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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
 
| Brian
 
|-
 
|-
||'''Monday, Nov. 12, B139'''
+
|Nov 5
| Kyle Hambrook
+
| Kevin O'Neill
| San Jose State University
+
| UC Davis
|[[#Kyle Hambrook Fourier Decay and Fourier Restriction for Fractal Measures on Curves ]]
+
|[[#Kevin O'Neill A Quantitative Stability Theorem for Convolution on the Heisenberg Group ]]
| Andreas
+
| Betsy
|-
 
|Nov 13
 
| Laurent Stolovitch
 
| Université de Nice - Sophia Antipolis
 
|[[#Laurent Stolovitch  |  Equivalence of Cauchy-Riemann manifolds and multisummability theory ]]
 
|Xianghong
 
 
|-
 
|-
|Nov 20
+
|Nov 12
| Grad Student Seminar
+
| Francesco di Plinio
|  
+
| Washington University in St. Louis
|[[#linktoabstract |   ]]
+
|[[#Francesco di Plinio |   Maximal directional integrals along algebraic and lacunary sets]]
|  
+
| Shaoming
 
|-
 
|-
|Nov 27
+
|Nov 13 (Wednesday)
| No Seminar
+
| Xiaochun Li
|  
+
| UIUC
|[[#linktoabstract |   ]]
+
|[[#Xiaochun Li |   Roth's type theorems on progressions]]
|  
+
| Brian, Shaoming
 
|-
 
|-
|Dec 4
+
|Nov 19
| No Seminar
+
| Joao Ramos
|[[#linktoabstract |   ]]
+
| University of Bonn
|  
+
|[[#Joao Ramos |   Fourier uncertainty principles, interpolation and uniqueness sets ]]
 +
| Joris, Shaoming
 
|-
 
|-
|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'''
+
|Friday, Jan 31, 4 pm, B239, Colloquium
| Alexei Poltoratski
+
| Lillian Pierce
| Texas A&M
+
| Duke University
|[[#Alexei Poltoratski Completeness of exponentials: Beurling-Malliavin and type problems ]]
+
|[[#Lillian Pierce On Bourgain’s counterexample for the Schrödinger maximal function ]]
| Denisov
+
| Andreas, Simon
 
|-
 
|-
|'''Friday, Feb 8'''
+
|Feb 4
| Aaron Naber
+
| Ruixiang Zhang
| Northwestern University
+
| UW Madison
|[[#linktoabstract A structure theory for spaces with lower Ricci curvature bounds ]]
+
|[[#Ruixiang Zhang Local smoothing for the wave equation in 2+1 dimensions ]]
| See colloquium website for location
+
| Andreas
 
|-
 
|-
|Feb 12
+
|Feb 11
| Shaoming Guo
+
| Zane Li
 +
| Indiana University
 +
|[[#Zane Li  |  A bilinear proof of decoupling for the moment curve ]]
 +
| Betsy
 +
|-
 +
|Feb 18
 +
| Sergey Denisov
 
| UW Madison
 
| UW Madison
|[[# Shaoming Guo | Polynomial Roth theorems in Salem sets   ]]
+
|[[#linktoabstract  De Branges canonical systems with finite logarithmic integral ]]
|  
+
| Brian
 
|-
 
|-
|'''Wed, Feb 13, B239'''
+
|Feb 25
| Dean Baskin
+
| Michel Alexis
| TAMU
+
| UW Madison
|[[# Dean Baskin Radiation fields for wave  equations ]]
+
|[[#Michel Alexis The Steklov problem for trigonometric polynomials orthogonal to a Muckenhoupt weight ]]
| Colloquium
+
| Sergey
 
|-
 
|-
|'''Friday, Feb 15'''
+
|Friday, Feb 28 (Colloquium)
| Lillian Pierce
+
| Brett Wick
| Duke
+
| Washington University - St. Louis
|[[# Lillian Pierce Short character sums ]]
+
|[[#MBrett Wick The Corona Theorem]]
| Colloquium
+
| Andreas
 
|-
 
|-
|'''Monday,  Feb 18, 3:30 p.m, B239.'''
+
|Mar 3
| Daniel Tataru
+
| William Green
| UC Berkeley
+
| Rose-Hulman Institute of Technology
|[[#lDaniel Tataru A Morawetz inequality for water waves ]]
+
|[[#William Green Dispersive estimates for the Dirac equation ]]
| PDE Seminar
+
| Betsy
 
|-
 
|-
|Feb 19
+
|Mar 10
| Wenjia Jing
+
| Ziming Shi
|Tsinghua University
+
| UW Madison
|Periodic homogenization of Dirichlet problems in perforated domains: a unified proof
+
|[[#linktoabstract |On the Sobolev space property of logarithmic modulus of holomorphic functions in C^n]]
| PDE Seminar
+
| Xianghong
 
|-
 
|-
|Feb 26
+
|Mar 17
| No Seminar
+
| Spring Break!
 
|
 
|
 
|
 
|
 +
|
 
|-
 
|-
|Mar 5
+
|Mar 24
| Loredana Lanzani
+
| Oscar Dominguez
| Syracuse University
+
| Universidad Complutense de Madrid
|[[# Loredana Lanzani  |  On regularity and irregularity of the Cauchy-Szegő projection in several complex variables ]]
+
|Canceled
| Xianghong
+
| Andreas
 
|-
 
|-
|Mar 12
+
|Mar 31
| Trevor Leslie
+
| Brian Street
| UW Madison
+
| University of Wisconsin-Madison
|[[# Trevor Leslie  |  Energy Equality for the Navier-Stokes Equations at the First Possible Blowup Time ]]
+
|Canceled
|
+
| Local
 
|-
 
|-
|Mar 19
+
|Apr 7
|Spring Break!!!
+
| Hong Wang
|  
+
| Institution
|
+
|Canceled
|
+
| Street
 
|-
 
|-
|Mar 26
+
|<b>Monday, Apr 13</b>
| No seminar
+
|Yumeng Ou
|  
+
|CUNY, Baruch College
|[[#linktoabstract  |    ]]
+
|Canceled
|  
+
|Ruixiang
 
|-
 
|-
|Apr 2
+
|Apr 14
| Stefan Steinerberger
+
| Tamás Titkos
| Yale
+
| BBS University of Applied Sciences & Rényi Institute
|[[#linktoabstract  |  Title ]]
+
|Canceled
| Shaoming, Andreas
+
| Brian
 
|-
 
|-
 
+
|Apr 21
|Apr 9
+
| Diogo Oliveira e Silva
| Franc Forstnerič
+
| University of Birmingham
| Unversity of Ljubljana
+
|Canceled
|[[#linktoabstract  |  Title ]]
+
| Betsy
| Xianghong, Andreas
 
 
|-
 
|-
|Apr 16
+
|Apr 28
| Andrew Zimmer
+
| No Seminar
| Louisiana State University
 
|[[#linktoabstract  |  Title ]]
 
| Xianghong
 
 
|-
 
|-
|Apr 23
+
|May 5
| Person
+
|Jonathan Hickman
| Institution
+
|University of Edinburgh
|[[#linktoabstract  |  Title ]]
+
|Canceled
| Sponsor
+
| Andreas
 
|-
 
|-
|Apr 30
+
|Nov 17, 2020
| Reserved
+
| Tamás Titkos
| Institution
+
| BBS University of Applied Sciences & Rényi Institute
|[[#linktoabstract  |  Title ]]
+
|
| Shaoming
+
| Brian
 
|-
 
|-
 
|}
 
|}
  
 
=Abstracts=
 
=Abstracts=
===Simon Marshall===
+
===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===
  
''Integrals of eigenfunctions on hyperbolic manifolds''
+
Title: Distance sets on Riemannian surfaces and microlocal decoupling inequalities
  
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 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===
  
===Hong Wang===
+
Title: L^p improving estimates for maximal spherical averages
  
''About Falconer distance problem in the plane''
+
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.
  
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.
 
  
===Polona Durcik===
 
  
''Singular Brascamp-Lieb inequalities and extended boxes in R^n''
+
===Joao Ramos===
  
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.
+
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
  
===Song-Ying Li===
+
$$ f(x) = \sum_{n \in \mathbb{Z}} f(n) \frac{\sin(\pi(x-n))}{\pi(x -n)}. $$
  
''Estimates for the first positive eigenvalue of Kohn Laplacian on a pseudo-Hermitian manifold''
+
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.
  
In this talk, I will present my recent works with my collaborators on the lower bound and upper bounds estimates
+
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 $f \equiv 0.$
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.
 
  
 +
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.
  
===Hanlong Fan===
+
===Xiaojun Huang===
  
''A generalization of the theorem of Weil and Kodaira on prescribing residues''
+
Title: A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries
  
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.
+
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.
  
===Kyle Hambrook===
+
===Xiaocheng Li===
  
''Fourier Decay and Fourier Restriction for Fractal Measures on Curves''
+
Title:  An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$
  
I will discuss my recent work on some problems concerning
+
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.
Fourier decay and Fourier restriction for fractal measures on curves.
 
  
===Laurent Stolovitch===
 
  
''Equivalence of Cauchy-Riemann manifolds and multisummability theory''
+
===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===
 +
 
 +
<b>Concentration and Growth of Laplace Eigenfunctions</b>
 +
 
 +
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
  
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: 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.
  
===Brian Cook===
+
===Dominique Kemp===
  
''Equidistribution results for integral points on affine homogenous algebraic varieties''
+
<b>Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature</b>
  
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.
+
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.
  
===Alexei Poltoratski===
 
  
''Completeness of exponentials: Beurling-Malliavin and type problems''
+
===Kevin O'Neill===
  
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.
+
<b>A Quantitative Stability Theorem for Convolution on the Heisenberg Group </b>
  
 +
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.
  
===Shaoming Guo===
+
===Francesco di Plinio===
  
''Polynomial Roth theorems in Salem sets''
+
<b>Maximal directional integrals along algebraic and lacunary sets </b>
  
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.  
+
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===
  
 +
<b>Linearization of neighborhoods of embeddings of complex compact manifolds </b>
  
 +
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.
  
===Dean Baskin===
+
===Bingyang Hu===
  
''Radiation fields for wave equations''
+
<b>Sparse bounds of singular Radon transforms</b>
  
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.
+
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.
  
 
===Lillian Pierce===
 
===Lillian Pierce===
 +
<b> On Bourgain’s counterexample for the Schrödinger maximal function </b>
 +
 +
In 1980, Carleson asked a question in harmonic analysis: to which Sobolev space H^s must an initial data function belong, for a pointwise a.e. convergence result to hold for the solution to the associated linear Schrödinger equation? Over the next decades, many people developed counterexamples to push the (necessary) range of s up, and positive results to push the (sufficient) range of s down. Now, these ranges are finally meeting: Bourgain’s 2016 counterexample showed s < n/(2(n+1)) fails, and Du and Zhang’s 2019 paper shows that s>n/(2(n+1)) suffices.
 +
In this talk, we will give an overview of how to rigorously derive Bourgain’s 2016 counterexample, based on simple facts from number theory. We will show how to build Bourgain’s counterexample starting from “zero knowledge," and how to gradually optimize the set-up to arrive at the final counterexample. The talk will be broadly accessible, particularly if we live up to the claim of starting from “zero knowledge.”
 +
 +
===Ruixiang Zhang===
 +
 +
<b> Local smoothing for the wave equation in 2+1 dimensions </b>
 +
 +
Sogge's local smoothing conjecture for the wave equation predicts that the local L^p space-time estimate  gains a fractional  derivative of order almost 1/p compared to the fixed time L^p estimates, when p>2n/(n-1). Jointly with Larry Guth and Hong Wang, we recently proved the conjecture in $\mathbb{R}^{2+1}$. I will talk about a sharp square function estimate we proved which implies the local smoothing conjecture in dimensions 2+1. A key ingredient in the proof is an incidence type theorem.
 +
 +
===Zane Li===
 +
 +
<b> A bilinear proof of decoupling for the moment curve</b>
 +
 +
We give a proof of decoupling for the moment curve that is inspired from nested efficient congruencing. We also discuss the relationship between Wooley's nested efficient congruencing and Bourgain-Demeter-Guth's decoupling proofs of Vinogradov's Mean Value Theorem. This talk is based on joint work with Shaoming Guo, Po-Lam Yung, and Pavel Zorin-Kranich.
 +
  
''Short character sums''
+
===Sergey Denisov===
  
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.
+
<b> De Branges canonical systems with finite logarithmic integral </b>
  
===Loredana Lanzani===
+
We consider measures  m on the real line for which logarithmic
 +
integral exists and give a complete characterization of all Hamiltonians
 +
in de Branges canonical system for which m  is the spectral measure.
 +
This characterization involves the matrix A_2 Muckenhoupt condition on a
 +
fixed scale. Our result provides a generalization of the classical
 +
theorem of Szego for polynomials orthogonal on the unit circle and
 +
complements the Krein-Wiener theorem. Based on the joint work with R.
 +
Bessonov.
  
''On regularity and irregularity of the Cauchy-Szegő projection in several complex variables''
 
  
This talk is a survey of my latest, and now final, collaboration with Eli Stein.
+
===Michel Alexis===
  
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.
+
<b>The Steklov problem for Trigonometric Polynomials orthogonal to a Muckenhoupt weight</b>
  
===Trevor Leslie===
+
Let $\{\varphi_n\}_{n=0}^{\infty}$ be the sequence of degree $n$ polynomials on $\mathbb{T}$, orthonormal with respect to a positive weight $w$. Steklov conjectured whenever $w \geq \delta> 0$ a.e.\ then $\{\varphi_n\}$ are uniformly bounded in $L^{\infty}$. While false, this conjecture brings us to ask the following: under what regularity conditions on $w$ are $\{\varphi_n\}$ uniformly bounded in $L^p (w)$ for some $p > 2$?
 +
 +
We discuss some answers to this question using the contraction principle and operator estimates for the Hilbert transform, in particular recent joint work with Alexander Aptakarev and Sergey Denisov for when $w$ is a Muckenhoupt weight.
  
''Energy Equality for the Navier-Stokes Equations at the First Possible Blowup Time''
+
===William Green===
  
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).
+
<b> Dispersive estimates for the Dirac equation </b>
  
 +
The Dirac equation was derived by Dirac in 1928 to model the behavior of subatomic particles moving at relativistic speeds.  Dirac formulated a hyberbolic system of partial differential equations
 +
That can be interpreted as a sort of square root of a system of Klein-Gordon equations.
 
   
 
   
 +
The Dirac equation is considerably less well studied than other dispersive equations such as the Schrodinger, wave or Klein-Gordon equations.  We will survey recent work on time-decay estimates for the solution operator.  Specifically the mapping properties of the solution operator between L^p spaces.  As in other dispersive equations, the existence of eigenvalues and/or resonances at the edge of the continuous spectrum affects the dynamics of the solution.  We classify the threshold eigenvalue and resonance structure in two and three spatial dimensions and study their effect on the time decay.  The talk with survey joint works with B. Erdogan (Illinois), M. Goldberg (Cincinnati) and E. Toprak (Rutgers).
 +
 +
===Yifei Pan===
 +
 +
<b>On the Sobolev space property of logarithmic modulus of holomorphic functions in C^n</b>
 +
 +
In this talk, I will present a proof of the following Sobolev space property of logarithmic modulus of holomorphic functions in C^n. If f is a holomorphic function on the unit ball B(0,1) in C^n vanishing at the origin (i.e., f(0) = 0) but it is not identically zero, then log |f| ∈ W^{1,p}(B(0, r)) for any p < 2, but log |f| is not in W^{1,2}(B(0, r)) (r < 1). As you may see, this result is rather simple to prove in the complex plane due to the discreteness of zeros of holomorphic functions. In higher dimensions, we are going to apply Hironaka’s resolution of singularity and then Harvey- Polking removable singularity method to prove the existence of weak derivatives of log |f(z)|. This is part of a joint project with Ziming Shi at Madison.
 +
 +
===Tamás Titkos===
 +
 +
<b>Isometries of Wasserstein spaces</b>
 +
 +
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.
 +
 +
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, these two groups are not isomorphic in general. Recently, Kloeckner described the isometry group of the quadratic Wasserstein space over the real line. It turned out that this group is extremely rich: it contains a flow of wild behaving isometries that distort the shape of measures. Following this line of investigation,  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 our recent manuscript \emph{"Isometric study of Wasserstein spaces -- The real line"} (to appear in Trans. Amer. Math. Soc., arXiv:2002.00859).
 +
 +
Joint work with György Pál Gehér (University of Reading) and Dániel Virosztek (IST Austria).
 +
 
=Extras=
 
=Extras=
 
[[Blank Analysis Seminar Template]]
 
[[Blank Analysis Seminar Template]]

Latest revision as of 07:02, 18 March 2020

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 Brian
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
Jan 21 No Seminar
Friday, Jan 31, 4 pm, B239, Colloquium Lillian Pierce Duke University On Bourgain’s counterexample for the Schrödinger maximal function Andreas, Simon
Feb 4 Ruixiang Zhang UW Madison Local smoothing for the wave equation in 2+1 dimensions Andreas
Feb 11 Zane Li Indiana University A bilinear proof of decoupling for the moment curve Betsy
Feb 18 Sergey Denisov UW Madison De Branges canonical systems with finite logarithmic integral Brian
Feb 25 Michel Alexis UW Madison The Steklov problem for trigonometric polynomials orthogonal to a Muckenhoupt weight Sergey
Friday, Feb 28 (Colloquium) Brett Wick Washington University - St. Louis The Corona Theorem Andreas
Mar 3 William Green Rose-Hulman Institute of Technology Dispersive estimates for the Dirac equation Betsy
Mar 10 Ziming Shi UW Madison On the Sobolev space property of logarithmic modulus of holomorphic functions in C^n Xianghong
Mar 17 Spring Break!
Mar 24 Oscar Dominguez Universidad Complutense de Madrid Canceled Andreas
Mar 31 Brian Street University of Wisconsin-Madison Canceled Local
Apr 7 Hong Wang Institution Canceled Street
Monday, Apr 13 Yumeng Ou CUNY, Baruch College Canceled Ruixiang
Apr 14 Tamás Titkos BBS University of Applied Sciences & Rényi Institute Canceled Brian
Apr 21 Diogo Oliveira e Silva University of Birmingham Canceled Betsy
Apr 28 No Seminar
May 5 Jonathan Hickman University of Edinburgh Canceled Andreas
Nov 17, 2020 Tamás Titkos BBS University of Applied Sciences & Rényi Institute Brian

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.

Lillian Pierce

On Bourgain’s counterexample for the Schrödinger maximal function

In 1980, Carleson asked a question in harmonic analysis: to which Sobolev space H^s must an initial data function belong, for a pointwise a.e. convergence result to hold for the solution to the associated linear Schrödinger equation? Over the next decades, many people developed counterexamples to push the (necessary) range of s up, and positive results to push the (sufficient) range of s down. Now, these ranges are finally meeting: Bourgain’s 2016 counterexample showed s < n/(2(n+1)) fails, and Du and Zhang’s 2019 paper shows that s>n/(2(n+1)) suffices. In this talk, we will give an overview of how to rigorously derive Bourgain’s 2016 counterexample, based on simple facts from number theory. We will show how to build Bourgain’s counterexample starting from “zero knowledge," and how to gradually optimize the set-up to arrive at the final counterexample. The talk will be broadly accessible, particularly if we live up to the claim of starting from “zero knowledge.”

Ruixiang Zhang

Local smoothing for the wave equation in 2+1 dimensions

Sogge's local smoothing conjecture for the wave equation predicts that the local L^p space-time estimate gains a fractional derivative of order almost 1/p compared to the fixed time L^p estimates, when p>2n/(n-1). Jointly with Larry Guth and Hong Wang, we recently proved the conjecture in $\mathbb{R}^{2+1}$. I will talk about a sharp square function estimate we proved which implies the local smoothing conjecture in dimensions 2+1. A key ingredient in the proof is an incidence type theorem.

Zane Li

A bilinear proof of decoupling for the moment curve

We give a proof of decoupling for the moment curve that is inspired from nested efficient congruencing. We also discuss the relationship between Wooley's nested efficient congruencing and Bourgain-Demeter-Guth's decoupling proofs of Vinogradov's Mean Value Theorem. This talk is based on joint work with Shaoming Guo, Po-Lam Yung, and Pavel Zorin-Kranich.


Sergey Denisov

De Branges canonical systems with finite logarithmic integral

We consider measures m on the real line for which logarithmic integral exists and give a complete characterization of all Hamiltonians in de Branges canonical system for which m is the spectral measure. This characterization involves the matrix A_2 Muckenhoupt condition on a fixed scale. Our result provides a generalization of the classical theorem of Szego for polynomials orthogonal on the unit circle and complements the Krein-Wiener theorem. Based on the joint work with R. Bessonov.


Michel Alexis

The Steklov problem for Trigonometric Polynomials orthogonal to a Muckenhoupt weight

Let $\{\varphi_n\}_{n=0}^{\infty}$ be the sequence of degree $n$ polynomials on $\mathbb{T}$, orthonormal with respect to a positive weight $w$. Steklov conjectured whenever $w \geq \delta> 0$ a.e.\ then $\{\varphi_n\}$ are uniformly bounded in $L^{\infty}$. While false, this conjecture brings us to ask the following: under what regularity conditions on $w$ are $\{\varphi_n\}$ uniformly bounded in $L^p (w)$ for some $p > 2$?

We discuss some answers to this question using the contraction principle and operator estimates for the Hilbert transform, in particular recent joint work with Alexander Aptakarev and Sergey Denisov for when $w$ is a Muckenhoupt weight.

William Green

Dispersive estimates for the Dirac equation

The Dirac equation was derived by Dirac in 1928 to model the behavior of subatomic particles moving at relativistic speeds. Dirac formulated a hyberbolic system of partial differential equations That can be interpreted as a sort of square root of a system of Klein-Gordon equations.

The Dirac equation is considerably less well studied than other dispersive equations such as the Schrodinger, wave or Klein-Gordon equations. We will survey recent work on time-decay estimates for the solution operator. Specifically the mapping properties of the solution operator between L^p spaces. As in other dispersive equations, the existence of eigenvalues and/or resonances at the edge of the continuous spectrum affects the dynamics of the solution. We classify the threshold eigenvalue and resonance structure in two and three spatial dimensions and study their effect on the time decay. The talk with survey joint works with B. Erdogan (Illinois), M. Goldberg (Cincinnati) and E. Toprak (Rutgers).

Yifei Pan

On the Sobolev space property of logarithmic modulus of holomorphic functions in C^n

In this talk, I will present a proof of the following Sobolev space property of logarithmic modulus of holomorphic functions in C^n. If f is a holomorphic function on the unit ball B(0,1) in C^n vanishing at the origin (i.e., f(0) = 0) but it is not identically zero, then log |f| ∈ W^{1,p}(B(0, r)) for any p < 2, but log |f| is not in W^{1,2}(B(0, r)) (r < 1). As you may see, this result is rather simple to prove in the complex plane due to the discreteness of zeros of holomorphic functions. In higher dimensions, we are going to apply Hironaka’s resolution of singularity and then Harvey- Polking removable singularity method to prove the existence of weak derivatives of log |f(z)|. This is part of a joint project with Ziming Shi at Madison.

Tamás Titkos

Isometries of Wasserstein spaces

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.

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, these two groups are not isomorphic in general. Recently, Kloeckner described the isometry group of the quadratic Wasserstein space over the real line. It turned out that this group is extremely rich: it contains a flow of wild behaving isometries that distort the shape of measures. Following this line of investigation, 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 our recent manuscript \emph{"Isometric study of Wasserstein spaces -- The real line"} (to appear in Trans. Amer. Math. Soc., arXiv:2002.00859).

Joint work with György Pál Gehér (University of Reading) and Dániel Virosztek (IST Austria).

Extras

Blank Analysis Seminar Template