Difference between revisions of "Analysis Seminar"
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− | The | + | The 2020-2021 Analysis Seminar will be organized by David Beltran and Andreas Seeger. |
+ | 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). | ||
− | + | 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 | |
− | + | If you'd like to suggest speakers for the spring semester please contact David and Andreas (dbeltran at math, seeger at math). | |
− | = Analysis Seminar Schedule = | + | |
+ | |||
+ | =[[Previous_Analysis_seminars]]= | ||
+ | |||
+ | https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars | ||
+ | |||
+ | = Current Analysis Seminar Schedule = | ||
{| cellpadding="8" | {| cellpadding="8" | ||
!align="left" | date | !align="left" | date | ||
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!align="left" | host(s) | !align="left" | host(s) | ||
|- | |- | ||
− | | | + | |September 22 |
− | | | + | |Alexei Poltoratski |
− | | | + | |UW Madison |
− | |[[# | + | |[[#Alexei Poltoratski | Dirac inner functions ]] |
− | | | + | | |
|- | |- | ||
− | | | + | |September 29 |
− | | | + | |Joris Roos |
− | | University of | + | |University of Massachusetts - Lowell |
− | |[[# | + | |[[#Polona Durcik and Joris Rooslinktoabstract | A triangular Hilbert transform with curvature, I ]] |
− | | | + | | |
|- | |- | ||
− | | | + | |Wednesday September 30, 4 p.m. |
− | | | + | |Polona Durcik |
− | | | + | |Chapman University |
− | |[[#Joris Roos | | + | |[[#Polona Durcik and Joris Roos | A triangular Hilbert transform with curvature, II ]] |
− | | | + | | |
|- | |- | ||
− | | | + | |October 6 |
− | | | + | |Andrew Zimmer |
− | | | + | |UW Madison |
− | |[[# | + | |[[#Andrew Zimmer | Complex analytic problems on domains with good intrinsic geometry ]] |
− | | | + | | |
|- | |- | ||
− | | | + | |October 13 |
− | | | + | |Hong Wang |
− | | | + | |Princeton/IAS |
− | |[[# | + | |[[#Hong Wang | Improved decoupling for the parabola ]] |
− | | | + | | |
|- | |- | ||
− | | | + | |October 20 |
− | | | + | |Kevin Luli |
− | | | + | |UC Davis |
− | |[[# | + | |[[#Kevin Luli | Smooth Nonnegative Interpolation ]] |
− | | | + | | |
|- | |- | ||
− | | | + | |October 21, 4.00 p.m. |
− | | | + | |Niclas Technau |
− | | UW Madison | + | |UW Madison |
− | |[[# | + | |[[#Niclas Technau | Number theoretic applications of oscillatory integrals ]] |
− | | | + | | |
|- | |- | ||
− | | | + | |October 27 |
− | | | + | |Terence Harris |
− | | University | + | | Cornell University |
− | |[[# | + | |[[#Terence Harris | Low dimensional pinned distance sets via spherical averages ]] |
− | | | + | | |
|- | |- | ||
− | | | + | |Monday, November 2, 4 p.m. |
− | | | + | |Yuval Wigderson |
− | | | + | |Stanford University |
− | |[[# | + | |[[#Yuval Wigderson | New perspectives on the uncertainty principle ]] |
− | | | + | | |
|- | |- | ||
− | | | + | |November 10, 10 a.m. |
− | | | + | |Óscar Domínguez |
− | | | + | | Universidad Complutense de Madrid |
− | |[[# | + | |[[#Oscar Dominguez | New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions ]] |
− | | | + | | |
|- | |- | ||
− | | | + | |November 17 |
− | | | + | |Tamas Titkos |
− | | | + | |BBS U of Applied Sciences and Renyi Institute |
− | |[[# | + | |[[#Tamas Titkos | Isometries of Wasserstein spaces ]] |
− | | | + | | |
|- | |- | ||
− | | | + | |November 24 |
− | | | + | |Shukun Wu |
− | | | + | |University of Illinois (Urbana-Champaign) |
− | |[[# | + | ||[[#Shukun Wu | On the Bochner-Riesz operator and the maximal Bochner-Riesz operator ]] |
− | | | + | | |
|- | |- | ||
− | | | + | |December 1 |
− | | | + | | Jonathan Hickman |
− | | | + | | The University of Edinburgh |
− | |[[# | + | |[[#Jonathan Hickman | Sobolev improving for averages over space curves ]] |
− | | | + | | |
|- | |- | ||
− | | | + | |February 2, 7:00 p.m. |
− | | | + | |Hanlong Fang |
− | | | + | |UW Madison |
− | |[[# | + | |[[#Hanlong Fang | Canonical blow-ups of Grassmann manifolds ]] |
− | | | + | | |
|- | |- | ||
− | | | + | |February 9 |
− | | | + | |Bingyang Hu |
+ | |Purdue University | ||
+ | |[[#linktoabstract | Title ]] | ||
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|- | |- | ||
− | | | + | |February 16 |
− | | | + | |Krystal Taylor |
− | | | + | |The Ohio State University |
|[[#linktoabstract | Title ]] | |[[#linktoabstract | Title ]] | ||
− | | | + | | |
|- | |- | ||
− | | | + | |February 23 |
− | | | + | |Dominique Maldague |
− | | | + | |MIT |
|[[#linktoabstract | Title ]] | |[[#linktoabstract | Title ]] | ||
− | | | + | | |
|- | |- | ||
− | | | + | |March 2 |
− | | | + | |Diogo Oliveira e Silva |
− | | | + | |University of Birmingham |
|[[#linktoabstract | Title ]] | |[[#linktoabstract | Title ]] | ||
− | | | + | | |
|- | |- | ||
− | | | + | |March 9 |
− | | | + | |Oleg Safronov |
− | | | + | |University of North Carolina Charlotte |
− | |[[# | + | |[[#linktoabstract | Relations between discrete and continuous spectra of differential operators ]] |
− | | | + | | |
|- | |- | ||
− | | | + | |March 16 |
− | | | + | |Ziming Shi |
− | | | + | |Rutgers University |
|[[#linktoabstract | Title ]] | |[[#linktoabstract | Title ]] | ||
− | | | + | | |
|- | |- | ||
− | | | + | |March 23 |
− | | | + | |TBA |
| | | | ||
+ | |[[#linktoabstract | Title ]] | ||
| | | | ||
− | |||
|- | |- | ||
− | | | + | |March 30, 10:00 a.m. |
− | | | + | |Etienne Le Masson |
− | | | + | |Cergy Paris University |
|[[#linktoabstract | Title ]] | |[[#linktoabstract | Title ]] | ||
− | | | + | | |
|- | |- | ||
− | | | + | |April 6 |
− | | | + | |TBA |
− | | | + | | |
|[[#linktoabstract | Title ]] | |[[#linktoabstract | Title ]] | ||
− | | | + | | |
|- | |- | ||
− | | | + | |April 13 |
− | | | + | |TBA |
− | | | + | | |
|[[#linktoabstract | Title ]] | |[[#linktoabstract | Title ]] | ||
− | | | + | | |
|- | |- | ||
− | | | + | |April 20 |
− | | | + | |Jongchon Kim |
− | | | + | | UBC |
− | |[[#linktoabstract | | + | |[[#linktoabstract | Title ]] |
− | | | + | | |
|- | |- | ||
− | | | + | |April 27 |
− | | | + | |TBA |
− | | | + | | |
− | |||
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|[[#linktoabstract | Title ]] | |[[#linktoabstract | Title ]] | ||
− | | | + | | |
|- | |- | ||
− | + | |May 4 | |
− | + | | | |
− | + | | | |
− | |May | ||
− | | | ||
− | | | ||
|[[#linktoabstract | Title ]] | |[[#linktoabstract | Title ]] | ||
− | |||
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|} | |} | ||
=Abstracts= | =Abstracts= | ||
− | === | + | ===Alexei Poltoratski=== |
− | Title: | + | Title: Dirac inner functions |
− | Abstract: | + | Abstract: My talk will focus on some new (and old) complex analytic objects arising from Dirac systems of differential equations. |
− | + | We will discuss connections between problems in complex function theory, spectral and scattering problems for differential | |
− | + | operators and the non-linear Fourier transform. | |
− | |||
− | |||
− | === | + | ===Polona Durcik and Joris Roos=== |
− | Title: | + | Title: A triangular Hilbert transform with curvature, I & II. |
− | Abstract: In | + | Abstract: The triangular Hilbert is a two-dimensional bilinear singular |
+ | originating in time-frequency analysis. No Lp bounds are currently | ||
+ | known for this operator. | ||
+ | In these two talks we discuss a recent joint work with Michael Christ | ||
+ | on a variant of the triangular Hilbert transform involving curvature. | ||
+ | This object is closely related to the bilinear Hilbert transform with | ||
+ | curvature and a maximally modulated singular integral of Stein-Wainger | ||
+ | type. As an application we also discuss a quantitative nonlinear Roth | ||
+ | type theorem on patterns in the Euclidean plane. | ||
+ | The second talk will focus on the proof of a key ingredient, a certain | ||
+ | regularity estimate for a local operator. | ||
− | === | + | ===Andrew Zimmer=== |
− | Title: | + | Title: Complex analytic problems on domains with good intrinsic geometry |
− | Abstract: | + | 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). |
− | |||
− | |||
+ | ===Hong Wang=== | ||
+ | Title: Improved decoupling for the parabola | ||
− | + | Abstract: In 2014, Bourgain and Demeter proved the $l^2$ decoupling estimates for the paraboloid with constant $R^{\epsilon}$. | |
+ | 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. | ||
− | + | ===Kevin Luli=== | |
− | + | Title: Smooth Nonnegative Interpolation | |
− | + | 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. | |
− | + | ===Niclas Technau=== | |
− | + | Title: Number theoretic applications of oscillatory integrals | |
− | + | 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. | |
− | === | + | ===Terence Harris=== |
− | Title: | + | Title: Low dimensional pinned distance sets via spherical averages |
− | Abstract: | + | 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. |
− | === | + | ===Yuval Wigderson=== |
− | Title: | + | Title: New perspectives on the uncertainty principle |
− | Abstract: | + | 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. |
+ | ===Oscar Dominguez=== | ||
− | + | Title: New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions | |
− | + | 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. | |
− | + | ===Tamas Titkos=== | |
− | + | Title: Isometries of Wasserstein spaces | |
− | + | Abstract: Due to its nice theoretical properties and an astonishing number of | |
+ | applications via optimal transport problems, probably the most | ||
+ | intensively studied metric nowadays is the p-Wasserstein metric. Given | ||
+ | a complete and separable metric space $X$ and a real number $p\geq1$, | ||
+ | one defines the p-Wasserstein space $\mathcal{W}_p(X)$ as the collection | ||
+ | of Borel probability measures with finite $p$-th moment, endowed with a | ||
+ | distance which is calculated by means of transport plans \cite{5}. | ||
− | + | The main aim of our research project is to reveal the structure of the | |
+ | isometry group $\mathrm{Isom}(\mathcal{W}_p(X))$. Although | ||
+ | $\mathrm{Isom}(X)$ embeds naturally into | ||
+ | $\mathrm{Isom}(\mathcal{W}_p(X))$ by push-forward, and this embedding | ||
+ | turned out to be surjective in many cases (see e.g. [1]), these two | ||
+ | groups are not isomorphic in general. Kloeckner in [2] described | ||
+ | the isometry group of the quadratic Wasserstein space | ||
+ | $\mathcal{W}_2(\mathbb{R}^n)$, and it turned out that the case of $n=1$ | ||
+ | is special in the sense that $\mathrm{Isom}(\mathcal{W}_2(\mathbb{R})$ | ||
+ | is extremely rich. Namely, it contains a large subgroup of wild behaving | ||
+ | isometries that distort the shape of measures. Following this line of | ||
+ | investigation, in \cite{3} we described | ||
+ | $\mathrm{Isom}(\mathcal{W}_p(\mathbb{R}))$ and | ||
+ | $\mathrm{Isom}(\mathcal{W}_p([0,1])$ for all $p\geq 1$. | ||
− | + | In this talk I will survey first some of the earlier results in the | |
+ | subject, and then I will present the key results of [3]. If time | ||
+ | permits, I will also report on our most recent manuscript [4] in | ||
+ | which we extended Kloeckner's multidimensional results. Joint work with Gy\"orgy P\'al Geh\'er (University of Reading) | ||
+ | and D\'aniel Virosztek (IST Austria). | ||
− | + | [1] J. Bertrand and B. Kloeckner, \emph{A geometric study of Wasserstein | |
+ | spaces: isometric rigidity in negative curvature}, International | ||
+ | Mathematics Research Notices, 2016 (5), 1368--1386. | ||
− | + | [2] B. Kloeckner, \emph{A geometric study of Wasserstein spaces: Euclidean | |
+ | spaces}, Annali della Scuola Normale Superiore di Pisa - Classe di | ||
+ | Scienze, Serie 5, Tome 9 (2010) no. 2, 297--323. | ||
− | + | [3] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{Isometric study of | |
+ | Wasserstein spaces – the real line}, Trans. Amer. Math. Soc., 373 | ||
+ | (2020), 5855--5883. | ||
− | + | [4] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{The isometry group of | |
+ | Wasserstein spaces: The Hilbertian case}, submitted manuscript. | ||
− | + | [5] C. Villani, \emph{Optimal Transport: Old and New,} | |
+ | (Grundlehren der mathematischen Wissenschaften) | ||
+ | Springer, 2009. | ||
− | + | ===Shukun Wu=== | |
+ | Title: On the Bochner-Riesz operator and the maximal Bochner-Riesz operator | ||
− | + | 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. | |
− | |||
− | + | ===Jonathan Hickman=== | |
− | + | Title: Sobolev improving for averages over space curves | |
− | + | 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. | |
− | + | ===Hanlong Fang=== | |
− | + | Title: Canonical blow-ups of Grassmann manifolds | |
− | + | 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. | |
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+ | =Extras= | ||
+ | [[Blank Analysis Seminar Template]] | ||
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− | + | Graduate Student Seminar: | |
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− | + | https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html | |
− |
Latest revision as of 12:07, 16 January 2021
The 2020-2021 Analysis Seminar will be organized by David Beltran and Andreas Seeger. 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).
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
If you'd like to suggest speakers for the spring semester please contact David and Andreas (dbeltran at math, seeger at math).
Previous_Analysis_seminars
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars
Current Analysis Seminar Schedule
date | speaker | institution | title | host(s) |
---|---|---|---|---|
September 22 | Alexei Poltoratski | UW Madison | Dirac inner functions | |
September 29 | Joris Roos | University of Massachusetts - Lowell | A triangular Hilbert transform with curvature, I | |
Wednesday September 30, 4 p.m. | Polona Durcik | Chapman University | A triangular Hilbert transform with curvature, II | |
October 6 | Andrew Zimmer | UW Madison | Complex analytic problems on domains with good intrinsic geometry | |
October 13 | Hong Wang | Princeton/IAS | Improved decoupling for the parabola | |
October 20 | Kevin Luli | UC Davis | Smooth Nonnegative Interpolation | |
October 21, 4.00 p.m. | Niclas Technau | UW Madison | Number theoretic applications of oscillatory integrals | |
October 27 | Terence Harris | Cornell University | Low dimensional pinned distance sets via spherical averages | |
Monday, November 2, 4 p.m. | Yuval Wigderson | Stanford University | New perspectives on the uncertainty principle | |
November 10, 10 a.m. | Óscar Domínguez | Universidad Complutense de Madrid | New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions | |
November 17 | Tamas Titkos | BBS U of Applied Sciences and Renyi Institute | Isometries of Wasserstein spaces | |
November 24 | Shukun Wu | University of Illinois (Urbana-Champaign) | On the Bochner-Riesz operator and the maximal Bochner-Riesz operator | |
December 1 | Jonathan Hickman | The University of Edinburgh | Sobolev improving for averages over space curves | |
February 2, 7:00 p.m. | Hanlong Fang | UW Madison | Canonical blow-ups of Grassmann manifolds | |
February 9 | Bingyang Hu | Purdue University | Title | |
February 16 | Krystal Taylor | The Ohio State University | Title | |
February 23 | Dominique Maldague | MIT | Title | |
March 2 | Diogo Oliveira e Silva | University of Birmingham | Title | |
March 9 | Oleg Safronov | University of North Carolina Charlotte | Relations between discrete and continuous spectra of differential operators | |
March 16 | Ziming Shi | Rutgers University | Title | |
March 23 | TBA | Title | ||
March 30, 10:00 a.m. | Etienne Le Masson | Cergy Paris University | Title | |
April 6 | TBA | Title | ||
April 13 | TBA | Title | ||
April 20 | Jongchon Kim | UBC | Title | |
April 27 | TBA | Title | ||
May 4 | Title |
Abstracts
Alexei Poltoratski
Title: Dirac inner functions
Abstract: My talk will focus on some new (and old) complex analytic objects arising from Dirac systems of differential equations. We will discuss connections between problems in complex function theory, spectral and scattering problems for differential operators and the non-linear Fourier transform.
Polona Durcik and Joris Roos
Title: A triangular Hilbert transform with curvature, I & II.
Abstract: The triangular Hilbert is a two-dimensional bilinear singular originating in time-frequency analysis. No Lp bounds are currently known for this operator. In these two talks we discuss a recent joint work with Michael Christ on a variant of the triangular Hilbert transform involving curvature. This object is closely related to the bilinear Hilbert transform with curvature and a maximally modulated singular integral of Stein-Wainger type. As an application we also discuss a quantitative nonlinear Roth type theorem on patterns in the Euclidean plane. The second talk will focus on the proof of a key ingredient, a certain regularity estimate for a local operator.
Andrew Zimmer
Title: Complex analytic problems on domains with good intrinsic geometry
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).
Hong Wang
Title: Improved decoupling for the parabola
Abstract: In 2014, Bourgain and Demeter proved the $l^2$ decoupling estimates for the paraboloid with constant $R^{\epsilon}$. 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.
Kevin Luli
Title: Smooth Nonnegative Interpolation
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.
Niclas Technau
Title: Number theoretic applications of oscillatory integrals
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.
Terence Harris
Title: Low dimensional pinned distance sets via spherical averages
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.
Yuval Wigderson
Title: New perspectives on the uncertainty principle
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.
Oscar Dominguez
Title: New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions
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.
Tamas Titkos
Title: Isometries of Wasserstein spaces
Abstract: Due to its nice theoretical properties and an astonishing number of applications via optimal transport problems, probably the most intensively studied metric nowadays is the p-Wasserstein metric. Given a complete and separable metric space $X$ and a real number $p\geq1$, one defines the p-Wasserstein space $\mathcal{W}_p(X)$ as the collection of Borel probability measures with finite $p$-th moment, endowed with a distance which is calculated by means of transport plans \cite{5}.
The main aim of our research project is to reveal the structure of the isometry group $\mathrm{Isom}(\mathcal{W}_p(X))$. Although $\mathrm{Isom}(X)$ embeds naturally into $\mathrm{Isom}(\mathcal{W}_p(X))$ by push-forward, and this embedding turned out to be surjective in many cases (see e.g. [1]), these two groups are not isomorphic in general. Kloeckner in [2] described the isometry group of the quadratic Wasserstein space $\mathcal{W}_2(\mathbb{R}^n)$, and it turned out that the case of $n=1$ is special in the sense that $\mathrm{Isom}(\mathcal{W}_2(\mathbb{R})$ is extremely rich. Namely, it contains a large subgroup of wild behaving isometries that distort the shape of measures. Following this line of investigation, in \cite{3} we described $\mathrm{Isom}(\mathcal{W}_p(\mathbb{R}))$ and $\mathrm{Isom}(\mathcal{W}_p([0,1])$ for all $p\geq 1$.
In this talk I will survey first some of the earlier results in the subject, and then I will present the key results of [3]. If time permits, I will also report on our most recent manuscript [4] in which we extended Kloeckner's multidimensional results. Joint work with Gy\"orgy P\'al Geh\'er (University of Reading) and D\'aniel Virosztek (IST Austria).
[1] J. Bertrand and B. Kloeckner, \emph{A geometric study of Wasserstein spaces: isometric rigidity in negative curvature}, International Mathematics Research Notices, 2016 (5), 1368--1386.
[2] B. Kloeckner, \emph{A geometric study of Wasserstein spaces: Euclidean spaces}, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Tome 9 (2010) no. 2, 297--323.
[3] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{Isometric study of Wasserstein spaces – the real line}, Trans. Amer. Math. Soc., 373 (2020), 5855--5883.
[4] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{The isometry group of Wasserstein spaces: The Hilbertian case}, submitted manuscript.
[5] C. Villani, \emph{Optimal Transport: Old and New,} (Grundlehren der mathematischen Wissenschaften) Springer, 2009.
Shukun Wu
Title: On the Bochner-Riesz operator and the maximal Bochner-Riesz operator
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.
Jonathan Hickman
Title: Sobolev improving for averages over space curves
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.
Hanlong Fang
Title: Canonical blow-ups of Grassmann manifolds
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.
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