Difference between revisions of "Analysis Seminar"

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'''Analysis Seminar
+
'''Fall 2019 and Spring 2020 Analysis Seminar Series
 
'''
 
'''
  
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!align="left" | host(s)
 
!align="left" | host(s)
 
|-
 
|-
|Sept 11
+
|Sept 10
| Simon Marshall
+
| José Madrid
| UW Madison
+
| UCLA
|[[#Simon Marshall | Integrals of eigenfunctions on hyperbolic manifolds ]]
+
|[[#José Madrid |   On the regularity of maximal operators on Sobolev Spaces ]]
|  
+
| Andreas, David
 
|-
 
|-
|'''Wednesday, Sept 12'''
+
|Sept 13 (Friday, B139)
| Gunther Uhlmann 
+
| Yakun Xi
| University of Washington
+
| University of Rochester
| Distinguished Lecture Series
+
|[[#Yakun Xi  |  Distance sets on Riemannian surfaces and microlocal decoupling inequalities ]]
| See colloquium website for location
+
| Shaoming
 
|-
 
|-
|'''Friday, Sept 14'''
+
|Sept 17
| Gunther Uhlmann 
+
| Joris Roos
| University of Washington
+
| UW Madison
| Distinguished Lecture Series
+
|[[#Joris Roos  |  L^p improving estimates for maximal spherical averages ]]
| See colloquium website for location
+
| Brian
 
|-
 
|-
|Sept 18
+
|Sept 20 (2:25 PM Friday, Room B139 VV)
| Grad Student Seminar
+
| Xiaojun Huang
|  
+
| Rutgers University–New Brunswick
|
+
|[[#linktoabstract  |  A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries ]]
|
+
| Xianghong
 
|-
 
|-
|Sept 25
+
|Oct 1
| Grad Student Seminar
+
| Xiaocheng Li
|
+
| UW Madison
|
+
|[[#Xiaocheng Li  |  An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$ ]]
|
+
| Simon
 
|-
 
|-
|Oct 9
+
|Oct 8
| Hong Wang
+
| Jeff Galkowski
| MIT
+
| Northeastern University
|[[#Hong Wang About Falconer distance problem in the plane ]]
+
|[[#Jeff Galkowski Concentration and Growth of Laplace Eigenfunctions ]]
| Ruixiang
+
| Betsy
 
|-
 
|-
|Oct 16
+
|Oct 15
| Polona Durcik
+
| David Beltran
| Caltech
+
| UW Madison
|[[#Polona Durcik Singular Brascamp-Lieb inequalities and extended boxes in R^n ]]
+
|[[#David Beltran Regularity of the centered fractional maximal function ]]
| Joris
+
| Brian
 
|-
 
|-
|Oct 23
+
|Oct 22
| Song-Ying Li
+
| Laurent Stolovitch
| UC Irvine
+
| University of Côte d'Azur
|[[#Song-Ying Li |   Estimates for the first positive eigenvalue of Kohn Laplacian on a pseudo-Hermitian manifold ]]
+
|[[#Laurent Stolovitch | Linearization of neighborhoods of embeddings of complex compact manifolds ]]
| Xianghong  
+
| Xianghong
 
|-
 
|-
|Oct 30
+
|<b>Wednesday Oct 23 in B129</b>
|Grad student seminar
+
|Dominique Kemp
|
+
|Indiana University
|
+
|[[#Dominique Kemp | Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature]]
|
+
|Betsy
 
|-
 
|-
|Nov 6
+
|Oct 29
| Hanlong Fang
+
| Bingyang Hu
 
| UW Madison
 
| UW Madison
|[[#Hanlong Fang |  A generalization of the theorem of Weil and Kodaira on prescribing residues ]]
+
|[[#Bingyang Hu  |    Sparse bounds of singular Radon transforms]]
| Brian
+
| Street
 +
|-
 +
|Nov 5
 +
| Kevin O'Neill
 +
| UC Davis
 +
|[[#Kevin O'Neill |  A Quantitative Stability Theorem for Convolution on the Heisenberg Group ]]
 +
| Betsy
 
|-
 
|-
||'''Monday, Nov. 12, B139'''
+
|Nov 12
| Kyle Hambrook
+
| Francesco di Plinio
| San Jose State University
+
| Washington University in St. Louis
|[[#Kyle Hambrook Fourier Decay and Fourier Restriction for Fractal Measures on Curves ]]
+
|[[#Francesco di Plinio Maximal directional integrals along algebraic and lacunary sets]]
| Andreas
+
| Shaoming
 
|-
 
|-
|Nov 13
+
|Nov 13 (Wednesday)
| Laurent Stolovitch
+
| Xiaochun Li
| Université de Nice - Sophia Antipolis
+
| UIUC
|[[#Laurent Stolovitch Equivalence of Cauchy-Riemann manifolds and multisummability theory ]]
+
|[[#Xiaochun Li Roth's type theorems on progressions]]
|Xianghong
+
| Brian, Shaoming
 
|-
 
|-
|Nov 20
+
|Nov 19
| Grad Student Seminar
+
| Joao Ramos
|  
+
| University of Bonn
|[[#linktoabstract |   ]]
+
|[[#Joao Ramos |   Fourier uncertainty principles, interpolation and uniqueness sets ]]
|  
+
| Joris, Shaoming
 
|-
 
|-
|Nov 27
+
|Nov 26
 
| No Seminar
 
| No Seminar
 
|  
 
|  
|[[#linktoabstract  |    ]]
+
|
 
|  
 
|  
 
|-
 
|-
|Dec 4
+
|Dec 3
 +
| Person
 +
| Institution
 +
|[[#linktoabstract  |  Title ]]
 +
| Sponsor
 +
|-
 +
|Dec 10
 
| No Seminar
 
| No Seminar
|[[#linktoabstract  |    ]]
 
 
|  
 
|  
 +
|
 +
|
 
|-
 
|-
|Jan 22
+
|Jan 21
| Brian Cook
 
| Kent
 
|[[#Brian Cook  |  Equidistribution results for integral points on affine homogenous algebraic varieties ]]
 
| Street
 
|-
 
|Jan 29
 
 
| No Seminar
 
| No Seminar
 
|  
 
|  
|[[#linktoabstract  |    ]]
+
|
 
|
 
|
 
|-
 
|-
|Feb 5, '''B239'''
+
|Jan 28
| Alexei Poltoratski
+
| Person
| Texas A&M
+
| Institution
|[[#Alexei Poltoratski  |  Completeness of exponentials: Beurling-Malliavin and type problems ]]
+
|[[#linktoabstract  |  Title ]]
| Denisov
+
| Sponsor
|-
 
|'''Friday, Feb 8'''
 
| Aaron Naber
 
| Northwestern University
 
|[[#linktoabstract  |  A structure theory for spaces with lower Ricci curvature bounds ]]
 
| See colloquium website for location
 
 
|-
 
|-
|Feb 12
+
|Feb 4
| Shaoming Guo
+
| Person
| UW Madison
+
| Institution
|[[#Shaoming Guo | Polynomial Roth theorems in Salem sets   ]]
+
|[[#linktoabstract  Title ]]
|  
+
| Sponsor
 
|-
 
|-
|'''Wed, Feb 13, B239'''
+
|Feb 11
| Dean Baskin
+
| Person
| TAMU
+
| Institution
|[[# Dean Baskin Radiation fields for wave  equations ]]
+
|[[#linktoabstract Title ]]
| Colloquium
+
| Sponsor
 
|-
 
|-
|'''Friday, Feb 15'''
+
|Feb 18
| Lillian Pierce
+
| Person
| Duke
+
| Institution
|[[#Lillian Pierce Short character sums ]]
+
|[[#linktoabstract Title ]]
| Colloquium
+
| Sponsor
 
|-
 
|-
|'''Monday,  Feb 18, 3:30 p.m, B239.'''
+
|Feb 25
| Daniel Tataru
+
| Person
| UC Berkeley
+
| Institution
|[[#Daniel Tataru A Morawetz inequality for water waves ]]
+
|[[#linktoabstract Title ]]
| PDE Seminar
+
| Sponsor
 
|-
 
|-
|Feb 19
+
|Mar 3
| Wenjia Jing
+
| Person
|Tsinghua University
+
| Institution
|Periodic homogenization of Dirichlet problems in perforated domains: a unified proof
+
|[[#linktoabstract |   Title ]]
| PDE Seminar
+
| Sponsor
|-
 
|Feb 26
 
| No Seminar
 
|
 
|
 
 
|-
 
|-
|Mar 5
+
|Mar 10
| Loredana Lanzani
+
| Yifei Pan
| Syracuse University
+
| Indiana University-Purdue University Fort Wayne
|[[#Loredana Lanzani On regularity and irregularity of the Cauchy-Szegő projection in several complex variables ]]
+
|[[#linktoabstract Title ]]
 
| Xianghong
 
| Xianghong
 
|-
 
|-
|Mar 12
+
|Mar 17
| Trevor Leslie
+
| Spring Break!
| UW Madison
+
|
|[[#Trevor Leslie  |  Energy Equality for the Navier-Stokes Equations at the First Possible Blowup Time ]]
 
 
|
 
|
|-
 
|Mar 19
 
|Spring Break!
 
 
|  
 
|  
|
 
|
 
 
|-
 
|-
|Mar 26
+
|Mar 24
| No seminar
+
| Oscar Dominguez
|  
+
| Universidad Complutense de Madrid
|[[#linktoabstract  |   ]]
+
|[[#linktoabstract  |   Title ]]
|  
+
| Andreas
 
|-
 
|-
|Apr 2
+
|Mar 31
| Stefan Steinerberger
+
| Reserved
| Yale
+
| Institution
|[[#Stefan Steinerberger  Wasserstein Distance as a Tool in Analysis ]]
+
|[[#linktoabstract  Title ]]
| Shaoming, Andreas
+
| Street
 
|-
 
|-
 
+
|Apr 7
|Apr 9
+
| Hong Wang
| Franc Forstnerič
+
| Institution
| Unversity of Ljubljana
+
|[[#linktoabstract Title ]]
|[[#Franc Forstnerič Minimal surfaces by way of complex analysis ]]
+
| Street
| Xianghong, Andreas
 
 
|-
 
|-
|Apr 16
+
|<b>Monday, Apr 13</b>
| Andrew Zimmer
+
|Yumeng Ou
| Louisiana State University
+
|CUNY, Baruch College
|[[#Andrew Zimmer The geometry of domains with negatively pinched Kaehler metrics ]]
+
|[[#linktoabstract TBA ]]
| Xianghong
+
|Zhang
 
|-
 
|-
|Apr 23
+
|Apr 14
| Brian Street
+
| Tamás Titkos
| University of Wisconsin-Madison
+
| BBS University of Applied Sciences & Rényi Institute
|[[#Brian Street Maximal Hypoellipticity ]]
+
|[[#linktoabstract Distance preserving maps on spaces of probability measures ]]
 
| Street
 
| Street
 
|-
 
|-
|Apr 30
+
|Apr 21
| Zhen Zeng
+
| Diogo Oliveira e Silva
| UPenn
+
| University of Birmingham
|[[#Zhen Zeng Decay property of multilinear oscillatory integrals ]]
+
|[[#linktoabstract Title ]]
| Shaoming
+
| Betsy
 
|-
 
|-
|[https://www.math.wisc.edu/seeger2019/?q=node/2.html  Madison Lectures in Fourier Analysis]
+
|Apr 28
 +
| No Seminar
 
|-
 
|-
|Summer
+
|May 5
|-
+
|Jonathan Hickman
|Sept  10
+
|University of Edinburgh
|Jose Madrid
+
|[[#linktoabstract  |   Title ]]
|UCLA
+
| Andreas
|
 
|Andreas, David
 
 
|-
 
|-
|Oct 15
 
|Bassam Shayya
 
|American University of Beirut
 
|
 
|Andreas, Betsy
 
 
 
|}
 
|}
  
 
=Abstracts=
 
=Abstracts=
===Simon Marshall===
+
===José Madrid===
  
''Integrals of eigenfunctions on hyperbolic manifolds''
+
Title: On the regularity of maximal operators on Sobolev Spaces
  
Let X be a compact hyperbolic manifold, and let Y be a totally geodesic closed submanifold in X.  I will discuss the problem of bounding the integral of a Laplace eigenfunction on X over Y, as the eigenvalue tends to infinity. I will present an upper bound for these integrals that is sharp on average, and briefly describe ongoing work with Farrell Brumley in which we attempt to produce eigenfunctions with very large periods.
+
Abstract:  In this talk, we will discuss the regularity properties (boundedness and
 +
continuity) of the classical and fractional maximal
 +
operators when these act on the Sobolev space W^{1,p}(\R^n). We will
 +
focus on the endpoint case p=1. We will talk about
 +
some recent results and current open problems.
  
 +
===Yakun Xi===
  
===Hong Wang===
+
Title: Distance sets on Riemannian surfaces and microlocal decoupling inequalities
  
''About Falconer distance problem in the plane''
+
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.
  
If E is a compact set of Hausdorff dimension greater than 5/4 on the plane, we prove that there is a point x\in E such that the set of distances between x and E has positive Lebesgue measure. Our result improves upon Wolff's theorem for dim E> 4/3. This is joint work with Larry Guth, Alex Iosevich and Yumeng Ou.
+
===Joris Roos===
  
===Polona Durcik===
+
Title: L^p improving estimates for maximal spherical averages
  
''Singular Brascamp-Lieb inequalities and extended boxes in R^n''
+
Abstract: For a given compact set of radii $E$ we will discuss $L^p$ improving properties of maximal spherical averages with a supremum over $E$.
 +
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 $E$.
 +
Joint work with Tess Anderson, Kevin Hughes and Andreas Seeger.
  
Brascamp-Lieb inequalities are L^p estimates for certain multilinear forms on functions on Euclidean spaces. In this talk we consider singular Brascamp-Lieb inequalities, which arise when one of the functions is replaced by a Calderon-Zygmund kernel. We focus on a family of multilinear forms in R^n with a certain cubical structure and discuss their connection to some patterns in positive density subsets in R^n. Based on joint works with V. Kovac and C. Thiele.
 
  
  
===Song-Ying Li===
+
===Joao Ramos===
  
''Estimates for the first positive eigenvalue of Kohn Laplacian on a pseudo-Hermitian manifold''
+
Title: Fourier uncertainty principles, interpolation and uniqueness sets
  
In this talk, I will present my recent works with my collaborators on the lower bound and upper bounds estimates
+
Abstract: A classical result in the theory of entire functions of exponential type, Shannon’s interpolation formula predicates that, given a function whose Fourier transform vanishes outside the interval $[-1/2,1/2]$, it is possible to recover it from its values at the integers. More specifically, it holds, in a suitable sense of convergence, that
for the first positive eigenvalues of Kohn Laplacian and sub-Laplacian on a strictly pseudoconvex pseudo-Hermitian CR manifold,
 
which include CR Lichnerowicz-Obata theorem for the lower and upper bounds for the first positive eigenvalue for the
 
Kohn Laplacian on strictly pseudoconvex hypersurfaces.
 
  
 +
$$ f(x) = \sum_{n \in \mathbb{Z}} f(n) \frac{\sin(\pi(x-n))}{\pi(x -n)}. $$
  
===Hanlong Fan===
+
This formula is unfortunately unavailable for arbitrary Schwartz functions on the real line, but a recent result of Radchenko and Viazovska provides us with an explicit construction of an interpolation basis for even Schwartz functions. It states, in a nutshell, that we can recover explicitly the function given its values at the squares of roots of integers.
  
''A generalization of the theorem of Weil and Kodaira on prescribing residues''
+
We will discuss a bit these two results, and explore, in connection to classical Fourier uncertainty results, the question of determining which pairs of sets $(A,B)$ satisfy that, if a Schwartz function $f$ vanishes on A and its Fourier transform vanishes on B, then $f \equiv 0.$
  
An old theorem of Weil and Kodaira says that: For a K\"ahler manifold X, there exists a closed meromorphic one-form with residue divisor D if and only if D is homologous to zero. In this talk, I will generalize Weil and Kodaira's criterion to non-K\"ahler manifolds.
+
In particular, we will give sufficient conditions on $(\alpha,\beta)$ pairs of positive numbers so that, if $f$ vanishes at $\pm n^{\alpha}$ and its Fourier transform vanishes at $\pm n^{\beta}$, then $f$ is identically zero.
  
===Kyle Hambrook===
+
===Xiaojun Huang===
  
''Fourier Decay and Fourier Restriction for Fractal Measures on Curves''
+
Title: A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries
  
I will discuss my recent work on some problems concerning
+
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.
Fourier decay and Fourier restriction for fractal measures on curves.
 
  
===Laurent Stolovitch===
+
===Xiaocheng Li===
  
''Equivalence of Cauchy-Riemann manifolds and multisummability theory''
+
Title:  An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$
  
We apply the multisummability theory from Dynamical Systems to CR-geometry. As the main result, we show that two real-analytic hypersurfaces in $\mathbb C^2$ are formally equivalent, if and only if they are $C^\infty$ CR-equivalent at the respective point. As a corollary, we prove that all formal equivalences between real-algebraic Levi-nonflat hypersurfaces in $\mathbb C^2$ are algebraic (and in particular convergent). This is a joint work with I. Kossovskiy and B. Lamel.
+
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.
  
  
===Brian Cook===
+
===Xiaochun Li===
  
''Equidistribution results for integral points on affine homogenous algebraic varieties''
+
Title:  Roth’s type theorems on progressions
  
Let Q be a homogenous integral polynomial of degree at least two. We consider certain results and questions concerning the distribution of the integral points on the level sets of Q.
+
Abstract:  The arithmetic progression problems were posed by Erd\”os-Turan, answered affirmatively by Semer\’edi. However, there are still many questions remained on precise quantitative description on how large a subset shall be in oredr to guarantee a progression in it. Involving with Fourier analysis, considerable work had been accomplished recently. We will give a survey on those progress, and report our recent progress on quantitative version of Roth’s type theorem on (polynomial) progressions of short length.
  
===Alexei Poltoratski===
+
===Jeff Galkowski===
  
''Completeness of exponentials: Beurling-Malliavin and type problems''
+
<b>Concentration and Growth of Laplace Eigenfunctions</b>
  
This talk is devoted to two old problems of harmonic analysis mentioned in the title. Both problems ask when a family of complex exponentials is complete (spans) an L^2-space. The Beruling-Malliavin problem was solved in the early 1960s and I will present its classical solution along with modern generalizations and applications. I will then discuss history and recent progress in the type problem, which stood open for more than 70 years.
+
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.
  
 +
===David Beltran===
  
===Shaoming Guo===
+
Title: Regularity of the centered fractional maximal function
  
''Polynomial Roth theorems in Salem sets''
+
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.
  
Let P(t) be a polynomial of one real variable. I will report a result on searching for patterns of the form (x, x+t, x+P(t)) within Salem sets, whose Hausdorff dimension is sufficiently close to one. Joint work with Fraser and Pramanik.  
+
This is joint work with José Madrid.
  
 +
===Dominique Kemp===
  
 +
<b>Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature</b>
  
 +
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.
  
===Dean Baskin===
 
  
''Radiation fields for wave equations''
+
===Kevin O'Neill===
  
Radiation fields are rescaled limits of solutions of wave equations near "null infinity" and capture the radiation pattern seen by a distant observer. They are intimately connected with the Fourier and Radon transforms and with scattering theory. In this talk, I will define and discuss radiation fields in a few contexts, with an emphasis on spacetimes that look flat near infinity. The main result is a connection between the asymptotic behavior of the radiation field and a family of quantum objects on an associated asymptotically hyperbolic space.
+
<b>A Quantitative Stability Theorem for Convolution on the Heisenberg Group </b>
  
===Lillian Pierce===
+
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.
  
''Short character sums''
 
  
A surprisingly diverse array of problems in analytic number theory have at their heart a problem of bounding (from above) an exponential sum, or its multiplicative cousin, a so-called character sum. For example, both understanding the Riemann zeta function or Dirichlet L-functions inside the critical strip, and also counting solutions to Diophantine equations via the circle method or power sieve methods, involve bounding such sums. In general, the sums of interest fall into one of two main regimes: complete sums or incomplete sums, with this latter regime including in particular “short sums.” Short sums are particularly useful, and particularly resistant to almost all known methods. In this talk, we will see what makes a sum “short,” sketch why it would be incredibly powerful to understand short sums, and discuss a curious proof from the 1950’s which is still the best way we know to bound short sums. We will end by describing new work which extends the ideas of this curious proof to bound short sums in much more general situations.
 
  
===Loredana Lanzani===
+
===Francesco di Plinio===
  
''On regularity and irregularity of the Cauchy-Szegő projection in several complex variables''
+
<b>Maximal directional integrals along algebraic and lacunary sets </b>
  
This talk is a survey of my latest, and now final, collaboration with Eli Stein.
+
I will discuss two recent results obtained in collaboration with (partly) Natalia Accomazzo and Ioannis Parissis (U Basque Country). The first is a sharp $L^2$ estimate for the maximal averaging operator associated to sets of directions from algebraic sets in R^n of arbitrary codimension. The proof uses a new scheme of polynomial partitioning on manifolds which extends ideas by Larry Guth. The second result is a sharp estimate in all dimensions for the maximal directional singular integrals along lacunary directions. This settles a question of Parcet and Rogers. The proof uses a combination of two-dimensional and $n$-dimensional coverings combining seemingly contrasting ideas  of Parcet-Rogers and of  Nagel-Stein-Wainger.
  
It is known that for bounded domains $D$ in $\mathbb C^n$ that are of class $C^2$ and are strongly pseudo-convex, the Cauchy-Szegő projection is bounded in $L^p(\text{b}D, d\Sigma)$ for $1<p<\infty$. (Here $d\Sigma$ is induced Lebesgue measure.) We show, using appropriate worm domains, that this fails for any $p\neq 2$, when we assume that the domain in question is only weakly pseudo-convex. Our starting point are the ideas of Kiselman-Barrett introduced more than 30 years ago in the analysis of the Bergman projection. However the study of the Cauchy-Szegő projection raises a number of new issues and obstacles that need to be overcome. We will also compare these results to the analogous problem for the Cauchy-Leray integral, where however the relevant counter-example is of much simpler nature.
+
   
  
===Trevor Leslie===
 
  
''Energy Equality for the Navier-Stokes Equations at the First Possible Blowup Time''
 
  
In this talk, we discuss the problem of energy equality for strong solutions of the Navier-Stokes Equations (NSE) at the first time where such solutions may lose regularity.  Our approach is motivated by a famous theorem of Caffarelli, Kohn, and Nirenberg, which states that the set of singular points associated to a suitable weak solution of the NSE has parabolic Hausdorff dimension of at most 1.  In particular, we furnish sufficient conditions for energy equality which depend on the dimension of the singularity set in addition to time and space integrability assumptions; in doing so we improve upon the classical results when attention is restricted to the first blowup time.  When our method is inconclusive, we are able to quantify the possible failure of energy equality in terms of the lower local dimension and the ''concentration dimension'' of a certain measure associated to the solution.  The work described is joint with Roman Shvydkoy (UIC).
 
  
===Stefan Steinerberger===
 
  
''Wasserstein Distance as a Tool in Analysis''
+
===Laurent Stolovitch===
 
 
Wasserstein Distance is a way of measuring the distance between two probability distributions (minimizing it is a main problem in Optimal Transport). We will give a gentle Introduction into what it means and then use it to prove (1) a completely elementary but possibly new and quite curious inequality for real-valued functions and (2) a statement along the following lines: linear combinations of eigenfunctions of elliptic operators corresponding to high frequencies oscillate a lot and vanish on a large set of co-dimension 1 (this is already interesting for trigonometric polynomials on the 2-torus, sums of finitely many sines and cosines, whose sum has to vanish on long lines) and (3) some statements in Basic Analytic Number Theory that drop out for free as a byproduct.
 
 
 
===Franc Forstnerič===
 
 
 
''Minimal surfaces by way of complex analysis''
 
 
 
After a brief historical introduction, I will present some recent developments in the theory of minimal surfaces in Euclidean spaces which have been obtained by complex analytic methods. The emphasis will be on results pertaining to the global theory of minimal surfaces including Runge and Mergelyan approximation, the conformal Calabi-Yau problem, properly immersed and embedded minimal surfaces, and a new result on the Gauss map of minimal surfaces.
 
 
 
===Andrew Zimmer===
 
 
 
''The geometry of domains with negatively pinched Kaehler metrics''
 
 
 
Every bounded pseudoconvex domain in C^n has a natural complete metric: the Kaehler-Einstein metric constructed by Cheng-Yau. When the boundary of the domain is strongly pseudoconvex, Cheng-Yau showed that the holomorphic sectional curvature of this metric is asymptotically a negative constant. In this talk I will describe some partial converses to this result, including the following: if a smoothly bounded convex domain has a complete Kaehler metric with close to constant negative holomorphic sectional curvature near the boundary, then the domain is strongly pseudoconvex. This is joint work with F. Bracci and H. Gaussier.
 
 
 
 
 
===Brian Street===
 
  
''Maximal Hypoellipticity''
+
<b>Linearization of neighborhoods of embeddings of complex compact manifolds </b>
  
In 1974, Folland and Stein introduced a generalization of ellipticity known as maximal hypoellipticity. This talk will be an introduction to this concept and some of the ways it generalizes ellipticity.
+
In this work, we address the following question due to Grauert: if a neighborhood M of a holomorphically embedded complex compact manifold C is formally equivalent to another one, are two neighborhoods biholomorphically equivalent? We shall present the case where the other neighborhood is the neighborhood of the zero section of the normal bundle of C in M. The solution to this problem involves "small divisors problems". This is joint work with X. Gong.
  
  
===Zhen Zeng===
+
===Bingyang Hu===
  
''Decay property of multilinear oscillatory integrals''
+
<b>Sparse bounds of singular Radon transforms</b>
  
In this talk, I will be talking about the conditions of the phase function $P$ and the linear mappings $\{\pi_i\}_{i=1}^n$ to ensure the asymptotic power decay properties of the following trilinear oscillatory integrals
+
In this talk, we will first briefly talk about the general theory of sparse domination, and then talk about the sparse bounds of singular Radon transforms, which strengths the $L^p$ boundedness of such operators due to Christ, Nagel, Stein and Wainger in 1999.
\[
 
I_{\lambda}(f_1,f_2,f_3)=\int_{\mathbb{R}^m}e^{i\lambda P(x)}\prod_{j=1}^3 f_j(\pi_j(x))\eta(x)dx, 
 
\]
 
which falls into the broad goal in the previous work of Christ, Li, Tao and Thiele.
 
  
 
=Extras=
 
=Extras=
 
[[Blank Analysis Seminar Template]]
 
[[Blank Analysis Seminar Template]]

Latest revision as of 22:45, 14 November 2019

Fall 2019 and Spring 2020 Analysis Seminar Series

The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.

If you wish to invite a speaker please contact Brian at street(at)math

Previous Analysis seminars

Analysis Seminar Schedule

date speaker institution title host(s)
Sept 10 José Madrid UCLA On the regularity of maximal operators on Sobolev Spaces Andreas, David
Sept 13 (Friday, B139) Yakun Xi University of Rochester Distance sets on Riemannian surfaces and microlocal decoupling inequalities Shaoming
Sept 17 Joris Roos UW Madison L^p improving estimates for maximal spherical averages Brian
Sept 20 (2:25 PM Friday, Room B139 VV) Xiaojun Huang Rutgers University–New Brunswick A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries Xianghong
Oct 1 Xiaocheng Li UW Madison An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$ Simon
Oct 8 Jeff Galkowski Northeastern University Concentration and Growth of Laplace Eigenfunctions Betsy
Oct 15 David Beltran UW Madison Regularity of the centered fractional maximal function Brian
Oct 22 Laurent Stolovitch University of Côte d'Azur Linearization of neighborhoods of embeddings of complex compact manifolds Xianghong
Wednesday Oct 23 in B129 Dominique Kemp Indiana University Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature Betsy
Oct 29 Bingyang Hu UW Madison Sparse bounds of singular Radon transforms Street
Nov 5 Kevin O'Neill UC Davis A Quantitative Stability Theorem for Convolution on the Heisenberg Group Betsy
Nov 12 Francesco di Plinio Washington University in St. Louis Maximal directional integrals along algebraic and lacunary sets Shaoming
Nov 13 (Wednesday) Xiaochun Li UIUC Roth's type theorems on progressions Brian, Shaoming
Nov 19 Joao Ramos University of Bonn Fourier uncertainty principles, interpolation and uniqueness sets Joris, Shaoming
Nov 26 No Seminar
Dec 3 Person Institution Title Sponsor
Dec 10 No Seminar
Jan 21 No Seminar
Jan 28 Person Institution Title Sponsor
Feb 4 Person Institution Title Sponsor
Feb 11 Person Institution Title Sponsor
Feb 18 Person Institution Title Sponsor
Feb 25 Person Institution Title Sponsor
Mar 3 Person Institution Title Sponsor
Mar 10 Yifei Pan Indiana University-Purdue University Fort Wayne Title Xianghong
Mar 17 Spring Break!
Mar 24 Oscar Dominguez Universidad Complutense de Madrid Title Andreas
Mar 31 Reserved Institution Title Street
Apr 7 Hong Wang Institution Title Street
Monday, Apr 13 Yumeng Ou CUNY, Baruch College TBA Zhang
Apr 14 Tamás Titkos BBS University of Applied Sciences & Rényi Institute Distance preserving maps on spaces of probability measures Street
Apr 21 Diogo Oliveira e Silva University of Birmingham Title Betsy
Apr 28 No Seminar
May 5 Jonathan Hickman University of Edinburgh Title Andreas

Abstracts

José Madrid

Title: On the regularity of maximal operators on Sobolev Spaces

Abstract: In this talk, we will discuss the regularity properties (boundedness and continuity) of the classical and fractional maximal operators when these act on the Sobolev space W^{1,p}(\R^n). We will focus on the endpoint case p=1. We will talk about some recent results and current open problems.

Yakun Xi

Title: Distance sets on Riemannian surfaces and microlocal decoupling inequalities

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.

Joris Roos

Title: L^p improving estimates for maximal spherical averages

Abstract: For a given compact set of radii $E$ we will discuss $L^p$ improving properties of maximal spherical averages with a supremum over $E$. 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 $E$. Joint work with Tess Anderson, Kevin Hughes and Andreas Seeger.


Joao Ramos

Title: Fourier uncertainty principles, interpolation and uniqueness sets

Abstract: A classical result in the theory of entire functions of exponential type, Shannon’s interpolation formula predicates that, given a function whose Fourier transform vanishes outside the interval $[-1/2,1/2]$, it is possible to recover it from its values at the integers. More specifically, it holds, in a suitable sense of convergence, that

$$ f(x) = \sum_{n \in \mathbb{Z}} f(n) \frac{\sin(\pi(x-n))}{\pi(x -n)}. $$

This formula is unfortunately unavailable for arbitrary Schwartz functions on the real line, but a recent result of Radchenko and Viazovska provides us with an explicit construction of an interpolation basis for even Schwartz functions. It states, in a nutshell, that we can recover explicitly the function given its values at the squares of roots of integers.

We will discuss a bit these two results, and explore, in connection to classical Fourier uncertainty results, the question of determining which pairs of sets $(A,B)$ satisfy that, if a Schwartz function $f$ vanishes on A and its Fourier transform vanishes on B, then $f \equiv 0.$

In particular, we will give sufficient conditions on $(\alpha,\beta)$ pairs of positive numbers so that, if $f$ vanishes at $\pm n^{\alpha}$ and its Fourier transform vanishes at $\pm n^{\beta}$, then $f$ is identically zero.

Xiaojun Huang

Title: A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries

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.

Xiaocheng Li

Title: An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$

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.


Xiaochun Li

Title: Roth’s type theorems on progressions

Abstract: The arithmetic progression problems were posed by Erd\”os-Turan, answered affirmatively by Semer\’edi. However, there are still many questions remained on precise quantitative description on how large a subset shall be in oredr to guarantee a progression in it. Involving with Fourier analysis, considerable work had been accomplished recently. We will give a survey on those progress, and report our recent progress on quantitative version of Roth’s type theorem on (polynomial) progressions of short length.

Jeff Galkowski

Concentration and Growth of Laplace Eigenfunctions

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.

David Beltran

Title: Regularity of the centered fractional maximal function

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.

This is joint work with José Madrid.

Dominique Kemp

Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature

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.


Kevin O'Neill

A Quantitative Stability Theorem for Convolution on the Heisenberg Group

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.


Francesco di Plinio

Maximal directional integrals along algebraic and lacunary sets

I will discuss two recent results obtained in collaboration with (partly) Natalia Accomazzo and Ioannis Parissis (U Basque Country). The first is a sharp $L^2$ estimate for the maximal averaging operator associated to sets of directions from algebraic sets in R^n of arbitrary codimension. The proof uses a new scheme of polynomial partitioning on manifolds which extends ideas by Larry Guth. The second result is a sharp estimate in all dimensions for the maximal directional singular integrals along lacunary directions. This settles a question of Parcet and Rogers. The proof uses a combination of two-dimensional and $n$-dimensional coverings combining seemingly contrasting ideas of Parcet-Rogers and of Nagel-Stein-Wainger.




Laurent Stolovitch

Linearization of neighborhoods of embeddings of complex compact manifolds

In this work, we address the following question due to Grauert: if a neighborhood M of a holomorphically embedded complex compact manifold C is formally equivalent to another one, are two neighborhoods biholomorphically equivalent? We shall present the case where the other neighborhood is the neighborhood of the zero section of the normal bundle of C in M. The solution to this problem involves "small divisors problems". This is joint work with X. Gong.


Bingyang Hu

Sparse bounds of singular Radon transforms

In this talk, we will first briefly talk about the general theory of sparse domination, and then talk about the sparse bounds of singular Radon transforms, which strengths the $L^p$ boundedness of such operators due to Christ, Nagel, Stein and Wainger in 1999.

Extras

Blank Analysis Seminar Template