br />
|Óscar Domínguez<br />
| Universidad Complutense de Madrid<br />
|[[#Oscar Dominguez | New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions ]]<br />
| <br />
|-<br />
|November 17<br />
|Tamas Titkos<br />
|BBS U of Applied Sciences and Renyi Institute<br />
|[[#Tamas Titkos | Isometries of Wasserstein spaces ]]<br />
| <br />
|-<br />
|November 24<br />
|Shukun Wu<br />
|University of Illinois (Urbana-Champaign)<br />
||[[#Shukun Wu | On the Bochner-Riesz operator and the maximal Bochner-Riesz operator ]] <br />
| <br />
|-<br />
|December 1<br />
| Jonathan Hickman<br />
| The University of Edinburgh<br />
|[[#Jonathan Hickman | Sobolev improving for averages over space curves ]]<br />
| <br />
|-<br />
|February 2, 7:00 p.m.<br />
|Hanlong Fang<br />
|UW Madison<br />
|[[#Hanlong Fang | Canonical blow-ups of Grassmann manifolds ]]<br />
| <br />
|-<br />
|February 9<br />
|Bingyang Hu<br />
|Purdue University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|February 16<br />
|Krystal Taylor<br />
|The Ohio State University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|February 23<br />
|Dominique Maldague<br />
|MIT<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 2<br />
|Diogo Oliveira e Silva<br />
|University of Birmingham<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 9<br />
|Oleg Safronov <br />
|University of North Carolina Charlotte<br />
|[[#linktoabstract | Relations between discrete and continuous spectra of differential operators ]]<br />
|<br />
|-<br />
|March 16<br />
|Ziming Shi<br />
|Rutgers University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 23<br />
|Xiumin Du<br />
|Northwestern University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 30, 10:00 a.m.<br />
|Etienne Le Masson<br />
|Cergy Paris University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 6<br />
|TBA<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 13<br />
|TBA<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 20<br />
|Jongchon Kim<br />
| UBC<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 27<br />
|Yumeng Ou<br />
|University of Pennsylvania<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|May 4<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|}<br />
<br />
=Abstracts=<br />
===Alexei Poltoratski===<br />
<br />
Title: Dirac inner functions<br />
<br />
Abstract: My talk will focus on some new (and old) complex analytic objects arising from Dirac systems of differential equations.<br />
We will discuss connections between problems in complex function theory, spectral and scattering problems for differential<br />
operators and the non-linear Fourier transform.<br />
<br />
===Polona Durcik and Joris Roos===<br />
<br />
Title: A triangular Hilbert transform with curvature, I & II.<br />
<br />
Abstract: The triangular Hilbert is a two-dimensional bilinear singular<br />
originating in time-frequency analysis. No Lp bounds are currently<br />
known for this operator.<br />
In these two talks we discuss a recent joint work with Michael Christ<br />
on a variant of the triangular Hilbert transform involving curvature.<br />
This object is closely related to the bilinear Hilbert transform with<br />
curvature and a maximally modulated singular integral of Stein-Wainger<br />
type. As an application we also discuss a quantitative nonlinear Roth<br />
type theorem on patterns in the Euclidean plane.<br />
The second talk will focus on the proof of a key ingredient, a certain<br />
regularity estimate for a local operator.<br />
<br />
===Andrew Zimmer===<br />
<br />
Title: Complex analytic problems on domains with good intrinsic geometry<br />
<br />
Abstract: In this talk, I will describe a new class of domains in complex Euclidean space which is defined in terms of the existence of a Kaehler metric with good geometric properties. This class is invariant under biholomorphism and includes many well-studied classes of domains such as strongly pseudoconvex domains, finite type domains in dimension two, convex domains, homogeneous domains, and embeddings of Teichmuller spaces. Further, certain analytic problems are tractable for domains in this family even when the boundary is non-smooth. In particular, it is possible to characterize the domains in this family where the dbar-Neumann operator on (0, q)-forms is compact (which generalizes an old result of Fu-Straube for convex domains).<br />
<br />
===Hong Wang===<br />
<br />
Title: Improved decoupling for the parabola<br />
<br />
Abstract: In 2014, Bourgain and Demeter proved the $l^2$ decoupling estimates for the paraboloid with constant $R^{\epsilon}$. <br />
We prove an $(l^2, L^6)$ decoupling inequality for the parabola with constant $(\log R)^c$. This is joint work with Larry Guth and Dominique Maldague.<br />
<br />
===Kevin Luli===<br />
<br />
Title: Smooth Nonnegative Interpolation<br />
<br />
Abstract: Suppose E is an arbitrary subset of R^n. Let f: E \rightarrow [0, \infty). How can we decide if f extends to a nonnegative function C^m function F defined on all of R^n? Suppose E is finite. Can we compute a nonnegative C^m function F on R^n that agrees with f on E with the least possible C^m norm? How many computer operations does this take? In this talk, I will explain recent results on these problems. Non-negativity is one of the most important shape preserving properties for interpolants. In real life applications, the range of the interpolant is imposed by nature. For example, probability density, the amount of snow, rain, humidity, chemical concentration are all nonnegative quantities and are of interest in natural sciences. Even in one dimension, the existing techniques can only handle nonnegative interpolation under special assumptions on the data set. Our results work without any assumptions on the data sets. <br />
<br />
===Niclas Technau===<br />
<br />
Title: Number theoretic applications of oscillatory integrals<br />
<br />
Abstract: We discuss how the analysis of oscillatory integrals can be used to solve number theoretic problems. More specifically, the focus will be on understanding fine-scale statistics of sequences on the unit circle. Further, we shall briefly explain a connection to quantum chaos.<br />
<br />
===Terence Harris===<br />
<br />
Title: Low dimensional pinned distance sets via spherical averages<br />
<br />
Abstract: An inequality is derived for the average t-energy of weighted pinned distance measures, where 0 < t < 1, in terms of the L^2 spherical averages of Fourier transforms of measures. This generalises the result of Liu (originally for Lebesgue measure) to pinned distance sets of dimension smaller than 1, and strengthens Mattila's result from 1987, originally for the full distance set.<br />
<br />
===Yuval Wigderson===<br />
<br />
Title: New perspectives on the uncertainty principle<br />
<br />
Abstract: The phrase ``uncertainty principle'' refers to a wide array of results in several disparate fields of mathematics, all of which capture the notion that a function and its Fourier transform cannot both be ``very localized''. The measure of localization varies from one uncertainty principle to the next, and well-studied notions include the variance (and higher moments), the entropy, the support-size, and the rate of decay at infinity. Similarly, the proofs of the various uncertainty principles rely on a range of tools, from the elementary to the very deep. In this talk, I'll describe how many of the uncertainty principles all follow from a single, simple result, whose proof uses only a basic property of the Fourier transform: that it and its inverse are bounded as operators $L^1 \to L^\infty$. Using this result, one can also prove new variants of the uncertainty principle, which apply to new measures of localization and to operators other than the Fourier transform. This is joint work with Avi Wigderson.<br />
<br />
===Oscar Dominguez===<br />
<br />
Title: New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions<br />
<br />
Abstract: The celebrated Bourgain--Brezis--Mironescu formula enables us to recover Sobolev spaces in terms of limits of Gagliardo seminorms. Very recently, Brezis, Van Schaftingen and Yung have proposed an alternative methodology to approach Sobolev spaces via limits of weak-type Gagliardo functionals. The goal of this talk is twofold. Firstly, we will show that the BvSY result is a special case of a more general phenomenon based on maximal inequalities. In particular, we shall derive not only analogs of the BvSY theorem for different kinds of function spaces (Lebesgue, Calderon, higher-order Sobolev, …), but also applications to ergodic theory, Fourier series, etc. In the second part of the talk, we shall investigate the fractional setting in the BvSY theorem. Our approach is based on new Garsia-type inequalities and an application of the Caffarelli--Silvestre extension. This is joint work with Mario Milman.<br />
<br />
===Tamas Titkos===<br />
<br />
Title: Isometries of Wasserstein spaces<br />
<br />
Abstract: Due to its nice theoretical properties and an astonishing number of<br />
applications via optimal transport problems, probably the most<br />
intensively studied metric nowadays is the p-Wasserstein metric. Given<br />
a complete and separable metric space $X$ and a real number $p\geq1$,<br />
one defines the p-Wasserstein space $\mathcal{W}_p(X)$ as the collection<br />
of Borel probability measures with finite $p$-th moment, endowed with a<br />
distance which is calculated by means of transport plans \cite{5}.<br />
<br />
The main aim of our research project is to reveal the structure of the<br />
isometry group $\mathrm{Isom}(\mathcal{W}_p(X))$. Although<br />
$\mathrm{Isom}(X)$ embeds naturally into<br />
$\mathrm{Isom}(\mathcal{W}_p(X))$ by push-forward, and this embedding<br />
turned out to be surjective in many cases (see e.g. [1]), these two<br />
groups are not isomorphic in general. Kloeckner in [2] described<br />
the isometry group of the quadratic Wasserstein space<br />
$\mathcal{W}_2(\mathbb{R}^n)$, and it turned out that the case of $n=1$<br />
is special in the sense that $\mathrm{Isom}(\mathcal{W}_2(\mathbb{R})$<br />
is extremely rich. Namely, it contains a large subgroup of wild behaving<br />
isometries that distort the shape of measures. Following this line of<br />
investigation, in \cite{3} we described<br />
$\mathrm{Isom}(\mathcal{W}_p(\mathbb{R}))$ and<br />
$\mathrm{Isom}(\mathcal{W}_p([0,1])$ for all $p\geq 1$.<br />
<br />
In this talk I will survey first some of the earlier results in the<br />
subject, and then I will present the key results of [3]. If time<br />
permits, I will also report on our most recent manuscript [4] in<br />
which we extended Kloeckner's multidimensional results. Joint work with Gy\"orgy P\'al Geh\'er (University of Reading)<br />
and D\'aniel Virosztek (IST Austria).<br />
<br />
[1] J. Bertrand and B. Kloeckner, \emph{A geometric study of Wasserstein<br />
spaces: isometric rigidity in negative curvature}, International<br />
Mathematics Research Notices, 2016 (5), 1368--1386.<br />
<br />
[2] B. Kloeckner, \emph{A geometric study of Wasserstein spaces: Euclidean<br />
spaces}, Annali della Scuola Normale Superiore di Pisa - Classe di<br />
Scienze, Serie 5, Tome 9 (2010) no. 2, 297--323.<br />
<br />
[3] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{Isometric study of<br />
Wasserstein spaces – the real line}, Trans. Amer. Math. Soc., 373<br />
(2020), 5855--5883.<br />
<br />
[4] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{The isometry group of<br />
Wasserstein spaces: The Hilbertian case}, submitted manuscript.<br />
<br />
[5] C. Villani, \emph{Optimal Transport: Old and New,}<br />
(Grundlehren der mathematischen Wissenschaften)<br />
Springer, 2009.<br />
<br />
===Shukun Wu===<br />
<br />
Title: On the Bochner-Riesz operator and the maximal Bochner-Riesz operator<br />
<br />
Abstract: The Bochner-Riesz problem is one of the most important problems in the field of Fourier analysis. It has a strong connection to other famous problems, such as the restriction conjecture and the Kakeya conjecture. In this talk, I will present some recent improvements to the Bochner-Riesz conjecture and the maximal Bochner-Riesz conjecture. The main methods we used are polynomial partitioning and the Bourgain Demeter l^2 decoupling theorem. <br />
<br />
<br />
===Jonathan Hickman===<br />
<br />
Title: Sobolev improving for averages over space curves<br />
<br />
Abstract: Consider the averaging operator given by convolution with arclength measure on compact piece of a smooth curve in R^n. A simple question is to precisely quantify the gain in regularity induced by this averaging, for instance by studying the L^p-Sobolev mapping properties of the operator. This talk will report on ongoing developments towards understanding this problem. In particular, we will explore some non-trivial necessary conditions on the gain in regularity. Joint with D. Beltran, S. Guo and A. Seeger.<br />
<br />
===Hanlong Fang===<br />
<br />
Title: Canonical blow-ups of Grassmann manifolds<br />
<br />
Abstract: We introduce certain canonical blow-ups \mathcal T_{s,p,n}, as well as their distinct submanifolds \mathcal M_{s,p,n}, of Grassmann manifolds G(p,n) by partitioning the Plücker coordinates with respect to a parameter s. Various geometric aspects of \mathcal T_{s,p,n} and \mathcal M_{s,p,n} are studied, for instance, the smoothness, the holomorphic symmetries, the (semi-)positivity of the anti-canonical bundles, the existence of Kähler-Einstein metrics, the functoriality, etc. In particular, we introduce the notion of homeward compactification, of which \mathcal T_{s,p,n} are examples, as a generalization of the wonderful compactification. <br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract:<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract:<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract:<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]<br />
<br />
<br />
Graduate Student Seminar:<br />
<br />
https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html</div>Beltranhttps://hilbert.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=20649Analysis Seminar2021-01-26T17:47:39Z<p>Beltran: /* Current Analysis Seminar Schedule */</p>
<hr />
<div><br />
The 2020-2021 Analysis Seminar will be organized by David Beltran and Andreas Seeger.<br />
It will be online for the entire academic year. The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar at different times, to accomodate speakers).<br />
<br />
Zoom links will be sent to those who have signed up for the Analysis Seminar List. For instructions how to sign up for seminar lists, see https://www.math.wisc.edu/node/230<br />
<br />
If you'd like to suggest speakers for the spring semester please contact David and Andreas (dbeltran at math, seeger at math).<br />
<br />
<br />
<br />
=[[Previous_Analysis_seminars]]=<br />
<br />
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars<br />
<br />
= Current Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 22<br />
|Alexei Poltoratski<br />
|UW Madison<br />
|[[#Alexei Poltoratski | Dirac inner functions ]]<br />
| <br />
|-<br />
|September 29<br />
|Joris Roos<br />
|University of Massachusetts - Lowell<br />
|[[#Polona Durcik and Joris Rooslinktoabstract | A triangular Hilbert transform with curvature, I ]]<br />
| <br />
|-<br />
|Wednesday September 30, 4 p.m.<br />
|Polona Durcik<br />
|Chapman University<br />
|[[#Polona Durcik and Joris Roos | A triangular Hilbert transform with curvature, II ]]<br />
| <br />
|-<br />
|October 6<br />
|Andrew Zimmer<br />
|UW Madison<br />
|[[#Andrew Zimmer | Complex analytic problems on domains with good intrinsic geometry ]]<br />
| <br />
|-<br />
|October 13<br />
|Hong Wang <br />
|Princeton/IAS<br />
|[[#Hong Wang | Improved decoupling for the parabola ]]<br />
| <br />
|-<br />
|October 20<br />
|Kevin Luli<br />
|UC Davis<br />
|[[#Kevin Luli | Smooth Nonnegative Interpolation ]]<br />
| <br />
|-<br />
|October 21, 4.00 p.m.<br />
|Niclas Technau<br />
|UW Madison<br />
|[[#Niclas Technau | Number theoretic applications of oscillatory integrals ]]<br />
| <br />
|-<br />
|October 27<br />
|Terence Harris<br />
| Cornell University<br />
|[[#Terence Harris | Low dimensional pinned distance sets via spherical averages ]]<br />
| <br />
|-<br />
|Monday, November 2, 4 p.m.<br />
|Yuval Wigderson<br />
|Stanford University<br />
|[[#Yuval Wigderson | New perspectives on the uncertainty principle ]]<br />
| <br />
|-<br />
|November 10, 10 a.m. <br />
|Óscar Domínguez<br />
| Universidad Complutense de Madrid<br />
|[[#Oscar Dominguez | New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions ]]<br />
| <br />
|-<br />
|November 17<br />
|Tamas Titkos<br />
|BBS U of Applied Sciences and Renyi Institute<br />
|[[#Tamas Titkos | Isometries of Wasserstein spaces ]]<br />
| <br />
|-<br />
|November 24<br />
|Shukun Wu<br />
|University of Illinois (Urbana-Champaign)<br />
||[[#Shukun Wu | On the Bochner-Riesz operator and the maximal Bochner-Riesz operator ]] <br />
| <br />
|-<br />
|December 1<br />
| Jonathan Hickman<br />
| The University of Edinburgh<br />
|[[#Jonathan Hickman | Sobolev improving for averages over space curves ]]<br />
| <br />
|-<br />
|February 2, 7:00 p.m.<br />
|Hanlong Fang<br />
|UW Madison<br />
|[[#Hanlong Fang | Canonical blow-ups of Grassmann manifolds ]]<br />
| <br />
|-<br />
|February 9<br />
|Bingyang Hu<br />
|Purdue University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|February 16<br />
|Krystal Taylor<br />
|The Ohio State University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|February 23<br />
|Dominique Maldague<br />
|MIT<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 2<br />
|Diogo Oliveira e Silva<br />
|University of Birmingham<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 9<br />
|Oleg Safronov <br />
|University of North Carolina Charlotte<br />
|[[#linktoabstract | Relations between discrete and continuous spectra of differential operators ]]<br />
|<br />
|-<br />
|March 16<br />
|Ziming Shi<br />
|Rutgers University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 23<br />
|TBA<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 30, 10:00 a.m.<br />
|Etienne Le Masson<br />
|Cergy Paris University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 6<br />
|TBA<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 13<br />
|TBA<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 20<br />
|Jongchon Kim<br />
| UBC<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 27<br />
|Yumeng Ou<br />
|University of Pennsylvania<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|May 4<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|}<br />
<br />
=Abstracts=<br />
===Alexei Poltoratski===<br />
<br />
Title: Dirac inner functions<br />
<br />
Abstract: My talk will focus on some new (and old) complex analytic objects arising from Dirac systems of differential equations.<br />
We will discuss connections between problems in complex function theory, spectral and scattering problems for differential<br />
operators and the non-linear Fourier transform.<br />
<br />
===Polona Durcik and Joris Roos===<br />
<br />
Title: A triangular Hilbert transform with curvature, I & II.<br />
<br />
Abstract: The triangular Hilbert is a two-dimensional bilinear singular<br />
originating in time-frequency analysis. No Lp bounds are currently<br />
known for this operator.<br />
In these two talks we discuss a recent joint work with Michael Christ<br />
on a variant of the triangular Hilbert transform involving curvature.<br />
This object is closely related to the bilinear Hilbert transform with<br />
curvature and a maximally modulated singular integral of Stein-Wainger<br />
type. As an application we also discuss a quantitative nonlinear Roth<br />
type theorem on patterns in the Euclidean plane.<br />
The second talk will focus on the proof of a key ingredient, a certain<br />
regularity estimate for a local operator.<br />
<br />
===Andrew Zimmer===<br />
<br />
Title: Complex analytic problems on domains with good intrinsic geometry<br />
<br />
Abstract: In this talk, I will describe a new class of domains in complex Euclidean space which is defined in terms of the existence of a Kaehler metric with good geometric properties. This class is invariant under biholomorphism and includes many well-studied classes of domains such as strongly pseudoconvex domains, finite type domains in dimension two, convex domains, homogeneous domains, and embeddings of Teichmuller spaces. Further, certain analytic problems are tractable for domains in this family even when the boundary is non-smooth. In particular, it is possible to characterize the domains in this family where the dbar-Neumann operator on (0, q)-forms is compact (which generalizes an old result of Fu-Straube for convex domains).<br />
<br />
===Hong Wang===<br />
<br />
Title: Improved decoupling for the parabola<br />
<br />
Abstract: In 2014, Bourgain and Demeter proved the $l^2$ decoupling estimates for the paraboloid with constant $R^{\epsilon}$. <br />
We prove an $(l^2, L^6)$ decoupling inequality for the parabola with constant $(\log R)^c$. This is joint work with Larry Guth and Dominique Maldague.<br />
<br />
===Kevin Luli===<br />
<br />
Title: Smooth Nonnegative Interpolation<br />
<br />
Abstract: Suppose E is an arbitrary subset of R^n. Let f: E \rightarrow [0, \infty). How can we decide if f extends to a nonnegative function C^m function F defined on all of R^n? Suppose E is finite. Can we compute a nonnegative C^m function F on R^n that agrees with f on E with the least possible C^m norm? How many computer operations does this take? In this talk, I will explain recent results on these problems. Non-negativity is one of the most important shape preserving properties for interpolants. In real life applications, the range of the interpolant is imposed by nature. For example, probability density, the amount of snow, rain, humidity, chemical concentration are all nonnegative quantities and are of interest in natural sciences. Even in one dimension, the existing techniques can only handle nonnegative interpolation under special assumptions on the data set. Our results work without any assumptions on the data sets. <br />
<br />
===Niclas Technau===<br />
<br />
Title: Number theoretic applications of oscillatory integrals<br />
<br />
Abstract: We discuss how the analysis of oscillatory integrals can be used to solve number theoretic problems. More specifically, the focus will be on understanding fine-scale statistics of sequences on the unit circle. Further, we shall briefly explain a connection to quantum chaos.<br />
<br />
===Terence Harris===<br />
<br />
Title: Low dimensional pinned distance sets via spherical averages<br />
<br />
Abstract: An inequality is derived for the average t-energy of weighted pinned distance measures, where 0 < t < 1, in terms of the L^2 spherical averages of Fourier transforms of measures. This generalises the result of Liu (originally for Lebesgue measure) to pinned distance sets of dimension smaller than 1, and strengthens Mattila's result from 1987, originally for the full distance set.<br />
<br />
===Yuval Wigderson===<br />
<br />
Title: New perspectives on the uncertainty principle<br />
<br />
Abstract: The phrase ``uncertainty principle'' refers to a wide array of results in several disparate fields of mathematics, all of which capture the notion that a function and its Fourier transform cannot both be ``very localized''. The measure of localization varies from one uncertainty principle to the next, and well-studied notions include the variance (and higher moments), the entropy, the support-size, and the rate of decay at infinity. Similarly, the proofs of the various uncertainty principles rely on a range of tools, from the elementary to the very deep. In this talk, I'll describe how many of the uncertainty principles all follow from a single, simple result, whose proof uses only a basic property of the Fourier transform: that it and its inverse are bounded as operators $L^1 \to L^\infty$. Using this result, one can also prove new variants of the uncertainty principle, which apply to new measures of localization and to operators other than the Fourier transform. This is joint work with Avi Wigderson.<br />
<br />
===Oscar Dominguez===<br />
<br />
Title: New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions<br />
<br />
Abstract: The celebrated Bourgain--Brezis--Mironescu formula enables us to recover Sobolev spaces in terms of limits of Gagliardo seminorms. Very recently, Brezis, Van Schaftingen and Yung have proposed an alternative methodology to approach Sobolev spaces via limits of weak-type Gagliardo functionals. The goal of this talk is twofold. Firstly, we will show that the BvSY result is a special case of a more general phenomenon based on maximal inequalities. In particular, we shall derive not only analogs of the BvSY theorem for different kinds of function spaces (Lebesgue, Calderon, higher-order Sobolev, …), but also applications to ergodic theory, Fourier series, etc. In the second part of the talk, we shall investigate the fractional setting in the BvSY theorem. Our approach is based on new Garsia-type inequalities and an application of the Caffarelli--Silvestre extension. This is joint work with Mario Milman.<br />
<br />
===Tamas Titkos===<br />
<br />
Title: Isometries of Wasserstein spaces<br />
<br />
Abstract: Due to its nice theoretical properties and an astonishing number of<br />
applications via optimal transport problems, probably the most<br />
intensively studied metric nowadays is the p-Wasserstein metric. Given<br />
a complete and separable metric space $X$ and a real number $p\geq1$,<br />
one defines the p-Wasserstein space $\mathcal{W}_p(X)$ as the collection<br />
of Borel probability measures with finite $p$-th moment, endowed with a<br />
distance which is calculated by means of transport plans \cite{5}.<br />
<br />
The main aim of our research project is to reveal the structure of the<br />
isometry group $\mathrm{Isom}(\mathcal{W}_p(X))$. Although<br />
$\mathrm{Isom}(X)$ embeds naturally into<br />
$\mathrm{Isom}(\mathcal{W}_p(X))$ by push-forward, and this embedding<br />
turned out to be surjective in many cases (see e.g. [1]), these two<br />
groups are not isomorphic in general. Kloeckner in [2] described<br />
the isometry group of the quadratic Wasserstein space<br />
$\mathcal{W}_2(\mathbb{R}^n)$, and it turned out that the case of $n=1$<br />
is special in the sense that $\mathrm{Isom}(\mathcal{W}_2(\mathbb{R})$<br />
is extremely rich. Namely, it contains a large subgroup of wild behaving<br />
isometries that distort the shape of measures. Following this line of<br />
investigation, in \cite{3} we described<br />
$\mathrm{Isom}(\mathcal{W}_p(\mathbb{R}))$ and<br />
$\mathrm{Isom}(\mathcal{W}_p([0,1])$ for all $p\geq 1$.<br />
<br />
In this talk I will survey first some of the earlier results in the<br />
subject, and then I will present the key results of [3]. If time<br />
permits, I will also report on our most recent manuscript [4] in<br />
which we extended Kloeckner's multidimensional results. Joint work with Gy\"orgy P\'al Geh\'er (University of Reading)<br />
and D\'aniel Virosztek (IST Austria).<br />
<br />
[1] J. Bertrand and B. Kloeckner, \emph{A geometric study of Wasserstein<br />
spaces: isometric rigidity in negative curvature}, International<br />
Mathematics Research Notices, 2016 (5), 1368--1386.<br />
<br />
[2] B. Kloeckner, \emph{A geometric study of Wasserstein spaces: Euclidean<br />
spaces}, Annali della Scuola Normale Superiore di Pisa - Classe di<br />
Scienze, Serie 5, Tome 9 (2010) no. 2, 297--323.<br />
<br />
[3] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{Isometric study of<br />
Wasserstein spaces – the real line}, Trans. Amer. Math. Soc., 373<br />
(2020), 5855--5883.<br />
<br />
[4] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{The isometry group of<br />
Wasserstein spaces: The Hilbertian case}, submitted manuscript.<br />
<br />
[5] C. Villani, \emph{Optimal Transport: Old and New,}<br />
(Grundlehren der mathematischen Wissenschaften)<br />
Springer, 2009.<br />
<br />
===Shukun Wu===<br />
<br />
Title: On the Bochner-Riesz operator and the maximal Bochner-Riesz operator<br />
<br />
Abstract: The Bochner-Riesz problem is one of the most important problems in the field of Fourier analysis. It has a strong connection to other famous problems, such as the restriction conjecture and the Kakeya conjecture. In this talk, I will present some recent improvements to the Bochner-Riesz conjecture and the maximal Bochner-Riesz conjecture. The main methods we used are polynomial partitioning and the Bourgain Demeter l^2 decoupling theorem. <br />
<br />
<br />
===Jonathan Hickman===<br />
<br />
Title: Sobolev improving for averages over space curves<br />
<br />
Abstract: Consider the averaging operator given by convolution with arclength measure on compact piece of a smooth curve in R^n. A simple question is to precisely quantify the gain in regularity induced by this averaging, for instance by studying the L^p-Sobolev mapping properties of the operator. This talk will report on ongoing developments towards understanding this problem. In particular, we will explore some non-trivial necessary conditions on the gain in regularity. Joint with D. Beltran, S. Guo and A. Seeger.<br />
<br />
===Hanlong Fang===<br />
<br />
Title: Canonical blow-ups of Grassmann manifolds<br />
<br />
Abstract: We introduce certain canonical blow-ups \mathcal T_{s,p,n}, as well as their distinct submanifolds \mathcal M_{s,p,n}, of Grassmann manifolds G(p,n) by partitioning the Plücker coordinates with respect to a parameter s. Various geometric aspects of \mathcal T_{s,p,n} and \mathcal M_{s,p,n} are studied, for instance, the smoothness, the holomorphic symmetries, the (semi-)positivity of the anti-canonical bundles, the existence of Kähler-Einstein metrics, the functoriality, etc. In particular, we introduce the notion of homeward compactification, of which \mathcal T_{s,p,n} are examples, as a generalization of the wonderful compactification. <br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract:<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract:<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract:<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]<br />
<br />
<br />
Graduate Student Seminar:<br />
<br />
https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html</div>Beltranhttps://hilbert.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=20419Analysis Seminar2020-12-03T21:49:26Z<p>Beltran: /* Current Analysis Seminar Schedule */</p>
<hr />
<div><br />
The 2020-2021 Analysis Seminar will be organized by David Beltran and Andreas Seeger.<br />
It will be online at least for the Fall semester, with details to be announced in September.<br />
The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar earlier, or on different days, to accomodate speakers).<br />
<br />
Zoom links will be sent to those who have signed up for the Analysis Seminar List. For instructions how to sign up for seminar lists, see https://www.math.wisc.edu/node/230<br />
<br />
If you'd like to suggest speakers for the spring semester please contact David and Andreas (dbeltran at math, seeger at math).<br />
<br />
<br />
<br />
=[[Previous_Analysis_seminars]]=<br />
<br />
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars<br />
<br />
= Current Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 22<br />
|Alexei Poltoratski<br />
|UW Madison<br />
|[[#Alexei Poltoratski | Dirac inner functions ]]<br />
| <br />
|-<br />
|September 29<br />
|Joris Roos<br />
|University of Massachusetts - Lowell<br />
|[[#Polona Durcik and Joris Rooslinktoabstract | A triangular Hilbert transform with curvature, I ]]<br />
| <br />
|-<br />
|Wednesday September 30, 4 p.m.<br />
|Polona Durcik<br />
|Chapman University<br />
|[[#Polona Durcik and Joris Roos | A triangular Hilbert transform with curvature, II ]]<br />
| <br />
|-<br />
|October 6<br />
|Andrew Zimmer<br />
|UW Madison<br />
|[[#Andrew Zimmer | Complex analytic problems on domains with good intrinsic geometry ]]<br />
| <br />
|-<br />
|October 13<br />
|Hong Wang <br />
|Princeton/IAS<br />
|[[#Hong Wang | Improved decoupling for the parabola ]]<br />
| <br />
|-<br />
|October 20<br />
|Kevin Luli<br />
|UC Davis<br />
|[[#Kevin Luli | Smooth Nonnegative Interpolation ]]<br />
| <br />
|-<br />
|October 21, 4.00 p.m.<br />
|Niclas Technau<br />
|UW Madison<br />
|[[#Niclas Technau | Number theoretic applications of oscillatory integrals ]]<br />
| <br />
|-<br />
|October 27<br />
|Terence Harris<br />
| Cornell University<br />
|[[#Terence Harris | Low dimensional pinned distance sets via spherical averages ]]<br />
| <br />
|-<br />
|Monday, November 2, 4 p.m.<br />
|Yuval Wigderson<br />
|Stanford University<br />
|[[#Yuval Wigderson | New perspectives on the uncertainty principle ]]<br />
| <br />
|-<br />
|November 10, 10 a.m. <br />
|Óscar Domínguez<br />
| Universidad Complutense de Madrid<br />
|[[#Oscar Dominguez | New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions ]]<br />
| <br />
|-<br />
|November 17<br />
|Tamas Titkos<br />
|BBS U of Applied Sciences and Renyi Institute<br />
|[[#Tamas Titkos | Isometries of Wasserstein spaces ]]<br />
| <br />
|-<br />
|November 24<br />
|Shukun Wu<br />
|University of Illinois (Urbana-Champaign)<br />
||[[#Shukun Wu | On the Bochner-Riesz operator and the maximal Bochner-Riesz operator ]] <br />
| <br />
|-<br />
|December 1<br />
| Jonathan Hickman<br />
| The University of Edinburgh<br />
|[[#Jonathan Hickman | Sobolev improving for averages over space curves ]]<br />
| <br />
|-<br />
|December 8<br />
|<br />
| <br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|February 2<br />
|Jongchon Kim<br />
| UBC<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|February 9<br />
|Bingyang Hu<br />
|Purdue University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|February 16<br />
|Krystal Taylor<br />
|The Ohio State University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|February 23<br />
|Dominique Maldague<br />
|MIT<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 2<br />
|Diogo Oliveira e Silva<br />
|University of Birmingham<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 9<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 16<br />
|Ziming Shi<br />
|Rutgers University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 23<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 30<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 6<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 13<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 20<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 27<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|May 4<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|}<br />
<br />
=Abstracts=<br />
===Alexei Poltoratski===<br />
<br />
Title: Dirac inner functions<br />
<br />
Abstract: My talk will focus on some new (and old) complex analytic objects arising from Dirac systems of differential equations.<br />
We will discuss connections between problems in complex function theory, spectral and scattering problems for differential<br />
operators and the non-linear Fourier transform.<br />
<br />
===Polona Durcik and Joris Roos===<br />
<br />
Title: A triangular Hilbert transform with curvature, I & II.<br />
<br />
Abstract: The triangular Hilbert is a two-dimensional bilinear singular<br />
originating in time-frequency analysis. No Lp bounds are currently<br />
known for this operator.<br />
In these two talks we discuss a recent joint work with Michael Christ<br />
on a variant of the triangular Hilbert transform involving curvature.<br />
This object is closely related to the bilinear Hilbert transform with<br />
curvature and a maximally modulated singular integral of Stein-Wainger<br />
type. As an application we also discuss a quantitative nonlinear Roth<br />
type theorem on patterns in the Euclidean plane.<br />
The second talk will focus on the proof of a key ingredient, a certain<br />
regularity estimate for a local operator.<br />
<br />
===Andrew Zimmer===<br />
<br />
Title: Complex analytic problems on domains with good intrinsic geometry<br />
<br />
Abstract: In this talk, I will describe a new class of domains in complex Euclidean space which is defined in terms of the existence of a Kaehler metric with good geometric properties. This class is invariant under biholomorphism and includes many well-studied classes of domains such as strongly pseudoconvex domains, finite type domains in dimension two, convex domains, homogeneous domains, and embeddings of Teichmuller spaces. Further, certain analytic problems are tractable for domains in this family even when the boundary is non-smooth. In particular, it is possible to characterize the domains in this family where the dbar-Neumann operator on (0, q)-forms is compact (which generalizes an old result of Fu-Straube for convex domains).<br />
<br />
===Hong Wang===<br />
<br />
Title: Improved decoupling for the parabola<br />
<br />
Abstract: In 2014, Bourgain and Demeter proved the $l^2$ decoupling estimates for the paraboloid with constant $R^{\epsilon}$. <br />
We prove an $(l^2, L^6)$ decoupling inequality for the parabola with constant $(\log R)^c$. This is joint work with Larry Guth and Dominique Maldague.<br />
<br />
===Kevin Luli===<br />
<br />
Title: Smooth Nonnegative Interpolation<br />
<br />
Abstract: Suppose E is an arbitrary subset of R^n. Let f: E \rightarrow [0, \infty). How can we decide if f extends to a nonnegative function C^m function F defined on all of R^n? Suppose E is finite. Can we compute a nonnegative C^m function F on R^n that agrees with f on E with the least possible C^m norm? How many computer operations does this take? In this talk, I will explain recent results on these problems. Non-negativity is one of the most important shape preserving properties for interpolants. In real life applications, the range of the interpolant is imposed by nature. For example, probability density, the amount of snow, rain, humidity, chemical concentration are all nonnegative quantities and are of interest in natural sciences. Even in one dimension, the existing techniques can only handle nonnegative interpolation under special assumptions on the data set. Our results work without any assumptions on the data sets. <br />
<br />
===Niclas Technau===<br />
<br />
Title: Number theoretic applications of oscillatory integrals<br />
<br />
Abstract: We discuss how the analysis of oscillatory integrals can be used to solve number theoretic problems. More specifically, the focus will be on understanding fine-scale statistics of sequences on the unit circle. Further, we shall briefly explain a connection to quantum chaos.<br />
<br />
===Terence Harris===<br />
<br />
Title: Low dimensional pinned distance sets via spherical averages<br />
<br />
Abstract: An inequality is derived for the average t-energy of weighted pinned distance measures, where 0 < t < 1, in terms of the L^2 spherical averages of Fourier transforms of measures. This generalises the result of Liu (originally for Lebesgue measure) to pinned distance sets of dimension smaller than 1, and strengthens Mattila's result from 1987, originally for the full distance set.<br />
<br />
===Yuval Wigderson===<br />
<br />
Title: New perspectives on the uncertainty principle<br />
<br />
Abstract: The phrase ``uncertainty principle'' refers to a wide array of results in several disparate fields of mathematics, all of which capture the notion that a function and its Fourier transform cannot both be ``very localized''. The measure of localization varies from one uncertainty principle to the next, and well-studied notions include the variance (and higher moments), the entropy, the support-size, and the rate of decay at infinity. Similarly, the proofs of the various uncertainty principles rely on a range of tools, from the elementary to the very deep. In this talk, I'll describe how many of the uncertainty principles all follow from a single, simple result, whose proof uses only a basic property of the Fourier transform: that it and its inverse are bounded as operators $L^1 \to L^\infty$. Using this result, one can also prove new variants of the uncertainty principle, which apply to new measures of localization and to operators other than the Fourier transform. This is joint work with Avi Wigderson.<br />
<br />
===Oscar Dominguez===<br />
<br />
Title: New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions<br />
<br />
Abstract: The celebrated Bourgain--Brezis--Mironescu formula enables us to recover Sobolev spaces in terms of limits of Gagliardo seminorms. Very recently, Brezis, Van Schaftingen and Yung have proposed an alternative methodology to approach Sobolev spaces via limits of weak-type Gagliardo functionals. The goal of this talk is twofold. Firstly, we will show that the BvSY result is a special case of a more general phenomenon based on maximal inequalities. In particular, we shall derive not only analogs of the BvSY theorem for different kinds of function spaces (Lebesgue, Calderon, higher-order Sobolev, …), but also applications to ergodic theory, Fourier series, etc. In the second part of the talk, we shall investigate the fractional setting in the BvSY theorem. Our approach is based on new Garsia-type inequalities and an application of the Caffarelli--Silvestre extension. This is joint work with Mario Milman.<br />
<br />
===Tamas Titkos===<br />
<br />
Title: Isometries of Wasserstein spaces<br />
<br />
Abstract: Due to its nice theoretical properties and an astonishing number of<br />
applications via optimal transport problems, probably the most<br />
intensively studied metric nowadays is the p-Wasserstein metric. Given<br />
a complete and separable metric space $X$ and a real number $p\geq1$,<br />
one defines the p-Wasserstein space $\mathcal{W}_p(X)$ as the collection<br />
of Borel probability measures with finite $p$-th moment, endowed with a<br />
distance which is calculated by means of transport plans \cite{5}.<br />
<br />
The main aim of our research project is to reveal the structure of the<br />
isometry group $\mathrm{Isom}(\mathcal{W}_p(X))$. Although<br />
$\mathrm{Isom}(X)$ embeds naturally into<br />
$\mathrm{Isom}(\mathcal{W}_p(X))$ by push-forward, and this embedding<br />
turned out to be surjective in many cases (see e.g. [1]), these two<br />
groups are not isomorphic in general. Kloeckner in [2] described<br />
the isometry group of the quadratic Wasserstein space<br />
$\mathcal{W}_2(\mathbb{R}^n)$, and it turned out that the case of $n=1$<br />
is special in the sense that $\mathrm{Isom}(\mathcal{W}_2(\mathbb{R})$<br />
is extremely rich. Namely, it contains a large subgroup of wild behaving<br />
isometries that distort the shape of measures. Following this line of<br />
investigation, in \cite{3} we described<br />
$\mathrm{Isom}(\mathcal{W}_p(\mathbb{R}))$ and<br />
$\mathrm{Isom}(\mathcal{W}_p([0,1])$ for all $p\geq 1$.<br />
<br />
In this talk I will survey first some of the earlier results in the<br />
subject, and then I will present the key results of [3]. If time<br />
permits, I will also report on our most recent manuscript [4] in<br />
which we extended Kloeckner's multidimensional results. Joint work with Gy\"orgy P\'al Geh\'er (University of Reading)<br />
and D\'aniel Virosztek (IST Austria).<br />
<br />
[1] J. Bertrand and B. Kloeckner, \emph{A geometric study of Wasserstein<br />
spaces: isometric rigidity in negative curvature}, International<br />
Mathematics Research Notices, 2016 (5), 1368--1386.<br />
<br />
[2] B. Kloeckner, \emph{A geometric study of Wasserstein spaces: Euclidean<br />
spaces}, Annali della Scuola Normale Superiore di Pisa - Classe di<br />
Scienze, Serie 5, Tome 9 (2010) no. 2, 297--323.<br />
<br />
[3] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{Isometric study of<br />
Wasserstein spaces – the real line}, Trans. Amer. Math. Soc., 373<br />
(2020), 5855--5883.<br />
<br />
[4] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{The isometry group of<br />
Wasserstein spaces: The Hilbertian case}, submitted manuscript.<br />
<br />
[5] C. Villani, \emph{Optimal Transport: Old and New,}<br />
(Grundlehren der mathematischen Wissenschaften)<br />
Springer, 2009.<br />
<br />
===Shukun Wu===<br />
<br />
Title: On the Bochner-Riesz operator and the maximal Bochner-Riesz operator<br />
<br />
Abstract: The Bochner-Riesz problem is one of the most important problems in the field of Fourier analysis. It has a strong connection to other famous problems, such as the restriction conjecture and the Kakeya conjecture. In this talk, I will present some recent improvements to the Bochner-Riesz conjecture and the maximal Bochner-Riesz conjecture. The main methods we used are polynomial partitioning and the Bourgain Demeter l^2 decoupling theorem. <br />
<br />
<br />
===Jonathan Hickman===<br />
<br />
Title: Sobolev improving for averages over space curves<br />
<br />
Abstract: Consider the averaging operator given by convolution with arclength measure on compact piece of a smooth curve in R^n. A simple question is to precisely quantify the gain in regularity induced by this averaging, for instance by studying the L^p-Sobolev mapping properties of the operator. This talk will report on ongoing developments towards understanding this problem. In particular, we will explore some non-trivial necessary conditions on the gain in regularity. Joint with D. Beltran, S. Guo and A. Seeger.<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract:<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract:<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract:<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract:<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]<br />
<br />
<br />
Graduate Student Seminar:<br />
<br />
https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html</div>Beltranhttps://hilbert.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=20194Analysis Seminar2020-10-22T17:23:30Z<p>Beltran: /* Current Analysis Seminar Schedule */</p>
<hr />
<div><br />
The 2020-2021 Analysis Seminar will be organized by David Beltran and Andreas Seeger.<br />
It will be online at least for the Fall semester, with details to be announced in September.<br />
The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar earlier, or on different days, to accomodate speakers).<br />
<br />
Zoom links will be sent to those who have signed up for the Analysis Seminar List. For instructions how to sign up for seminar lists, see https://www.math.wisc.edu/node/230<br />
<br />
If you'd like to suggest speakers for the spring semester please contact David and Andreas (dbeltran at math, seeger at math).<br />
<br />
<br />
<br />
=[[Previous_Analysis_seminars]]=<br />
<br />
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars<br />
<br />
= Current Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 22<br />
|Alexei Poltoratski<br />
|UW Madison<br />
|[[#Alexei Poltoratski | Dirac inner functions ]]<br />
| <br />
|-<br />
|September 29<br />
|Joris Roos<br />
|University of Massachusetts - Lowell<br />
|[[#Polona Durcik and Joris Rooslinktoabstract | A triangular Hilbert transform with curvature, I ]]<br />
| <br />
|-<br />
|Wednesday September 30, 4 p.m.<br />
|Polona Durcik<br />
|Chapman University<br />
|[[#Polona Durcik and Joris Roos | A triangular Hilbert transform with curvature, II ]]<br />
| <br />
|-<br />
|October 6<br />
|Andrew Zimmer<br />
|UW Madison<br />
|[[#Andrew Zimmer | Complex analytic problems on domains with good intrinsic geometry ]]<br />
| <br />
|-<br />
|October 13<br />
|Hong Wang <br />
|Princeton/IAS<br />
|[[#Hong Wang | Improved decoupling for the parabola ]]<br />
| <br />
|-<br />
|October 20<br />
|Kevin Luli<br />
|UC Davis<br />
|[[#Kevin Luli | Smooth Nonnegative Interpolation ]]<br />
| <br />
|-<br />
|October 21, 4.00 p.m.<br />
|Niclas Technau<br />
|UW Madison<br />
|[[#Niclas Technau | Number theoretic applications of oscillatory integrals ]]<br />
| <br />
|-<br />
|October 27<br />
|Terence Harris<br />
| Cornell University<br />
|[[#Terence Harris | Low dimensional pinned distance sets via spherical averages ]]<br />
| <br />
|-<br />
|Monday, November 2, 4 p.m.<br />
|Yuval Wigderson<br />
|Stanford University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|November 10<br />
|Óscar Domínguez<br />
| Universidad Complutense de Madrid<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|November 17<br />
|Tamas Titkos<br />
|BBS U of Applied Sciences and Renyi Institute<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|November 24<br />
|Shukun Wu<br />
|University of Illinois (Urbana-Champaign)<br />
||[[#linktoabstract | Title ]] <br />
| <br />
|-<br />
|December 1<br />
| Jonathan Hickman<br />
| The University of Edinburgh<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|December 8<br />
|Alejandra Gaitán<br />
| Purdue University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|February 2<br />
|Jongchon Kim<br />
| UBC<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|February 9<br />
|Bingyang Hu<br />
|Purdue University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|February 16<br />
|Krystal Taylor<br />
|The Ohio State University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|February 23<br />
|Dominique Maldague<br />
|MIT<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 2<br />
|Diogo Oliveira e Silva<br />
|University of Birmingham<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 9<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 16<br />
|Ziming Shi<br />
|Rutgers University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 23<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 30<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 6<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 13<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 20<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 27<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|May 4<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|}<br />
<br />
=Abstracts=<br />
===Alexei Poltoratski===<br />
<br />
Title: Dirac inner functions<br />
<br />
Abstract: My talk will focus on some new (and old) complex analytic objects arising from Dirac systems of differential equations.<br />
We will discuss connections between problems in complex function theory, spectral and scattering problems for differential<br />
operators and the non-linear Fourier transform.<br />
<br />
===Polona Durcik and Joris Roos===<br />
<br />
Title: A triangular Hilbert transform with curvature, I & II.<br />
<br />
Abstract: The triangular Hilbert is a two-dimensional bilinear singular<br />
originating in time-frequency analysis. No Lp bounds are currently<br />
known for this operator.<br />
In these two talks we discuss a recent joint work with Michael Christ<br />
on a variant of the triangular Hilbert transform involving curvature.<br />
This object is closely related to the bilinear Hilbert transform with<br />
curvature and a maximally modulated singular integral of Stein-Wainger<br />
type. As an application we also discuss a quantitative nonlinear Roth<br />
type theorem on patterns in the Euclidean plane.<br />
The second talk will focus on the proof of a key ingredient, a certain<br />
regularity estimate for a local operator.<br />
<br />
===Andrew Zimmer===<br />
<br />
Title: Complex analytic problems on domains with good intrinsic geometry<br />
<br />
Abstract: In this talk, I will describe a new class of domains in complex Euclidean space which is defined in terms of the existence of a Kaehler metric with good geometric properties. This class is invariant under biholomorphism and includes many well-studied classes of domains such as strongly pseudoconvex domains, finite type domains in dimension two, convex domains, homogeneous domains, and embeddings of Teichmuller spaces. Further, certain analytic problems are tractable for domains in this family even when the boundary is non-smooth. In particular, it is possible to characterize the domains in this family where the dbar-Neumann operator on (0, q)-forms is compact (which generalizes an old result of Fu-Straube for convex domains).<br />
<br />
===Hong Wang===<br />
<br />
Title: Improved decoupling for the parabola<br />
<br />
Abstract: In 2014, Bourgain and Demeter proved the $l^2$ decoupling estimates for the paraboloid with constant $R^{\epsilon}$. <br />
We prove an $(l^2, L^6)$ decoupling inequality for the parabola with constant $(\log R)^c$. This is joint work with Larry Guth and Dominique Maldague.<br />
<br />
===Kevin Luli===<br />
<br />
Title: Smooth Nonnegative Interpolation<br />
<br />
Abstract: Suppose E is an arbitrary subset of R^n. Let f: E \rightarrow [0, \infty). How can we decide if f extends to a nonnegative function C^m function F defined on all of R^n? Suppose E is finite. Can we compute a nonnegative C^m function F on R^n that agrees with f on E with the least possible C^m norm? How many computer operations does this take? In this talk, I will explain recent results on these problems. Non-negativity is one of the most important shape preserving properties for interpolants. In real life applications, the range of the interpolant is imposed by nature. For example, probability density, the amount of snow, rain, humidity, chemical concentration are all nonnegative quantities and are of interest in natural sciences. Even in one dimension, the existing techniques can only handle nonnegative interpolation under special assumptions on the data set. Our results work without any assumptions on the data sets. <br />
<br />
===Niclas Technau===<br />
<br />
Title: Number theoretic applications of oscillatory integrals<br />
<br />
Abstract: We discuss how the analysis of oscillatory integrals can be used to solve number theoretic problems. More specifically, the focus will be on understanding fine-scale statistics of sequences on the unit circle. Further, we shall briefly explain a connection to quantum chaos.<br />
<br />
===Terence Harris===<br />
<br />
Title: Low dimensional pinned distance sets via spherical averages<br />
<br />
Abstract: An inequality is derived for the average t-energy of weighted pinned distance measures, where 0 < t < 1, in terms of the L^2 spherical averages of Fourier transforms of measures. This generalises the result of Liu (originally for Lebesgue measure) to pinned distance sets of dimension smaller than 1, and strengthens Mattila's result from 1987, originally for the full distance set.<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]<br />
<br />
<br />
Graduate Student Seminar:<br />
<br />
https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html</div>Beltranhttps://hilbert.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=20193Analysis Seminar2020-10-22T14:04:53Z<p>Beltran: /* Current Analysis Seminar Schedule */</p>
<hr />
<div><br />
The 2020-2021 Analysis Seminar will be organized by David Beltran and Andreas Seeger.<br />
It will be online at least for the Fall semester, with details to be announced in September.<br />
The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar earlier, or on different days, to accomodate speakers).<br />
<br />
Zoom links will be sent to those who have signed up for the Analysis Seminar List. For instructions how to sign up for seminar lists, see https://www.math.wisc.edu/node/230<br />
<br />
If you'd like to suggest speakers for the spring semester please contact David and Andreas (dbeltran at math, seeger at math).<br />
<br />
<br />
<br />
=[[Previous_Analysis_seminars]]=<br />
<br />
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars<br />
<br />
= Current Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 22<br />
|Alexei Poltoratski<br />
|UW Madison<br />
|[[#Alexei Poltoratski | Dirac inner functions ]]<br />
| <br />
|-<br />
|September 29<br />
|Joris Roos<br />
|University of Massachusetts - Lowell<br />
|[[#Polona Durcik and Joris Rooslinktoabstract | A triangular Hilbert transform with curvature, I ]]<br />
| <br />
|-<br />
|Wednesday September 30, 4 p.m.<br />
|Polona Durcik<br />
|Chapman University<br />
|[[#Polona Durcik and Joris Roos | A triangular Hilbert transform with curvature, II ]]<br />
| <br />
|-<br />
|October 6<br />
|Andrew Zimmer<br />
|UW Madison<br />
|[[#Andrew Zimmer | Complex analytic problems on domains with good intrinsic geometry ]]<br />
| <br />
|-<br />
|October 13<br />
|Hong Wang <br />
|Princeton/IAS<br />
|[[#Hong Wang | Improved decoupling for the parabola ]]<br />
| <br />
|-<br />
|October 20<br />
|Kevin Luli<br />
|UC Davis<br />
|[[#Kevin Luli | Smooth Nonnegative Interpolation ]]<br />
| <br />
|-<br />
|October 21, 4.00 p.m.<br />
|Niclas Technau<br />
|UW Madison<br />
|[[#Niclas Technau | Number theoretic applications of oscillatory integrals ]]<br />
| <br />
|-<br />
|October 27<br />
|Terence Harris<br />
| Cornell University<br />
|[[#Terence Harris | Low dimensional pinned distance sets via spherical averages ]]<br />
| <br />
|-<br />
|Monday, November 2, 4 p.m.<br />
|Yuval Wigderson<br />
|Stanford University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|November 10<br />
|Óscar Domínguez<br />
| Universidad Complutense de Madrid<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|November 17<br />
|Tamas Titkos<br />
|BBS U of Applied Sciences and Renyi Institute<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|November 24<br />
|Shukun Wu<br />
|University of Illinois (Urbana-Champaign)<br />
||[[#linktoabstract | Title ]] <br />
| <br />
|-<br />
|December 1<br />
| Jonathan Hickman<br />
| The University of Edinburgh<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|December 8<br />
|Alejandra Gaitán<br />
| Purdue University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|February 2<br />
|Jongchon Kim<br />
| UBC<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|February 9<br />
|Bingyang Hu<br />
|Purdue University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|February 16<br />
|Krystal Taylor<br />
|The Ohio State University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|February 23<br />
|Dominique Maldague<br />
|MIT<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 2<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 9<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 16<br />
|Ziming Shi<br />
|Rutgers University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 23<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 30<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 6<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 13<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 20<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 27<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|May 4<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|}<br />
<br />
=Abstracts=<br />
===Alexei Poltoratski===<br />
<br />
Title: Dirac inner functions<br />
<br />
Abstract: My talk will focus on some new (and old) complex analytic objects arising from Dirac systems of differential equations.<br />
We will discuss connections between problems in complex function theory, spectral and scattering problems for differential<br />
operators and the non-linear Fourier transform.<br />
<br />
===Polona Durcik and Joris Roos===<br />
<br />
Title: A triangular Hilbert transform with curvature, I & II.<br />
<br />
Abstract: The triangular Hilbert is a two-dimensional bilinear singular<br />
originating in time-frequency analysis. No Lp bounds are currently<br />
known for this operator.<br />
In these two talks we discuss a recent joint work with Michael Christ<br />
on a variant of the triangular Hilbert transform involving curvature.<br />
This object is closely related to the bilinear Hilbert transform with<br />
curvature and a maximally modulated singular integral of Stein-Wainger<br />
type. As an application we also discuss a quantitative nonlinear Roth<br />
type theorem on patterns in the Euclidean plane.<br />
The second talk will focus on the proof of a key ingredient, a certain<br />
regularity estimate for a local operator.<br />
<br />
===Andrew Zimmer===<br />
<br />
Title: Complex analytic problems on domains with good intrinsic geometry<br />
<br />
Abstract: In this talk, I will describe a new class of domains in complex Euclidean space which is defined in terms of the existence of a Kaehler metric with good geometric properties. This class is invariant under biholomorphism and includes many well-studied classes of domains such as strongly pseudoconvex domains, finite type domains in dimension two, convex domains, homogeneous domains, and embeddings of Teichmuller spaces. Further, certain analytic problems are tractable for domains in this family even when the boundary is non-smooth. In particular, it is possible to characterize the domains in this family where the dbar-Neumann operator on (0, q)-forms is compact (which generalizes an old result of Fu-Straube for convex domains).<br />
<br />
===Hong Wang===<br />
<br />
Title: Improved decoupling for the parabola<br />
<br />
Abstract: In 2014, Bourgain and Demeter proved the $l^2$ decoupling estimates for the paraboloid with constant $R^{\epsilon}$. <br />
We prove an $(l^2, L^6)$ decoupling inequality for the parabola with constant $(\log R)^c$. This is joint work with Larry Guth and Dominique Maldague.<br />
<br />
===Kevin Luli===<br />
<br />
Title: Smooth Nonnegative Interpolation<br />
<br />
Abstract: Suppose E is an arbitrary subset of R^n. Let f: E \rightarrow [0, \infty). How can we decide if f extends to a nonnegative function C^m function F defined on all of R^n? Suppose E is finite. Can we compute a nonnegative C^m function F on R^n that agrees with f on E with the least possible C^m norm? How many computer operations does this take? In this talk, I will explain recent results on these problems. Non-negativity is one of the most important shape preserving properties for interpolants. In real life applications, the range of the interpolant is imposed by nature. For example, probability density, the amount of snow, rain, humidity, chemical concentration are all nonnegative quantities and are of interest in natural sciences. Even in one dimension, the existing techniques can only handle nonnegative interpolation under special assumptions on the data set. Our results work without any assumptions on the data sets. <br />
<br />
===Niclas Technau===<br />
<br />
Title: Number theoretic applications of oscillatory integrals<br />
<br />
Abstract: We discuss how the analysis of oscillatory integrals can be used to solve number theoretic problems. More specifically, the focus will be on understanding fine-scale statistics of sequences on the unit circle. Further, we shall briefly explain a connection to quantum chaos.<br />
<br />
===Terence Harris===<br />
<br />
Title: Low dimensional pinned distance sets via spherical averages<br />
<br />
Abstract: An inequality is derived for the average t-energy of weighted pinned distance measures, where 0 < t < 1, in terms of the L^2 spherical averages of Fourier transforms of measures. This generalises the result of Liu (originally for Lebesgue measure) to pinned distance sets of dimension smaller than 1, and strengthens Mattila's result from 1987, originally for the full distance set.<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]<br />
<br />
<br />
Graduate Student Seminar:<br />
<br />
https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html</div>Beltranhttps://hilbert.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=20168Analysis Seminar2020-10-18T23:32:32Z<p>Beltran: /* Name */</p>
<hr />
<div><br />
The 2020-2021 Analysis Seminar will be organized by David Beltran and Andreas Seeger.<br />
It will be online at least for the Fall semester, with details to be announced in September.<br />
The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar earlier, or on different days, to accomodate speakers).<br />
<br />
Zoom links will be sent to those who have signed up for the Analysis Seminar List. For instructions how to sign up for seminar lists, see https://www.math.wisc.edu/node/230<br />
<br />
If you'd like to suggest speakers for the spring semester please contact David and Andreas (dbeltran at math, seeger at math).<br />
<br />
<br />
<br />
=[[Previous_Analysis_seminars]]=<br />
<br />
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars<br />
<br />
= Current Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 22<br />
|Alexei Poltoratski<br />
|UW Madison<br />
|[[#Alexei Poltoratski | Dirac inner functions ]]<br />
| <br />
|-<br />
|September 29<br />
|Joris Roos<br />
|University of Massachusetts - Lowell<br />
|[[#Polona Durcik and Joris Rooslinktoabstract | A triangular Hilbert transform with curvature, I ]]<br />
| <br />
|-<br />
|Wednesday September 30, 4 p.m.<br />
|Polona Durcik<br />
|Chapman University<br />
|[[#Polona Durcik and Joris Roos | A triangular Hilbert transform with curvature, II ]]<br />
| <br />
|-<br />
|October 6<br />
|Andrew Zimmer<br />
|UW Madison<br />
|[[#Andrew Zimmer | Complex analytic problems on domains with good intrinsic geometry ]]<br />
| <br />
|-<br />
|October 13<br />
|Hong Wang <br />
|Princeton/IAS<br />
|[[#Hong Wang | Improved decoupling for the parabola ]]<br />
| <br />
|-<br />
|October 20<br />
|Kevin Luli<br />
|UC Davis<br />
|[[#Kevin Luli | Smooth Nonnegative Interpolation ]]<br />
| <br />
|-<br />
|October 21, 4.00 p.m.<br />
|Niclas Technau<br />
|UW Madison<br />
|[[#Niclas Technau | Number theoretic applications of oscillatory integrals ]]<br />
| <br />
|-<br />
|October 27<br />
|Terence Harris<br />
| Cornell University<br />
|[[#Terence Harris | Low dimensional pinned distance sets via spherical averages ]]<br />
| <br />
|-<br />
|Monday, November 2, 4 p.m.<br />
|Yuval Wigderson<br />
|Stanford University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|November 10<br />
|Óscar Domínguez<br />
| Universidad Complutense de Madrid<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|November 17<br />
|Tamas Titkos<br />
|BBS U of Applied Sciences and Renyi Institute<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|November 24<br />
|Shukun Wu<br />
|University of Illinois (Urbana-Champaign)<br />
||[[#linktoabstract | Title ]] <br />
| <br />
|-<br />
|December 1<br />
| Jonathan Hickman<br />
| The University of Edinburgh<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|December 8<br />
|Alejandra Gaitán<br />
| Purdue University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|February 2<br />
|Jongchon Kim<br />
| UBC<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|February 9<br />
|Bingyang Hu<br />
|Purdue University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|February 16<br />
|Krystal Taylor<br />
|The Ohio State University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|February 23<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 2<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 9<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 16<br />
|Ziming Shi<br />
|Rutgers University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 23<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 30<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 6<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 13<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 20<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 27<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|May 4<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|}<br />
<br />
=Abstracts=<br />
===Alexei Poltoratski===<br />
<br />
Title: Dirac inner functions<br />
<br />
Abstract: My talk will focus on some new (and old) complex analytic objects arising from Dirac systems of differential equations.<br />
We will discuss connections between problems in complex function theory, spectral and scattering problems for differential<br />
operators and the non-linear Fourier transform.<br />
<br />
===Polona Durcik and Joris Roos===<br />
<br />
Title: A triangular Hilbert transform with curvature, I & II.<br />
<br />
Abstract: The triangular Hilbert is a two-dimensional bilinear singular<br />
originating in time-frequency analysis. No Lp bounds are currently<br />
known for this operator.<br />
In these two talks we discuss a recent joint work with Michael Christ<br />
on a variant of the triangular Hilbert transform involving curvature.<br />
This object is closely related to the bilinear Hilbert transform with<br />
curvature and a maximally modulated singular integral of Stein-Wainger<br />
type. As an application we also discuss a quantitative nonlinear Roth<br />
type theorem on patterns in the Euclidean plane.<br />
The second talk will focus on the proof of a key ingredient, a certain<br />
regularity estimate for a local operator.<br />
<br />
===Andrew Zimmer===<br />
<br />
Title: Complex analytic problems on domains with good intrinsic geometry<br />
<br />
Abstract: In this talk, I will describe a new class of domains in complex Euclidean space which is defined in terms of the existence of a Kaehler metric with good geometric properties. This class is invariant under biholomorphism and includes many well-studied classes of domains such as strongly pseudoconvex domains, finite type domains in dimension two, convex domains, homogeneous domains, and embeddings of Teichmuller spaces. Further, certain analytic problems are tractable for domains in this family even when the boundary is non-smooth. In particular, it is possible to characterize the domains in this family where the dbar-Neumann operator on (0, q)-forms is compact (which generalizes an old result of Fu-Straube for convex domains).<br />
<br />
===Hong Wang===<br />
<br />
Title: Improved decoupling for the parabola<br />
<br />
Abstract: In 2014, Bourgain and Demeter proved the $l^2$ decoupling estimates for the paraboloid with constant $R^{\epsilon}$. <br />
We prove an $(l^2, L^6)$ decoupling inequality for the parabola with constant $(\log R)^c$. This is joint work with Larry Guth and Dominique Maldague.<br />
<br />
===Kevin Luli===<br />
<br />
Title: Smooth Nonnegative Interpolation<br />
<br />
Abstract: Suppose E is an arbitrary subset of R^n. Let f: E \rightarrow [0, \infty). How can we decide if f extends to a nonnegative function C^m function F defined on all of R^n? Suppose E is finite. Can we compute a nonnegative C^m function F on R^n that agrees with f on E with the least possible C^m norm? How many computer operations does this take? In this talk, I will explain recent results on these problems. Non-negativity is one of the most important shape preserving properties for interpolants. In real life applications, the range of the interpolant is imposed by nature. For example, probability density, the amount of snow, rain, humidity, chemical concentration are all nonnegative quantities and are of interest in natural sciences. Even in one dimension, the existing techniques can only handle nonnegative interpolation under special assumptions on the data set. Our results work without any assumptions on the data sets. <br />
<br />
===Niclas Technau===<br />
<br />
Title: Number theoretic applications of oscillatory integrals<br />
<br />
Abstract: We discuss how the analysis of oscillatory integrals can be used to solve number theoretic problems. More specifically, the focus will be on understanding fine-scale statistics of sequences on the unit circle. Further, we shall briefly explain a connection to quantum chaos.<br />
<br />
===Terence Harris===<br />
<br />
Title: Low dimensional pinned distance sets via spherical averages<br />
<br />
Abstract: An inequality is derived for the average t-energy of weighted pinned distance measures, where 0 < t < 1, in terms of the L^2 spherical averages of Fourier transforms of measures. This generalises the result of Liu (originally for Lebesgue measure) to pinned distance sets of dimension smaller than 1, and strengthens Mattila's result from 1987, originally for the full distance set.<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]<br />
<br />
<br />
Graduate Student Seminar:<br />
<br />
https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html</div>Beltranhttps://hilbert.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=20167Analysis Seminar2020-10-18T23:31:50Z<p>Beltran: /* Current Analysis Seminar Schedule */</p>
<hr />
<div><br />
The 2020-2021 Analysis Seminar will be organized by David Beltran and Andreas Seeger.<br />
It will be online at least for the Fall semester, with details to be announced in September.<br />
The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar earlier, or on different days, to accomodate speakers).<br />
<br />
Zoom links will be sent to those who have signed up for the Analysis Seminar List. For instructions how to sign up for seminar lists, see https://www.math.wisc.edu/node/230<br />
<br />
If you'd like to suggest speakers for the spring semester please contact David and Andreas (dbeltran at math, seeger at math).<br />
<br />
<br />
<br />
=[[Previous_Analysis_seminars]]=<br />
<br />
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars<br />
<br />
= Current Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 22<br />
|Alexei Poltoratski<br />
|UW Madison<br />
|[[#Alexei Poltoratski | Dirac inner functions ]]<br />
| <br />
|-<br />
|September 29<br />
|Joris Roos<br />
|University of Massachusetts - Lowell<br />
|[[#Polona Durcik and Joris Rooslinktoabstract | A triangular Hilbert transform with curvature, I ]]<br />
| <br />
|-<br />
|Wednesday September 30, 4 p.m.<br />
|Polona Durcik<br />
|Chapman University<br />
|[[#Polona Durcik and Joris Roos | A triangular Hilbert transform with curvature, II ]]<br />
| <br />
|-<br />
|October 6<br />
|Andrew Zimmer<br />
|UW Madison<br />
|[[#Andrew Zimmer | Complex analytic problems on domains with good intrinsic geometry ]]<br />
| <br />
|-<br />
|October 13<br />
|Hong Wang <br />
|Princeton/IAS<br />
|[[#Hong Wang | Improved decoupling for the parabola ]]<br />
| <br />
|-<br />
|October 20<br />
|Kevin Luli<br />
|UC Davis<br />
|[[#Kevin Luli | Smooth Nonnegative Interpolation ]]<br />
| <br />
|-<br />
|October 21, 4.00 p.m.<br />
|Niclas Technau<br />
|UW Madison<br />
|[[#Niclas Technau | Number theoretic applications of oscillatory integrals ]]<br />
| <br />
|-<br />
|October 27<br />
|Terence Harris<br />
| Cornell University<br />
|[[#Terence Harris | Low dimensional pinned distance sets via spherical averages ]]<br />
| <br />
|-<br />
|Monday, November 2, 4 p.m.<br />
|Yuval Wigderson<br />
|Stanford University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|November 10<br />
|Óscar Domínguez<br />
| Universidad Complutense de Madrid<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|November 17<br />
|Tamas Titkos<br />
|BBS U of Applied Sciences and Renyi Institute<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|November 24<br />
|Shukun Wu<br />
|University of Illinois (Urbana-Champaign)<br />
||[[#linktoabstract | Title ]] <br />
| <br />
|-<br />
|December 1<br />
| Jonathan Hickman<br />
| The University of Edinburgh<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|December 8<br />
|Alejandra Gaitán<br />
| Purdue University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|February 2<br />
|Jongchon Kim<br />
| UBC<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|February 9<br />
|Bingyang Hu<br />
|Purdue University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|February 16<br />
|Krystal Taylor<br />
|The Ohio State University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|February 23<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 2<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 9<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 16<br />
|Ziming Shi<br />
|Rutgers University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 23<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 30<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 6<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 13<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 20<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 27<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|May 4<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|}<br />
<br />
=Abstracts=<br />
===Alexei Poltoratski===<br />
<br />
Title: Dirac inner functions<br />
<br />
Abstract: My talk will focus on some new (and old) complex analytic objects arising from Dirac systems of differential equations.<br />
We will discuss connections between problems in complex function theory, spectral and scattering problems for differential<br />
operators and the non-linear Fourier transform.<br />
<br />
===Polona Durcik and Joris Roos===<br />
<br />
Title: A triangular Hilbert transform with curvature, I & II.<br />
<br />
Abstract: The triangular Hilbert is a two-dimensional bilinear singular<br />
originating in time-frequency analysis. No Lp bounds are currently<br />
known for this operator.<br />
In these two talks we discuss a recent joint work with Michael Christ<br />
on a variant of the triangular Hilbert transform involving curvature.<br />
This object is closely related to the bilinear Hilbert transform with<br />
curvature and a maximally modulated singular integral of Stein-Wainger<br />
type. As an application we also discuss a quantitative nonlinear Roth<br />
type theorem on patterns in the Euclidean plane.<br />
The second talk will focus on the proof of a key ingredient, a certain<br />
regularity estimate for a local operator.<br />
<br />
===Andrew Zimmer===<br />
<br />
Title: Complex analytic problems on domains with good intrinsic geometry<br />
<br />
Abstract: In this talk, I will describe a new class of domains in complex Euclidean space which is defined in terms of the existence of a Kaehler metric with good geometric properties. This class is invariant under biholomorphism and includes many well-studied classes of domains such as strongly pseudoconvex domains, finite type domains in dimension two, convex domains, homogeneous domains, and embeddings of Teichmuller spaces. Further, certain analytic problems are tractable for domains in this family even when the boundary is non-smooth. In particular, it is possible to characterize the domains in this family where the dbar-Neumann operator on (0, q)-forms is compact (which generalizes an old result of Fu-Straube for convex domains).<br />
<br />
===Hong Wang===<br />
<br />
Title: Improved decoupling for the parabola<br />
<br />
Abstract: In 2014, Bourgain and Demeter proved the $l^2$ decoupling estimates for the paraboloid with constant $R^{\epsilon}$. <br />
We prove an $(l^2, L^6)$ decoupling inequality for the parabola with constant $(\log R)^c$. This is joint work with Larry Guth and Dominique Maldague.<br />
<br />
===Kevin Luli===<br />
<br />
Title: Smooth Nonnegative Interpolation<br />
<br />
Abstract: Suppose E is an arbitrary subset of R^n. Let f: E \rightarrow [0, \infty). How can we decide if f extends to a nonnegative function C^m function F defined on all of R^n? Suppose E is finite. Can we compute a nonnegative C^m function F on R^n that agrees with f on E with the least possible C^m norm? How many computer operations does this take? In this talk, I will explain recent results on these problems. Non-negativity is one of the most important shape preserving properties for interpolants. In real life applications, the range of the interpolant is imposed by nature. For example, probability density, the amount of snow, rain, humidity, chemical concentration are all nonnegative quantities and are of interest in natural sciences. Even in one dimension, the existing techniques can only handle nonnegative interpolation under special assumptions on the data set. Our results work without any assumptions on the data sets. <br />
<br />
===Niclas Technau===<br />
<br />
Title: Number theoretic applications of oscillatory integrals<br />
<br />
Abstract: We discuss how the analysis of oscillatory integrals can be used to solve number theoretic problems. More specifically, the focus will be on understanding fine-scale statistics of sequences on the unit circle. Further, we shall briefly explain a connection to quantum chaos.<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]<br />
<br />
<br />
Graduate Student Seminar:<br />
<br />
https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html</div>Beltranhttps://hilbert.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=20150Analysis Seminar2020-10-16T12:33:10Z<p>Beltran: /* Current Analysis Seminar Schedule */</p>
<hr />
<div><br />
The 2020-2021 Analysis Seminar will be organized by David Beltran and Andreas Seeger.<br />
It will be online at least for the Fall semester, with details to be announced in September.<br />
The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar earlier, or on different days, to accomodate speakers).<br />
<br />
Zoom links will be sent to those who have signed up for the Analysis Seminar List. For instructions how to sign up for seminar lists, see https://www.math.wisc.edu/node/230<br />
<br />
If you'd like to suggest speakers for the spring semester please contact David and Andreas (dbeltran at math, seeger at math).<br />
<br />
<br />
<br />
=[[Previous_Analysis_seminars]]=<br />
<br />
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars<br />
<br />
= Current Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 22<br />
|Alexei Poltoratski<br />
|UW Madison<br />
|[[#Alexei Poltoratski | Dirac inner functions ]]<br />
| <br />
|-<br />
|September 29<br />
|Joris Roos<br />
|University of Massachusetts - Lowell<br />
|[[#Polona Durcik and Joris Rooslinktoabstract | A triangular Hilbert transform with curvature, I ]]<br />
| <br />
|-<br />
|Wednesday September 30, 4 p.m.<br />
|Polona Durcik<br />
|Chapman University<br />
|[[#Polona Durcik and Joris Roos | A triangular Hilbert transform with curvature, II ]]<br />
| <br />
|-<br />
|October 6<br />
|Andrew Zimmer<br />
|UW Madison<br />
|[[#Andrew Zimmer | Complex analytic problems on domains with good intrinsic geometry ]]<br />
| <br />
|-<br />
|October 13<br />
|Hong Wang <br />
|Princeton/IAS<br />
|[[#Hong Wang | Improved decoupling for the parabola ]]<br />
| <br />
|-<br />
|October 20<br />
|Kevin Luli<br />
|UC Davis<br />
|[[#Kevin Luli | Smooth Nonnegative Interpolation ]]<br />
| <br />
|-<br />
|October 21, 4.00 p.m.<br />
|Niclas Technau<br />
|UW Madison<br />
|[[#Niclas Technau | Number theoretic applications of oscillatory integrals ]]<br />
| <br />
|-<br />
|October 27<br />
|Terence Harris<br />
| Cornell University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|Monday, November 2, 4 p.m.<br />
|Yuval Wigderson<br />
|Stanford University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|November 10<br />
|Óscar Domínguez<br />
| Universidad Complutense de Madrid<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|November 17<br />
|Tamas Titkos<br />
|BBS U of Applied Sciences and Renyi Institute<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|November 24<br />
|Shukun Wu<br />
|University of Illinois (Urbana-Champaign)<br />
||[[#linktoabstract | Title ]] <br />
| <br />
|-<br />
|December 1<br />
| Jonathan Hickman<br />
| The University of Edinburgh<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|December 8<br />
|Alejandra Gaitán<br />
| Purdue University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|February 2<br />
|Jongchon Kim<br />
| UBC<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|February 9<br />
|Bingyang Hu<br />
|Purdue University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|February 16<br />
|Krystal Taylor<br />
|The Ohio State University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|February 23<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 2<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 9<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 16<br />
|Ziming Shi<br />
|Rutgers University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 23<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 30<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 6<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 13<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 20<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 27<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|May 4<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|}<br />
<br />
=Abstracts=<br />
===Alexei Poltoratski===<br />
<br />
Title: Dirac inner functions<br />
<br />
Abstract: My talk will focus on some new (and old) complex analytic objects arising from Dirac systems of differential equations.<br />
We will discuss connections between problems in complex function theory, spectral and scattering problems for differential<br />
operators and the non-linear Fourier transform.<br />
<br />
===Polona Durcik and Joris Roos===<br />
<br />
Title: A triangular Hilbert transform with curvature, I & II.<br />
<br />
Abstract: The triangular Hilbert is a two-dimensional bilinear singular<br />
originating in time-frequency analysis. No Lp bounds are currently<br />
known for this operator.<br />
In these two talks we discuss a recent joint work with Michael Christ<br />
on a variant of the triangular Hilbert transform involving curvature.<br />
This object is closely related to the bilinear Hilbert transform with<br />
curvature and a maximally modulated singular integral of Stein-Wainger<br />
type. As an application we also discuss a quantitative nonlinear Roth<br />
type theorem on patterns in the Euclidean plane.<br />
The second talk will focus on the proof of a key ingredient, a certain<br />
regularity estimate for a local operator.<br />
<br />
===Andrew Zimmer===<br />
<br />
Title: Complex analytic problems on domains with good intrinsic geometry<br />
<br />
Abstract: In this talk, I will describe a new class of domains in complex Euclidean space which is defined in terms of the existence of a Kaehler metric with good geometric properties. This class is invariant under biholomorphism and includes many well-studied classes of domains such as strongly pseudoconvex domains, finite type domains in dimension two, convex domains, homogeneous domains, and embeddings of Teichmuller spaces. Further, certain analytic problems are tractable for domains in this family even when the boundary is non-smooth. In particular, it is possible to characterize the domains in this family where the dbar-Neumann operator on (0, q)-forms is compact (which generalizes an old result of Fu-Straube for convex domains).<br />
<br />
===Hong Wang===<br />
<br />
Title: Improved decoupling for the parabola<br />
<br />
Abstract: In 2014, Bourgain and Demeter proved the $l^2$ decoupling estimates for the paraboloid with constant $R^{\epsilon}$. <br />
We prove an $(l^2, L^6)$ decoupling inequality for the parabola with constant $(\log R)^c$. This is joint work with Larry Guth and Dominique Maldague.<br />
<br />
===Kevin Luli===<br />
<br />
Title: Smooth Nonnegative Interpolation<br />
<br />
Abstract: Suppose E is an arbitrary subset of R^n. Let f: E \rightarrow [0, \infty). How can we decide if f extends to a nonnegative function C^m function F defined on all of R^n? Suppose E is finite. Can we compute a nonnegative C^m function F on R^n that agrees with f on E with the least possible C^m norm? How many computer operations does this take? In this talk, I will explain recent results on these problems. Non-negativity is one of the most important shape preserving properties for interpolants. In real life applications, the range of the interpolant is imposed by nature. For example, probability density, the amount of snow, rain, humidity, chemical concentration are all nonnegative quantities and are of interest in natural sciences. Even in one dimension, the existing techniques can only handle nonnegative interpolation under special assumptions on the data set. Our results work without any assumptions on the data sets. <br />
<br />
===Niclas Technau===<br />
<br />
Title: Number theoretic applications of oscillatory integrals<br />
<br />
Abstract: We discuss how the analysis of oscillatory integrals can be used to solve number theoretic problems. More specifically, the focus will be on understanding fine-scale statistics of sequences on the unit circle. Further, we shall briefly explain a connection to quantum chaos.<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]<br />
<br />
<br />
Graduate Student Seminar:<br />
<br />
https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html</div>Beltranhttps://hilbert.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=20149Analysis Seminar2020-10-16T00:21:45Z<p>Beltran: /* Current Analysis Seminar Schedule */</p>
<hr />
<div><br />
The 2020-2021 Analysis Seminar will be organized by David Beltran and Andreas Seeger.<br />
It will be online at least for the Fall semester, with details to be announced in September.<br />
The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar earlier, or on different days, to accomodate speakers).<br />
<br />
Zoom links will be sent to those who have signed up for the Analysis Seminar List. For instructions how to sign up for seminar lists, see https://www.math.wisc.edu/node/230<br />
<br />
If you'd like to suggest speakers for the spring semester please contact David and Andreas (dbeltran at math, seeger at math).<br />
<br />
<br />
<br />
=[[Previous_Analysis_seminars]]=<br />
<br />
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars<br />
<br />
= Current Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 22<br />
|Alexei Poltoratski<br />
|UW Madison<br />
|[[#Alexei Poltoratski | Dirac inner functions ]]<br />
| <br />
|-<br />
|September 29<br />
|Joris Roos<br />
|University of Massachusetts - Lowell<br />
|[[#Polona Durcik and Joris Rooslinktoabstract | A triangular Hilbert transform with curvature, I ]]<br />
| <br />
|-<br />
|Wednesday September 30, 4 p.m.<br />
|Polona Durcik<br />
|Chapman University<br />
|[[#Polona Durcik and Joris Roos | A triangular Hilbert transform with curvature, II ]]<br />
| <br />
|-<br />
|October 6<br />
|Andrew Zimmer<br />
|UW Madison<br />
|[[#Andrew Zimmer | Complex analytic problems on domains with good intrinsic geometry ]]<br />
| <br />
|-<br />
|October 13<br />
|Hong Wang <br />
|Princeton/IAS<br />
|[[#Hong Wang | Improved decoupling for the parabola ]]<br />
| <br />
|-<br />
|October 20<br />
|Kevin Luli<br />
|UC Davis<br />
|[[#Kevin Luli | Smooth Nonnegative Interpolation ]]<br />
| <br />
|-<br />
|October 21, 4.00 p.m.<br />
|Niclas Technau<br />
|UW Madison<br />
|[[#Niclas Technau | Number theoretic applications of oscillatory integrals ]]<br />
| <br />
|-<br />
|October 27<br />
|Terence Harris<br />
| Cornell University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|Monday, November 2, 4 p.m.<br />
|Yuval Wigderson<br />
|Stanford University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|November 10<br />
|Óscar Domínguez<br />
| Universidad Complutense de Madrid<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|November 17<br />
|Tamas Titkos<br />
|BBS U of Applied Sciences and Renyi Institute<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|November 24<br />
|Shukun Wu<br />
|University of Illinois (Urbana-Champaign)<br />
||[[#linktoabstract | Title ]] <br />
| <br />
|-<br />
|December 1<br />
| Jonathan Hickman<br />
| The University of Edinburgh<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|December 8<br />
|Alejandra Gaitán<br />
| Purdue University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|February 2<br />
|Jongchon Kim<br />
| UBC<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|February 9<br />
|Bingyang Hu<br />
|Purdue University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|February 16<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|February 23<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 2<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 9<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 16<br />
|Ziming Shi<br />
|Rutgers University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 23<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 30<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 6<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 13<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 20<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 27<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|May 4<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|}<br />
<br />
=Abstracts=<br />
===Alexei Poltoratski===<br />
<br />
Title: Dirac inner functions<br />
<br />
Abstract: My talk will focus on some new (and old) complex analytic objects arising from Dirac systems of differential equations.<br />
We will discuss connections between problems in complex function theory, spectral and scattering problems for differential<br />
operators and the non-linear Fourier transform.<br />
<br />
===Polona Durcik and Joris Roos===<br />
<br />
Title: A triangular Hilbert transform with curvature, I & II.<br />
<br />
Abstract: The triangular Hilbert is a two-dimensional bilinear singular<br />
originating in time-frequency analysis. No Lp bounds are currently<br />
known for this operator.<br />
In these two talks we discuss a recent joint work with Michael Christ<br />
on a variant of the triangular Hilbert transform involving curvature.<br />
This object is closely related to the bilinear Hilbert transform with<br />
curvature and a maximally modulated singular integral of Stein-Wainger<br />
type. As an application we also discuss a quantitative nonlinear Roth<br />
type theorem on patterns in the Euclidean plane.<br />
The second talk will focus on the proof of a key ingredient, a certain<br />
regularity estimate for a local operator.<br />
<br />
===Andrew Zimmer===<br />
<br />
Title: Complex analytic problems on domains with good intrinsic geometry<br />
<br />
Abstract: In this talk, I will describe a new class of domains in complex Euclidean space which is defined in terms of the existence of a Kaehler metric with good geometric properties. This class is invariant under biholomorphism and includes many well-studied classes of domains such as strongly pseudoconvex domains, finite type domains in dimension two, convex domains, homogeneous domains, and embeddings of Teichmuller spaces. Further, certain analytic problems are tractable for domains in this family even when the boundary is non-smooth. In particular, it is possible to characterize the domains in this family where the dbar-Neumann operator on (0, q)-forms is compact (which generalizes an old result of Fu-Straube for convex domains).<br />
<br />
===Hong Wang===<br />
<br />
Title: Improved decoupling for the parabola<br />
<br />
Abstract: In 2014, Bourgain and Demeter proved the $l^2$ decoupling estimates for the paraboloid with constant $R^{\epsilon}$. <br />
We prove an $(l^2, L^6)$ decoupling inequality for the parabola with constant $(\log R)^c$. This is joint work with Larry Guth and Dominique Maldague.<br />
<br />
===Kevin Luli===<br />
<br />
Title: Smooth Nonnegative Interpolation<br />
<br />
Abstract: Suppose E is an arbitrary subset of R^n. Let f: E \rightarrow [0, \infty). How can we decide if f extends to a nonnegative function C^m function F defined on all of R^n? Suppose E is finite. Can we compute a nonnegative C^m function F on R^n that agrees with f on E with the least possible C^m norm? How many computer operations does this take? In this talk, I will explain recent results on these problems. Non-negativity is one of the most important shape preserving properties for interpolants. In real life applications, the range of the interpolant is imposed by nature. For example, probability density, the amount of snow, rain, humidity, chemical concentration are all nonnegative quantities and are of interest in natural sciences. Even in one dimension, the existing techniques can only handle nonnegative interpolation under special assumptions on the data set. Our results work without any assumptions on the data sets. <br />
<br />
===Niclas Technau===<br />
<br />
Title: Number theoretic applications of oscillatory integrals<br />
<br />
Abstract: We discuss how the analysis of oscillatory integrals can be used to solve number theoretic problems. More specifically, the focus will be on understanding fine-scale statistics of sequences on the unit circle. Further, we shall briefly explain a connection to quantum chaos.<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]<br />
<br />
<br />
Graduate Student Seminar:<br />
<br />
https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html</div>Beltranhttps://hilbert.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=20091Analysis Seminar2020-10-07T12:51:11Z<p>Beltran: /* Current Analysis Seminar Schedule */</p>
<hr />
<div><br />
The 2020-2021 Analysis Seminar will be organized by David Beltran and Andreas Seeger.<br />
It will be online at least for the Fall semester, with details to be announced in September.<br />
The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar earlier, or on different days, to accomodate speakers).<br />
<br />
Zoom links will be sent to those who have signed up for the Analysis Seminar List. For instructions how to sign up for seminar lists, see https://www.math.wisc.edu/node/230<br />
<br />
If you'd like to suggest speakers for the spring semester please contact David and Andreas (dbeltran at math, seeger at math).<br />
<br />
<br />
<br />
=[[Previous_Analysis_seminars]]=<br />
<br />
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars<br />
<br />
= Current Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 22<br />
|Alexei Poltoratski<br />
|UW Madison<br />
|[[#Alexei Poltoratski | Dirac inner functions ]]<br />
| <br />
|-<br />
|September 29<br />
|Joris Roos<br />
|University of Massachusetts - Lowell<br />
|[[#Polona Durcik and Joris Rooslinktoabstract | A triangular Hilbert transform with curvature, I ]]<br />
| <br />
|-<br />
|Wednesday September 30, 4 p.m.<br />
|Polona Durcik<br />
|Chapman University<br />
|[[#Polona Durcik and Joris Roos | A triangular Hilbert transform with curvature, II ]]<br />
| <br />
|-<br />
|October 6<br />
|Andrew Zimmer<br />
|UW Madison<br />
|[[#Andrew Zimmer | Complex analytic problems on domains with good intrinsic geometry ]]<br />
| <br />
|-<br />
|October 13<br />
|Hong Wang <br />
|Princeton/IAS<br />
|[[#Hong Wang | Improved decoupling for the parabola ]]<br />
| <br />
|-<br />
|October 20<br />
|Kevin Luli<br />
|UC Davis<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|October 27<br />
|Terence Harris<br />
| Cornell University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|Monday, November 2, 4 p.m.<br />
|Yuval Wigderson<br />
|Stanford University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|November 10<br />
|Óscar Domínguez<br />
| Universidad Complutense de Madrid<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|November 17<br />
|Tamas Titkos<br />
|BBS U of Applied Sciences and Renyi Institute<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|November 24<br />
|Shukun Wu<br />
|University of Illinois (Urbana-Champaign)<br />
||[[#linktoabstract | Title ]] <br />
| <br />
|-<br />
|December 1<br />
| Jonathan Hickman<br />
| The University of Edinburgh<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|December 8<br />
|Alejandra Gaitán<br />
| Purdue University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|February 2<br />
|Jongchon Kim<br />
| UBC<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|February 9<br />
|Bingyang Hu<br />
|Purdue University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|February 16<br />
|David Beltran<br />
|UW - Madison<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|February 23<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 2<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 9<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 16<br />
|Ziming Shi<br />
|Rutgers University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 23<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 30<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 6<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 13<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 20<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 27<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|May 4<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|}<br />
<br />
=Abstracts=<br />
===Alexei Poltoratski===<br />
<br />
Title: Dirac inner functions<br />
<br />
Abstract: My talk will focus on some new (and old) complex analytic objects arising from Dirac systems of differential equations.<br />
We will discuss connections between problems in complex function theory, spectral and scattering problems for differential<br />
operators and the non-linear Fourier transform.<br />
<br />
<br />
===Polona Durcik and Joris Roos===<br />
<br />
Title: A triangular Hilbert transform with curvature, I & II.<br />
<br />
Abstract: The triangular Hilbert is a two-dimensional bilinear singular<br />
originating in time-frequency analysis. No Lp bounds are currently<br />
known for this operator.<br />
In these two talks we discuss a recent joint work with Michael Christ<br />
on a variant of the triangular Hilbert transform involving curvature.<br />
This object is closely related to the bilinear Hilbert transform with<br />
curvature and a maximally modulated singular integral of Stein-Wainger<br />
type. As an application we also discuss a quantitative nonlinear Roth<br />
type theorem on patterns in the Euclidean plane.<br />
The second talk will focus on the proof of a key ingredient, a certain<br />
regularity estimate for a local operator.<br />
<br />
===Andrew Zimmer===<br />
<br />
Title: Complex analytic problems on domains with good intrinsic geometry<br />
<br />
Abstract: In this talk, I will describe a new class of domains in complex Euclidean space which is defined in terms of the existence of a Kaehler metric with good geometric properties. This class is invariant under biholomorphism and includes many well-studied classes of domains such as strongly pseudoconvex domains, finite type domains in dimension two, convex domains, homogeneous domains, and embeddings of Teichmuller spaces. Further, certain analytic problems are tractable for domains in this family even when the boundary is non-smooth. In particular, it is possible to characterize the domains in this family where the dbar-Neumann operator on (0, q)-forms is compact (which generalizes an old result of Fu-Straube for convex domains).<br />
<br />
===Hong Wang===<br />
<br />
Title: Improved decoupling for the parabola<br />
<br />
Abstract: In 2014, Bourgain and Demeter proved the $l^2$ decoupling estimates for the paraboloid with constant $R^{\epsilon}$. <br />
We prove an $(l^2, L^6)$ decoupling inequality for the parabola with constant $(\log R)^c$. This is joint work with Larry Guth and Dominique Maldague.<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]<br />
<br />
<br />
Graduate Student Seminar:<br />
<br />
https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html</div>Beltranhttps://hilbert.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=20090Analysis Seminar2020-10-07T12:50:15Z<p>Beltran: /* Hong Wang */</p>
<hr />
<div><br />
The 2020-2021 Analysis Seminar will be organized by David Beltran and Andreas Seeger.<br />
It will be online at least for the Fall semester, with details to be announced in September.<br />
The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar earlier, or on different days, to accomodate speakers).<br />
<br />
Zoom links will be sent to those who have signed up for the Analysis Seminar List. For instructions how to sign up for seminar lists, see https://www.math.wisc.edu/node/230<br />
<br />
If you'd like to suggest speakers for the spring semester please contact David and Andreas (dbeltran at math, seeger at math).<br />
<br />
<br />
<br />
=[[Previous_Analysis_seminars]]=<br />
<br />
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars<br />
<br />
= Current Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 22<br />
|Alexei Poltoratski<br />
|UW Madison<br />
|[[#Alexei Poltoratski | Dirac inner functions ]]<br />
| <br />
|-<br />
|September 29<br />
|Joris Roos<br />
|University of Massachusetts - Lowell<br />
|[[#Polona Durcik and Joris Rooslinktoabstract | A triangular Hilbert transform with curvature, I ]]<br />
| <br />
|-<br />
|Wednesday September 30, 4 p.m.<br />
|Polona Durcik<br />
|Chapman University<br />
|[[#Polona Durcik and Joris Roos | A triangular Hilbert transform with curvature, II ]]<br />
| <br />
|-<br />
|October 6<br />
|Andrew Zimmer<br />
|UW Madison<br />
|[[#Andrew Zimmer | Complex analytic problems on domains with good intrinsic geometry ]]<br />
| <br />
|-<br />
|October 13<br />
|Hong Wang <br />
|Princeton/IAS<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|October 20<br />
|Kevin Luli<br />
|UC Davis<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|October 27<br />
|Terence Harris<br />
| Cornell University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|Monday, November 2, 4 p.m.<br />
|Yuval Wigderson<br />
|Stanford University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|November 10<br />
|Óscar Domínguez<br />
| Universidad Complutense de Madrid<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|November 17<br />
|Tamas Titkos<br />
|BBS U of Applied Sciences and Renyi Institute<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|November 24<br />
|Shukun Wu<br />
|University of Illinois (Urbana-Champaign)<br />
||[[#linktoabstract | Title ]] <br />
| <br />
|-<br />
|December 1<br />
| Jonathan Hickman<br />
| The University of Edinburgh<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|December 8<br />
|Alejandra Gaitán<br />
| Purdue University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|February 2<br />
|Jongchon Kim<br />
| UBC<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|February 9<br />
|Bingyang Hu<br />
|Purdue University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|February 16<br />
|David Beltran<br />
|UW - Madison<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|February 23<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 2<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 9<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 16<br />
|Ziming Shi<br />
|Rutgers University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 23<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 30<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 6<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 13<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 20<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 27<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|May 4<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|}<br />
<br />
=Abstracts=<br />
===Alexei Poltoratski===<br />
<br />
Title: Dirac inner functions<br />
<br />
Abstract: My talk will focus on some new (and old) complex analytic objects arising from Dirac systems of differential equations.<br />
We will discuss connections between problems in complex function theory, spectral and scattering problems for differential<br />
operators and the non-linear Fourier transform.<br />
<br />
<br />
===Polona Durcik and Joris Roos===<br />
<br />
Title: A triangular Hilbert transform with curvature, I & II.<br />
<br />
Abstract: The triangular Hilbert is a two-dimensional bilinear singular<br />
originating in time-frequency analysis. No Lp bounds are currently<br />
known for this operator.<br />
In these two talks we discuss a recent joint work with Michael Christ<br />
on a variant of the triangular Hilbert transform involving curvature.<br />
This object is closely related to the bilinear Hilbert transform with<br />
curvature and a maximally modulated singular integral of Stein-Wainger<br />
type. As an application we also discuss a quantitative nonlinear Roth<br />
type theorem on patterns in the Euclidean plane.<br />
The second talk will focus on the proof of a key ingredient, a certain<br />
regularity estimate for a local operator.<br />
<br />
===Andrew Zimmer===<br />
<br />
Title: Complex analytic problems on domains with good intrinsic geometry<br />
<br />
Abstract: In this talk, I will describe a new class of domains in complex Euclidean space which is defined in terms of the existence of a Kaehler metric with good geometric properties. This class is invariant under biholomorphism and includes many well-studied classes of domains such as strongly pseudoconvex domains, finite type domains in dimension two, convex domains, homogeneous domains, and embeddings of Teichmuller spaces. Further, certain analytic problems are tractable for domains in this family even when the boundary is non-smooth. In particular, it is possible to characterize the domains in this family where the dbar-Neumann operator on (0, q)-forms is compact (which generalizes an old result of Fu-Straube for convex domains).<br />
<br />
===Hong Wang===<br />
<br />
Title: Improved decoupling for the parabola<br />
<br />
Abstract: In 2014, Bourgain and Demeter proved the $l^2$ decoupling estimates for the paraboloid with constant $R^{\epsilon}$. <br />
We prove an $(l^2, L^6)$ decoupling inequality for the parabola with constant $(\log R)^c$. This is joint work with Larry Guth and Dominique Maldague.<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]<br />
<br />
<br />
Graduate Student Seminar:<br />
<br />
https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html</div>Beltranhttps://hilbert.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=20037Analysis Seminar2020-09-30T17:41:24Z<p>Beltran: /* Current Analysis Seminar Schedule */</p>
<hr />
<div><br />
The 2020-2021 Analysis Seminar will be organized by David Beltran and Andreas Seeger.<br />
It will be online at least for the Fall semester, with details to be announced in September.<br />
The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar earlier, or on different days, to accomodate speakers).<br />
<br />
Zoom links will be sent to those who have signed up for the Analysis Seminar List. For instructions how to sign up for seminar lists, see https://www.math.wisc.edu/node/230<br />
<br />
If you'd like to suggest speakers for the spring semester please contact David and Andreas (dbeltran at math, seeger at math).<br />
<br />
<br />
<br />
=[[Previous_Analysis_seminars]]=<br />
<br />
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars<br />
<br />
= Current Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 22<br />
|Alexei Poltoratski<br />
|UW Madison<br />
|[[#Alexei Poltoratski | Dirac inner functions ]]<br />
| <br />
|-<br />
|September 29<br />
|Joris Roos<br />
|University of Massachusetts - Lowell<br />
|[[#Polona Durcik and Joris Rooslinktoabstract | A triangular Hilbert transform with curvature, I ]]<br />
| <br />
|-<br />
|Wednesday September 30, 4 p.m.<br />
|Polona Durcik<br />
|Chapman University<br />
|[[#Polona Durcik and Joris Roos | A triangular Hilbert transform with curvature, II ]]<br />
| <br />
|-<br />
|October 6<br />
|Andrew Zimmer<br />
|UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|October 13<br />
|Hong Wang <br />
|Princeton/IAS<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|October 20<br />
|Kevin Luli<br />
|UC Davis<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|October 27<br />
|Terence Harris<br />
| Cornell University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|Monday, November 2, 4 p.m.<br />
|Yuval Wigderson<br />
|Stanford University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|November 10<br />
|Óscar Domínguez<br />
| Universidad Complutense de Madrid<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|November 17<br />
|Tamas Titkos<br />
|BBS U of Applied Sciences and Renyi Institute<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|November 24<br />
|Shukun Wu<br />
|University of Illinois (Urbana-Champaign)<br />
||[[#linktoabstract | Title ]] <br />
| <br />
|-<br />
|December 1<br />
| Jonathan Hickman<br />
| The University of Edinburgh<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|December 8<br />
|Alejandra Gaitán<br />
| Purdue University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|February 2<br />
|Jongchon Kim<br />
| UBC<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|February 9<br />
|Bingyang Hu<br />
|Purdue University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|February 16<br />
|David Beltran<br />
|UW - Madison<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|February 23<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 2<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 9<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 16<br />
|Ziming Shi<br />
|Rutgers University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 23<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 30<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 6<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 13<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 20<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 27<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|May 4<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|}<br />
<br />
=Abstracts=<br />
===Alexei Poltoratski===<br />
<br />
Title: Dirac inner functions<br />
<br />
Abstract: My talk will focus on some new (and old) complex analytic objects arising from Dirac systems of differential equations.<br />
We will discuss connections between problems in complex function theory, spectral and scattering problems for differential<br />
operators and the non-linear Fourier transform.<br />
<br />
<br />
===Polona Durcik and Joris Roos===<br />
<br />
Title: A triangular Hilbert transform with curvature, I & II.<br />
<br />
Abstract: The triangular Hilbert is a two-dimensional bilinear singular<br />
originating in time-frequency analysis. No Lp bounds are currently<br />
known for this operator.<br />
In these two talks we discuss a recent joint work with Michael Christ<br />
on a variant of the triangular Hilbert transform involving curvature.<br />
This object is closely related to the bilinear Hilbert transform with<br />
curvature and a maximally modulated singular integral of Stein-Wainger<br />
type. As an application we also discuss a quantitative nonlinear Roth<br />
type theorem on patterns in the Euclidean plane.<br />
The second talk will focus on the proof of a key ingredient, a certain<br />
regularity estimate for a local operator.<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]<br />
<br />
<br />
Graduate Student Seminar:<br />
<br />
https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html</div>Beltranhttps://hilbert.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=20036Analysis Seminar2020-09-30T17:40:54Z<p>Beltran: /* Current Analysis Seminar Schedule */</p>
<hr />
<div><br />
The 2020-2021 Analysis Seminar will be organized by David Beltran and Andreas Seeger.<br />
It will be online at least for the Fall semester, with details to be announced in September.<br />
The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar earlier, or on different days, to accomodate speakers).<br />
<br />
Zoom links will be sent to those who have signed up for the Analysis Seminar List. For instructions how to sign up for seminar lists, see https://www.math.wisc.edu/node/230<br />
<br />
If you'd like to suggest speakers for the spring semester please contact David and Andreas (dbeltran at math, seeger at math).<br />
<br />
<br />
<br />
=[[Previous_Analysis_seminars]]=<br />
<br />
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars<br />
<br />
= Current Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 22<br />
|Alexei Poltoratski<br />
|UW Madison<br />
|[[#Alexei Poltoratski | Dirac inner functions ]]<br />
| <br />
|-<br />
|September 29<br />
|Joris Roos<br />
|University of Massachusetts - Lowell<br />
|[[#Polona Durcik and Joris Rooslinktoabstract | A triangular Hilbert transform with curvature, I ]]<br />
| <br />
|-<br />
|Wednesday September 30, 4 p.m.<br />
|Polona Durcik<br />
|Chapman University<br />
|[[#Polona Durcik and Joris Roos | A triangular Hilbert transform with curvature, II ]]<br />
| <br />
|-<br />
|October 6<br />
|Andrew Zimmer<br />
|UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|October 13<br />
|Hong Wang <br />
|Princeton/IAS<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|October 20<br />
|Kevin Luli<br />
|UC Davis<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|October 27<br />
|Terence Harris<br />
| Cornell University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|Monday, November 2, 4 p.m.<br />
|Yuval Wigderson<br />
|Stanford University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|November 10<br />
|Óscar Domínguez<br />
| Universidad Complutense de Madrid<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|November 17<br />
|Tamas Titkos<br />
|BBS U of Applied Sciences and Renyi Institute<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|November 24<br />
|Shukun Wu<br />
|University of Illinois (Urbana-Champaign)<br />
||[[#linktoabstract | Title ]] <br />
| <br />
|-<br />
|December 1<br />
| Jonathan Hickman<br />
| The University of Edinburgh<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|December 8<br />
|Alejandra Gaitán<br />
|Purdue University<br />
| <br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|February 2<br />
|Jongchon Kim<br />
| UBC<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|February 9<br />
|Bingyang Hu<br />
|Purdue University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|February 16<br />
|David Beltran<br />
|UW - Madison<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|February 23<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 2<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 9<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 16<br />
|Ziming Shi<br />
|Rutgers University<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 23<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|March 30<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 6<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 13<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 20<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|April 27<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|<br />
|-<br />
|May 4<br />
|<br />
|<br />
|[[#linktoabstract | Title ]]<br />
|}<br />
<br />
=Abstracts=<br />
===Alexei Poltoratski===<br />
<br />
Title: Dirac inner functions<br />
<br />
Abstract: My talk will focus on some new (and old) complex analytic objects arising from Dirac systems of differential equations.<br />
We will discuss connections between problems in complex function theory, spectral and scattering problems for differential<br />
operators and the non-linear Fourier transform.<br />
<br />
<br />
===Polona Durcik and Joris Roos===<br />
<br />
Title: A triangular Hilbert transform with curvature, I & II.<br />
<br />
Abstract: The triangular Hilbert is a two-dimensional bilinear singular<br />
originating in time-frequency analysis. No Lp bounds are currently<br />
known for this operator.<br />
In these two talks we discuss a recent joint work with Michael Christ<br />
on a variant of the triangular Hilbert transform involving curvature.<br />
This object is closely related to the bilinear Hilbert transform with<br />
curvature and a maximally modulated singular integral of Stein-Wainger<br />
type. As an application we also discuss a quantitative nonlinear Roth<br />
type theorem on patterns in the Euclidean plane.<br />
The second talk will focus on the proof of a key ingredient, a certain<br />
regularity estimate for a local operator.<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]<br />
<br />
<br />
Graduate Student Seminar:<br />
<br />
https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html</div>Beltranhttps://hilbert.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=19838Analysis Seminar2020-09-17T21:54:05Z<p>Beltran: /* Current Analysis Seminar Schedule */</p>
<hr />
<div><br />
The 2020-2021 Analysis Seminar will be organized by David Beltran and Andreas Seeger.<br />
It will be online at least for the Fall semester, with details to be announced in September.<br />
The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar earlier to accomodate speakers).<br />
<br />
Zoom links will be sent to those who have signed up for the Analysis Seminar List. For instructions how to sign up for seminar lists, see https://www.math.wisc.edu/node/230<br />
<br />
If you'd like to suggest speakers for the spring semester please contact David and Andreas (dbeltran at math, seeger at math).<br />
<br />
<br />
<br />
=[[Previous_Analysis_seminars]]=<br />
<br />
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars<br />
<br />
= Current Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 22<br />
|Alexei Poltoratski<br />
|UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|September 29<br />
|Polona Durcik<br />
| Chapman University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|September 30<br />
|Joris Roos<br />
|University of Massachusetts - Lowell<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|October 6<br />
|Andrew Zimmer<br />
|UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|October 13<br />
|Hong Wang <br />
|Princeton/IAS<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|October 20<br />
|Kevin Luli<br />
|UC Davis<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|October 27<br />
|Terence Harris<br />
| Cornell University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|November 3<br />
|No seminar<br />
| <br />
|<br />
| <br />
|-<br />
|November 10<br />
|Óscar Domínguez<br />
| Universidad Complutense de Madrid<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|November 17<br />
|Tamas Titkos<br />
|BBS U of Applied Sciences and Renyi Institute<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|November 24<br />
|Shukun Wu<br />
|University of Illinois (Urbana-Champaign)<br />
||[[#linktoabstract | Title ]] <br />
| <br />
|-<br />
|December 1<br />
| Jonathan Hickman<br />
| The University of Edinburgh<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|December 8<br />
|<br />
| <br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|February 2<br />
|Jongchon Kim<br />
| UBC<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|February 9<br />
|Bingyang Hu<br />
|Purdue University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|February 16<br />
|David Beltran<br />
|UW - Madison<br />
|[[#linktoabstract | Title ]]<br />
|}<br />
<br />
=Abstracts=<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]<br />
<br />
<br />
Graduate Student Seminar:<br />
<br />
https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html</div>Beltranhttps://hilbert.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=19837Analysis Seminar2020-09-17T21:39:18Z<p>Beltran: /* Current Analysis Seminar Schedule */</p>
<hr />
<div><br />
The 2020-2021 Analysis Seminar will be organized by David Beltran and Andreas Seeger.<br />
It will be online at least for the Fall semester, with details to be announced in September.<br />
The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar earlier to accomodate speakers).<br />
<br />
Zoom links will be sent to those who have signed up for the Analysis Seminar List. For instructions how to sign up for seminar lists, see https://www.math.wisc.edu/node/230<br />
<br />
If you'd like to suggest speakers for the spring semester please contact David and Andreas (dbeltran at math, seeger at math).<br />
<br />
<br />
<br />
=[[Previous_Analysis_seminars]]=<br />
<br />
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars<br />
<br />
= Current Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 22<br />
|Alexei Poltoratski<br />
|UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|September 29<br />
|Polona Durcik<br />
| Chapman University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|September 30<br />
|Joris Roos<br />
|University of Massachusetts - Lowell<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|October 6<br />
|Andrew Zimmer<br />
|UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|October 13<br />
|Hong Wang <br />
|Princeton/IAS<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|October 20<br />
|Kevin Luli<br />
|UC Davis<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|October 27<br />
|Terence Harris<br />
| Cornell University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|November 3<br />
|No seminar<br />
| <br />
|<br />
| <br />
|-<br />
|November 10<br />
|Óscar Domínguez<br />
| Universidad Complutense de Madrid<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|November 17<br />
|Tamas Titkos<br />
|BBS U of Applied Sciences and Renyi Institute<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|November 24<br />
|Shukun Wu<br />
|University of Illinois (Urbana-Champaign)<br />
||[[#linktoabstract | Title ]] <br />
| <br />
|-<br />
|December 1<br />
| Jonathan Hickman<br />
| The University of Edinburgh<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|December 8<br />
|<br />
| <br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|February 2<br />
|Jongchon Kim<br />
| UBC<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|February 9<br />
|Bingyang Hu<br />
|Purdue University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
<br />
|}<br />
<br />
=Abstracts=<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]<br />
<br />
<br />
Graduate Student Seminar:<br />
<br />
https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html</div>Beltranhttps://hilbert.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=19540Analysis Seminar2020-08-12T22:35:43Z<p>Beltran: /* Current Analysis Seminar Schedule */</p>
<hr />
<div><br />
The 2020-2021 Analysis Seminar will be organized by David Beltran and Andreas Seeger.<br />
It will be online at least for the Fall semester, with details to be announced in September.<br />
<br />
If you'd like to suggest speakers for the spring semester please contact David and Andreas (dbeltran at math, seeger at math).<br />
<br />
<br />
<br />
=[[Previous_Analysis_seminars]]=<br />
<br />
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars<br />
<br />
= Current Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 22<br />
|Alexei Poltoratski<br />
|UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|September 29<br />
|Polona Durcik<br />
| Chapman University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|September 30<br />
|Joris Roos<br />
|University of Massachusetts - Lowell<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|October 6<br />
|Andrew Zimmer<br />
|UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|October 13<br />
|Hong Wang <br />
|Princeton/IAS<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|October 20<br />
|Kevin Luli<br />
|UC Davis<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|October 27<br />
|Terence Harris<br />
| Cornell University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|November 3<br />
|No seminar<br />
| <br />
|<br />
| <br />
|-<br />
|November 10<br />
|Óscar Domínguez<br />
| Universidad Complutense de Madrid<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|November 17<br />
|Tamas Titkos<br />
|BBS U of Applied Sciences and Renyi Institute<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|November 24<br />
| No seminar<br />
| <br />
|<br />
| <br />
|-<br />
|December 1<br />
| Jonathan Hickman<br />
| The University of Edinburgh<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|December 8<br />
|David Beltran<br />
| UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
<br />
|}<br />
<br />
=Abstracts=<br />
===Name===<br />
<br />
Title<br />
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Abstract<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
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Abstract<br />
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===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Beltranhttps://hilbert.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=19531Analysis Seminar2020-08-12T14:34:57Z<p>Beltran: /* Current Analysis Seminar Schedule */</p>
<hr />
<div><br />
The 2020-2021 Analysis Seminar will be organized by David Beltran and Andreas Seeger.<br />
It will be online at least for the Fall semester, with details to be announced in September.<br />
<br />
If you'd like to suggest speakers for the spring semester please contact David and Andreas (dbeltran at math, seeger at math).<br />
<br />
<br />
<br />
=[[Previous_Analysis_seminars]]=<br />
<br />
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars<br />
<br />
= Current Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 22<br />
|Alexei Poltoratski<br />
|UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|September 29<br />
|Polona Durcik<br />
| Chapman University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|September 30<br />
|Joris Roos<br />
|University of Massachusetts - Lowell<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|October 6<br />
|Andrew Zimmer<br />
|UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|October 13<br />
|Hong Wang <br />
|Princeton/IAS<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|October 20<br />
|Kevin Luli<br />
|UC Davis<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|October 27<br />
|Terence Harris<br />
| Cornell University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|November 3<br />
|No seminar<br />
| <br />
|<br />
| <br />
|-<br />
|November 10<br />
|Óscar Domínguez<br />
| Universidad Complutense de Madrid<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|November 17<br />
|Tamas Titkos<br />
|BBS U of Applied Sciences and Renyi Institute<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|November 24<br />
| No seminar<br />
| <br />
|<br />
| <br />
|-<br />
|December 1<br />
| TBA<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|December 8<br />
|TBA<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
<br />
|}<br />
<br />
=Abstracts=<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Beltranhttps://hilbert.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=19521Analysis Seminar2020-08-05T18:19:50Z<p>Beltran: /* Current Analysis Seminar Schedule */</p>
<hr />
<div><br />
The 2020-2021 Analysis Seminar will be organized by David Beltran and Andreas Seeger.<br />
It will be online at least for the Fall semester, with details to be announced in September.<br />
<br />
If you'd like to suggest speakers for the spring semester please contact David and Andreas (dbeltran at math, seeger at math).<br />
<br />
<br />
<br />
=[[Previous_Analysis_seminars]]=<br />
<br />
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars<br />
<br />
= Current Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 22<br />
|Alexei Poltoratski<br />
|UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|September 29<br />
|Polona Durcik<br />
| Chapman University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|September 30<br />
|Joris Roos<br />
|University of Massachusetts - Lowell<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|October 6<br />
|Andrew Zimmer<br />
|UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|October 13<br />
|Hong Wang <br />
|Princeton/IAS<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|October 20<br />
|Kevin Luli<br />
|UC Davis<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|October 27<br />
|Terence Harris<br />
| Cornell University<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|November 3<br />
|No seminar<br />
| <br />
|<br />
| <br />
|-<br />
|November 10<br />
|TBA<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|November 17<br />
|Tamas Titkos<br />
|BBS U of Applied Sciences and Renyi Institute<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|November 24<br />
| No seminar<br />
| <br />
|<br />
| <br />
|-<br />
|December 1<br />
| TBA<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
|December 8<br />
|TBA<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| <br />
|-<br />
<br />
|}<br />
<br />
=Abstracts=<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
<br />
===Name===<br />
<br />
Title<br />
<br />
Abstract<br />
<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Beltranhttps://hilbert.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=18126Analysis Seminar2019-10-07T21:06:02Z<p>Beltran: /* David Beltran */</p>
<hr />
<div>'''Fall 2019 and Spring 2020 Analysis Seminar Series<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Brian at street(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sept 10<br />
| José Madrid<br />
| UCLA<br />
|[[#José Madrid | On the regularity of maximal operators on Sobolev Spaces ]]<br />
| Andreas, David<br />
|-<br />
|Sept 13 (Friday, B139)<br />
| Yakun Xi<br />
| University of Rochester<br />
|[[#Yakun Xi | Distance sets on Riemannian surfaces and microlocal decoupling inequalities ]]<br />
| Shaoming<br />
|-<br />
|Sept 17<br />
| Joris Roos<br />
| UW Madison<br />
|[[#Joris Roos | L^p improving estimates for maximal spherical averages ]]<br />
| Brian<br />
|-<br />
|Sept 20 (2:25 PM Friday, Room B139 VV)<br />
| Xiaojun Huang<br />
| Rutgers University–New Brunswick<br />
|[[#linktoabstract | A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries ]]<br />
| Xianghong<br />
|-<br />
|Sept 24<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Oct 1<br />
| Xiaocheng Li<br />
| UW Madison<br />
|[[#Xiaocheng Li | An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$ ]]<br />
| Simon<br />
|-<br />
|Oct 8<br />
| Jeff Galkowski<br />
| Northeastern University<br />
|[[#Jeff Galkowski | Concentration and Growth of Laplace Eigenfunctions ]]<br />
| Betsy<br />
|-<br />
|Oct 15<br />
| David Beltran<br />
| UW Madison<br />
|[[#David Beltran | Regularity of the centered fractional maximal function ]]<br />
| Brian<br />
|-<br />
|Oct 22<br />
| Laurent Stolovitch<br />
| University of Nice Sophia-Antipolis<br />
|[[#linktoabstract | Title ]]<br />
| Xianghong<br />
|-<br />
|<b>Wednesday Oct 23 in B129</b><br />
|Dominique Kemp<br />
|Indiana University<br />
|[[#Dominique Kemp | Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature]]<br />
|Betsy<br />
|-<br />
|Oct 29<br />
| Bingyang Hu<br />
| UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Nov 5<br />
| Kevin O'Neill<br />
| UC Davis<br />
|[[#Kevin O'Neill | A Quantitative Stability Theorem for Convolution on the Heisenberg Group ]]<br />
| Betsy<br />
|-<br />
|Nov 12<br />
| Francesco di Plinio<br />
| Washington University in St. Louis<br />
|[[#linktoabstract | Title ]]<br />
| Shaoming<br />
|-<br />
|Nov 19<br />
| Joao Ramos<br />
| University of Bonn<br />
|[[#linktoabstract | Title ]]<br />
| Joris, Shaoming<br />
|-<br />
|Nov 26<br />
| No Seminar<br />
| <br />
|<br />
| <br />
|-<br />
|Dec 3<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Dec 10<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Jan 21<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Jan 28<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 4<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 11<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 18<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 25<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 3<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 10<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 17<br />
| Spring Break!<br />
|<br />
|<br />
| <br />
|-<br />
|Mar 24<br />
| Oscar Dominguez<br />
| Universidad Complutense de Madrid<br />
|[[#linktoabstract | Title ]]<br />
| Andreas<br />
|-<br />
|Mar 31<br />
| Reserved<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Apr 7<br />
| Hong Wang<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Apr 14<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Apr 21<br />
| Diogo Oliveira e Silva<br />
| University of Birmingham<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Apr 28<br />
| No Seminar<br />
|-<br />
|May 5<br />
|Jonathan Hickman<br />
|University of Edinburgh<br />
|[[#linktoabstract | Title ]]<br />
| Andreas<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===José Madrid===<br />
<br />
Title: On the regularity of maximal operators on Sobolev Spaces<br />
<br />
Abstract: In this talk, we will discuss the regularity properties (boundedness and<br />
continuity) of the classical and fractional maximal<br />
operators when these act on the Sobolev space W^{1,p}(\R^n). We will<br />
focus on the endpoint case p=1. We will talk about<br />
some recent results and current open problems.<br />
<br />
===Yakun Xi===<br />
<br />
Title: Distance sets on Riemannian surfaces and microlocal decoupling inequalities <br />
<br />
Abstract: In this talk, we discuss the generalization of the Falconer distance problem to the Riemannian setting. In particular, we extend the recent result of Guth-Iosevich-Ou-Wang for the distance set in the plane to general Riemannian surfaces. The key new ingredient is a family of refined decoupling inequalities associated with phase functions that satisfy Carleson-Sj\”olin condition. This is joint work with Iosevich and Liu.<br />
<br />
===Joris Roos===<br />
<br />
Title: L^p improving estimates for maximal spherical averages<br />
<br />
Abstract: For a given compact set of radii $E$ we will discuss $L^p$ improving properties of maximal spherical averages with a supremum over $E$.<br />
Our results are sharp up to endpoints for a large class of $E$. A new feature is that the optimal exponents depend on both, the upper Minkowski dimension and the Assouad dimension of the set $E$.<br />
Joint work with Tess Anderson, Kevin Hughes and Andreas Seeger.<br />
<br />
===Xiaojun Huang===<br />
<br />
Title: A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries<br />
<br />
Abstract: This is a joint work with Ming Xiao. We discuss how to construct a hyperbolic metric over a Stein space with spherical boundary. The technique we use is to employ holomorphic continuation along curves for multiple valued functions.<br />
<br />
<br />
<br />
<br />
===Xiaocheng Li===<br />
<br />
Title: An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$<br />
<br />
Abstract: We prove an estimate for spherical functions $\phi_\lambda(a)$ on $\mathrm{SL}(3,\mathbb{R})$, establishing uniform decay in the spectral parameter $\lambda$ when the group parameter $a$ is restricted to a compact subset of the abelian subgroup $\mathrm{A}$. In the case of $\mathrm{SL}(3,\mathbb{R})$, it improves a result by J.J. Duistermaat, J.A.C. Kolk and V.S. Varadarajan by removing the limitation that $a$ should remain regular. As in their work, we estimate the oscillatory integral that appears in the integral formula for spherical functions by the method of stationary phase. However, the major difference is that we investigate the stability of the singularities arising from the linearized phase function by classifying their local normal forms when the parameters $\lambda$ and $a$ vary.<br />
<br />
===Jeff Galkowski===<br />
<br />
<b>Concentration and Growth of Laplace Eigenfunctions</b><br />
<br />
In this talk we will discuss a new approach to understanding eigenfunction concentration. We characterize the features that cause an eigenfunction to saturate the standard supremum bounds in terms of the distribution of L^2 mass along geodesic tubes emanating from a point. We also show that the phenomena behind extreme supremum norm growth is identical to that underlying extreme growth of eigenfunctions when averaged along submanifolds. Finally, we use these ideas to understand a variety of measures of concentration; in each case obtaining quantitative improvements over the known bounds.<br />
<br />
===David Beltran===<br />
<br />
Title: Regularity of the centered fractional maximal function<br />
<br />
Abstract: I will report some recent progress regarding the boundedness of the map $f \mapsto |\nabla M_\beta f|$ from the endpoint space $W^{1,1}(\mathbb{R}^d)$ to $L^{d/(d-\beta)}(\mathbb{R}^d)$, where $M_\beta$ denotes the fractional version of the centered Hardy--Littlewood maximal function. A key step in our analysis is a relation between the centered and non-centered fractional maximal functions at the derivative level, which allows to exploit the known techniques in the non-centered case.<br />
<br />
This is joint work with José Madrid.<br />
<br />
===Dominique Kemp===<br />
<br />
<b>Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature</b><br />
<br />
The celebrated l^2 decoupling theorem of Jean Bourgain and Ciprian Demeter presented a new perspective on a range of problems related to hypersurfaces with nonzero Gaussian curvature, such as exponential sum estimates, additive energy estimates, local smoothing, and counting solutions to Diophantine inequalities. The same authors also extended their theory to the n-dimensional cone. Following their steps, we prove optimal l^2 decoupling results for the remaining class of zero-curvature two-dimensional surfaces without umbilical points (the so-called tangent surfaces). We are also able to prove a decoupling theorem for the real analytic surfaces of revolution. These results should be viewed as partial progress toward the goal of proving a decoupling theorem for arbitrary real analytic hypersurfaces.<br />
<br />
<br />
===Kevin O'Neill===<br />
<br />
<b>A Quantitative Stability Theorem for Convolution on the Heisenberg Group </b><br />
<br />
Although convolution on Euclidean space and the Heisenberg group satisfy the same $L^p$ bounds with the same optimal constants, the former has maximizers while the latter does not. However, as work of Christ has shown, it is still possible to characterize near-maximizers. Specifically, any near-maximizing triple of the trilinear form for convolution on the Heisenberg group must be close to a particular type of triple of ordered Gaussians after adjusting by symmetry. In this talk, we will use the expansion method to prove a quantitative version of this characterization.<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Beltranhttps://hilbert.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=18125Analysis Seminar2019-10-07T21:05:28Z<p>Beltran: /* Abstracts */</p>
<hr />
<div>'''Fall 2019 and Spring 2020 Analysis Seminar Series<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Brian at street(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sept 10<br />
| José Madrid<br />
| UCLA<br />
|[[#José Madrid | On the regularity of maximal operators on Sobolev Spaces ]]<br />
| Andreas, David<br />
|-<br />
|Sept 13 (Friday, B139)<br />
| Yakun Xi<br />
| University of Rochester<br />
|[[#Yakun Xi | Distance sets on Riemannian surfaces and microlocal decoupling inequalities ]]<br />
| Shaoming<br />
|-<br />
|Sept 17<br />
| Joris Roos<br />
| UW Madison<br />
|[[#Joris Roos | L^p improving estimates for maximal spherical averages ]]<br />
| Brian<br />
|-<br />
|Sept 20 (2:25 PM Friday, Room B139 VV)<br />
| Xiaojun Huang<br />
| Rutgers University–New Brunswick<br />
|[[#linktoabstract | A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries ]]<br />
| Xianghong<br />
|-<br />
|Sept 24<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Oct 1<br />
| Xiaocheng Li<br />
| UW Madison<br />
|[[#Xiaocheng Li | An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$ ]]<br />
| Simon<br />
|-<br />
|Oct 8<br />
| Jeff Galkowski<br />
| Northeastern University<br />
|[[#Jeff Galkowski | Concentration and Growth of Laplace Eigenfunctions ]]<br />
| Betsy<br />
|-<br />
|Oct 15<br />
| David Beltran<br />
| UW Madison<br />
|[[#David Beltran | Regularity of the centered fractional maximal function ]]<br />
| Brian<br />
|-<br />
|Oct 22<br />
| Laurent Stolovitch<br />
| University of Nice Sophia-Antipolis<br />
|[[#linktoabstract | Title ]]<br />
| Xianghong<br />
|-<br />
|<b>Wednesday Oct 23 in B129</b><br />
|Dominique Kemp<br />
|Indiana University<br />
|[[#Dominique Kemp | Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature]]<br />
|Betsy<br />
|-<br />
|Oct 29<br />
| Bingyang Hu<br />
| UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Nov 5<br />
| Kevin O'Neill<br />
| UC Davis<br />
|[[#Kevin O'Neill | A Quantitative Stability Theorem for Convolution on the Heisenberg Group ]]<br />
| Betsy<br />
|-<br />
|Nov 12<br />
| Francesco di Plinio<br />
| Washington University in St. Louis<br />
|[[#linktoabstract | Title ]]<br />
| Shaoming<br />
|-<br />
|Nov 19<br />
| Joao Ramos<br />
| University of Bonn<br />
|[[#linktoabstract | Title ]]<br />
| Joris, Shaoming<br />
|-<br />
|Nov 26<br />
| No Seminar<br />
| <br />
|<br />
| <br />
|-<br />
|Dec 3<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Dec 10<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Jan 21<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Jan 28<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 4<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 11<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 18<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 25<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 3<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 10<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 17<br />
| Spring Break!<br />
|<br />
|<br />
| <br />
|-<br />
|Mar 24<br />
| Oscar Dominguez<br />
| Universidad Complutense de Madrid<br />
|[[#linktoabstract | Title ]]<br />
| Andreas<br />
|-<br />
|Mar 31<br />
| Reserved<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Apr 7<br />
| Hong Wang<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Apr 14<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Apr 21<br />
| Diogo Oliveira e Silva<br />
| University of Birmingham<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Apr 28<br />
| No Seminar<br />
|-<br />
|May 5<br />
|Jonathan Hickman<br />
|University of Edinburgh<br />
|[[#linktoabstract | Title ]]<br />
| Andreas<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===José Madrid===<br />
<br />
Title: On the regularity of maximal operators on Sobolev Spaces<br />
<br />
Abstract: In this talk, we will discuss the regularity properties (boundedness and<br />
continuity) of the classical and fractional maximal<br />
operators when these act on the Sobolev space W^{1,p}(\R^n). We will<br />
focus on the endpoint case p=1. We will talk about<br />
some recent results and current open problems.<br />
<br />
===Yakun Xi===<br />
<br />
Title: Distance sets on Riemannian surfaces and microlocal decoupling inequalities <br />
<br />
Abstract: In this talk, we discuss the generalization of the Falconer distance problem to the Riemannian setting. In particular, we extend the recent result of Guth-Iosevich-Ou-Wang for the distance set in the plane to general Riemannian surfaces. The key new ingredient is a family of refined decoupling inequalities associated with phase functions that satisfy Carleson-Sj\”olin condition. This is joint work with Iosevich and Liu.<br />
<br />
===Joris Roos===<br />
<br />
Title: L^p improving estimates for maximal spherical averages<br />
<br />
Abstract: For a given compact set of radii $E$ we will discuss $L^p$ improving properties of maximal spherical averages with a supremum over $E$.<br />
Our results are sharp up to endpoints for a large class of $E$. A new feature is that the optimal exponents depend on both, the upper Minkowski dimension and the Assouad dimension of the set $E$.<br />
Joint work with Tess Anderson, Kevin Hughes and Andreas Seeger.<br />
<br />
===Xiaojun Huang===<br />
<br />
Title: A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries<br />
<br />
Abstract: This is a joint work with Ming Xiao. We discuss how to construct a hyperbolic metric over a Stein space with spherical boundary. The technique we use is to employ holomorphic continuation along curves for multiple valued functions.<br />
<br />
<br />
<br />
<br />
===Xiaocheng Li===<br />
<br />
Title: An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$<br />
<br />
Abstract: We prove an estimate for spherical functions $\phi_\lambda(a)$ on $\mathrm{SL}(3,\mathbb{R})$, establishing uniform decay in the spectral parameter $\lambda$ when the group parameter $a$ is restricted to a compact subset of the abelian subgroup $\mathrm{A}$. In the case of $\mathrm{SL}(3,\mathbb{R})$, it improves a result by J.J. Duistermaat, J.A.C. Kolk and V.S. Varadarajan by removing the limitation that $a$ should remain regular. As in their work, we estimate the oscillatory integral that appears in the integral formula for spherical functions by the method of stationary phase. However, the major difference is that we investigate the stability of the singularities arising from the linearized phase function by classifying their local normal forms when the parameters $\lambda$ and $a$ vary.<br />
<br />
===Jeff Galkowski===<br />
<br />
<b>Concentration and Growth of Laplace Eigenfunctions</b><br />
<br />
In this talk we will discuss a new approach to understanding eigenfunction concentration. We characterize the features that cause an eigenfunction to saturate the standard supremum bounds in terms of the distribution of L^2 mass along geodesic tubes emanating from a point. We also show that the phenomena behind extreme supremum norm growth is identical to that underlying extreme growth of eigenfunctions when averaged along submanifolds. Finally, we use these ideas to understand a variety of measures of concentration; in each case obtaining quantitative improvements over the known bounds.<br />
<br />
===David Beltran===<br />
<br />
Title: Regularity of the centered fractional maximal function<br />
<br />
Abstract: I will report some recent progress regarding the boundedness of the map $f \mapsto |\nabla M_\beta f|$ from the endpoint space $W^{1,1}(\R^d)$ to $L^{d/(d-\beta)}(\R^d)$, where $M_\beta$ denotes the fractional version of the centered Hardy--Littlewood maximal function. A key step in our analysis is a relation between the centered and non-centered fractional maximal functions at the derivative level, which allows to exploit the known techniques in the non-centered case.<br />
<br />
This is joint work with José Madrid.<br />
<br />
===Dominique Kemp===<br />
<br />
<b>Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature</b><br />
<br />
The celebrated l^2 decoupling theorem of Jean Bourgain and Ciprian Demeter presented a new perspective on a range of problems related to hypersurfaces with nonzero Gaussian curvature, such as exponential sum estimates, additive energy estimates, local smoothing, and counting solutions to Diophantine inequalities. The same authors also extended their theory to the n-dimensional cone. Following their steps, we prove optimal l^2 decoupling results for the remaining class of zero-curvature two-dimensional surfaces without umbilical points (the so-called tangent surfaces). We are also able to prove a decoupling theorem for the real analytic surfaces of revolution. These results should be viewed as partial progress toward the goal of proving a decoupling theorem for arbitrary real analytic hypersurfaces.<br />
<br />
<br />
===Kevin O'Neill===<br />
<br />
<b>A Quantitative Stability Theorem for Convolution on the Heisenberg Group </b><br />
<br />
Although convolution on Euclidean space and the Heisenberg group satisfy the same $L^p$ bounds with the same optimal constants, the former has maximizers while the latter does not. However, as work of Christ has shown, it is still possible to characterize near-maximizers. Specifically, any near-maximizing triple of the trilinear form for convolution on the Heisenberg group must be close to a particular type of triple of ordered Gaussians after adjusting by symmetry. In this talk, we will use the expansion method to prove a quantitative version of this characterization.<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Beltranhttps://hilbert.math.wisc.edu/wiki/index.php?title=Analysis_Seminar&diff=18124Analysis Seminar2019-10-07T20:53:47Z<p>Beltran: /* Analysis Seminar Schedule */</p>
<hr />
<div>'''Fall 2019 and Spring 2020 Analysis Seminar Series<br />
'''<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Brian at street(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= Analysis Seminar Schedule =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
|align="left" | '''institution'''<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sept 10<br />
| José Madrid<br />
| UCLA<br />
|[[#José Madrid | On the regularity of maximal operators on Sobolev Spaces ]]<br />
| Andreas, David<br />
|-<br />
|Sept 13 (Friday, B139)<br />
| Yakun Xi<br />
| University of Rochester<br />
|[[#Yakun Xi | Distance sets on Riemannian surfaces and microlocal decoupling inequalities ]]<br />
| Shaoming<br />
|-<br />
|Sept 17<br />
| Joris Roos<br />
| UW Madison<br />
|[[#Joris Roos | L^p improving estimates for maximal spherical averages ]]<br />
| Brian<br />
|-<br />
|Sept 20 (2:25 PM Friday, Room B139 VV)<br />
| Xiaojun Huang<br />
| Rutgers University–New Brunswick<br />
|[[#linktoabstract | A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries ]]<br />
| Xianghong<br />
|-<br />
|Sept 24<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Oct 1<br />
| Xiaocheng Li<br />
| UW Madison<br />
|[[#Xiaocheng Li | An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$ ]]<br />
| Simon<br />
|-<br />
|Oct 8<br />
| Jeff Galkowski<br />
| Northeastern University<br />
|[[#Jeff Galkowski | Concentration and Growth of Laplace Eigenfunctions ]]<br />
| Betsy<br />
|-<br />
|Oct 15<br />
| David Beltran<br />
| UW Madison<br />
|[[#David Beltran | Regularity of the centered fractional maximal function ]]<br />
| Brian<br />
|-<br />
|Oct 22<br />
| Laurent Stolovitch<br />
| University of Nice Sophia-Antipolis<br />
|[[#linktoabstract | Title ]]<br />
| Xianghong<br />
|-<br />
|<b>Wednesday Oct 23 in B129</b><br />
|Dominique Kemp<br />
|Indiana University<br />
|[[#Dominique Kemp | Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature]]<br />
|Betsy<br />
|-<br />
|Oct 29<br />
| Bingyang Hu<br />
| UW Madison<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Nov 5<br />
| Kevin O'Neill<br />
| UC Davis<br />
|[[#Kevin O'Neill | A Quantitative Stability Theorem for Convolution on the Heisenberg Group ]]<br />
| Betsy<br />
|-<br />
|Nov 12<br />
| Francesco di Plinio<br />
| Washington University in St. Louis<br />
|[[#linktoabstract | Title ]]<br />
| Shaoming<br />
|-<br />
|Nov 19<br />
| Joao Ramos<br />
| University of Bonn<br />
|[[#linktoabstract | Title ]]<br />
| Joris, Shaoming<br />
|-<br />
|Nov 26<br />
| No Seminar<br />
| <br />
|<br />
| <br />
|-<br />
|Dec 3<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Dec 10<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Jan 21<br />
| No Seminar<br />
| <br />
|<br />
|<br />
|-<br />
|Jan 28<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 4<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 11<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 18<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Feb 25<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 3<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 10<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Mar 17<br />
| Spring Break!<br />
|<br />
|<br />
| <br />
|-<br />
|Mar 24<br />
| Oscar Dominguez<br />
| Universidad Complutense de Madrid<br />
|[[#linktoabstract | Title ]]<br />
| Andreas<br />
|-<br />
|Mar 31<br />
| Reserved<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Apr 7<br />
| Hong Wang<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Street<br />
|-<br />
|Apr 14<br />
| Person<br />
| Institution<br />
|[[#linktoabstract | Title ]]<br />
| Sponsor<br />
|-<br />
|Apr 21<br />
| Diogo Oliveira e Silva<br />
| University of Birmingham<br />
|[[#linktoabstract | Title ]]<br />
| Betsy<br />
|-<br />
|Apr 28<br />
| No Seminar<br />
|-<br />
|May 5<br />
|Jonathan Hickman<br />
|University of Edinburgh<br />
|[[#linktoabstract | Title ]]<br />
| Andreas<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
===José Madrid===<br />
<br />
Title: On the regularity of maximal operators on Sobolev Spaces<br />
<br />
Abstract: In this talk, we will discuss the regularity properties (boundedness and<br />
continuity) of the classical and fractional maximal<br />
operators when these act on the Sobolev space W^{1,p}(\R^n). We will<br />
focus on the endpoint case p=1. We will talk about<br />
some recent results and current open problems.<br />
<br />
===Yakun Xi===<br />
<br />
Title: Distance sets on Riemannian surfaces and microlocal decoupling inequalities <br />
<br />
Abstract: In this talk, we discuss the generalization of the Falconer distance problem to the Riemannian setting. In particular, we extend the recent result of Guth-Iosevich-Ou-Wang for the distance set in the plane to general Riemannian surfaces. The key new ingredient is a family of refined decoupling inequalities associated with phase functions that satisfy Carleson-Sj\”olin condition. This is joint work with Iosevich and Liu.<br />
<br />
===Joris Roos===<br />
<br />
Title: L^p improving estimates for maximal spherical averages<br />
<br />
Abstract: For a given compact set of radii $E$ we will discuss $L^p$ improving properties of maximal spherical averages with a supremum over $E$.<br />
Our results are sharp up to endpoints for a large class of $E$. A new feature is that the optimal exponents depend on both, the upper Minkowski dimension and the Assouad dimension of the set $E$.<br />
Joint work with Tess Anderson, Kevin Hughes and Andreas Seeger.<br />
<br />
===Xiaojun Huang===<br />
<br />
Title: A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries<br />
<br />
Abstract: This is a joint work with Ming Xiao. We discuss how to construct a hyperbolic metric over a Stein space with spherical boundary. The technique we use is to employ holomorphic continuation along curves for multiple valued functions.<br />
<br />
<br />
<br />
<br />
===Xiaocheng Li===<br />
<br />
Title: An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$<br />
<br />
Abstract: We prove an estimate for spherical functions $\phi_\lambda(a)$ on $\mathrm{SL}(3,\mathbb{R})$, establishing uniform decay in the spectral parameter $\lambda$ when the group parameter $a$ is restricted to a compact subset of the abelian subgroup $\mathrm{A}$. In the case of $\mathrm{SL}(3,\mathbb{R})$, it improves a result by J.J. Duistermaat, J.A.C. Kolk and V.S. Varadarajan by removing the limitation that $a$ should remain regular. As in their work, we estimate the oscillatory integral that appears in the integral formula for spherical functions by the method of stationary phase. However, the major difference is that we investigate the stability of the singularities arising from the linearized phase function by classifying their local normal forms when the parameters $\lambda$ and $a$ vary.<br />
<br />
===Jeff Galkowski===<br />
<br />
<b>Concentration and Growth of Laplace Eigenfunctions</b><br />
<br />
In this talk we will discuss a new approach to understanding eigenfunction concentration. We characterize the features that cause an eigenfunction to saturate the standard supremum bounds in terms of the distribution of L^2 mass along geodesic tubes emanating from a point. We also show that the phenomena behind extreme supremum norm growth is identical to that underlying extreme growth of eigenfunctions when averaged along submanifolds. Finally, we use these ideas to understand a variety of measures of concentration; in each case obtaining quantitative improvements over the known bounds.<br />
<br />
<br />
<br />
===Dominique Kemp===<br />
<br />
<b>Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature</b><br />
<br />
The celebrated l^2 decoupling theorem of Jean Bourgain and Ciprian Demeter presented a new perspective on a range of problems related to hypersurfaces with nonzero Gaussian curvature, such as exponential sum estimates, additive energy estimates, local smoothing, and counting solutions to Diophantine inequalities. The same authors also extended their theory to the n-dimensional cone. Following their steps, we prove optimal l^2 decoupling results for the remaining class of zero-curvature two-dimensional surfaces without umbilical points (the so-called tangent surfaces). We are also able to prove a decoupling theorem for the real analytic surfaces of revolution. These results should be viewed as partial progress toward the goal of proving a decoupling theorem for arbitrary real analytic hypersurfaces.<br />
<br />
<br />
===Kevin O'Neill===<br />
<br />
<b>A Quantitative Stability Theorem for Convolution on the Heisenberg Group </b><br />
<br />
Although convolution on Euclidean space and the Heisenberg group satisfy the same $L^p$ bounds with the same optimal constants, the former has maximizers while the latter does not. However, as work of Christ has shown, it is still possible to characterize near-maximizers. Specifically, any near-maximizing triple of the trilinear form for convolution on the Heisenberg group must be close to a particular type of triple of ordered Gaussians after adjusting by symmetry. In this talk, we will use the expansion method to prove a quantitative version of this characterization.<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Beltran