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'''Analysis Seminar
+
'''Fall 2019 and Spring 2020 Analysis Seminar Series
 
'''
 
'''
  
 
The seminar will  meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.
 
The seminar will  meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.
  
If you wish to invite a speaker please  contact  Betsy at stovall(at)math
+
If you wish to invite a speaker please  contact  Brian at street(at)math
  
 
===[[Previous Analysis seminars]]===
 
===[[Previous Analysis seminars]]===
  
= 2017-2018 Analysis Seminar Schedule =
+
= Analysis Seminar Schedule =
 
{| cellpadding="8"
 
{| cellpadding="8"
 
!align="left" | date   
 
!align="left" | date   
Line 16: Line 16:
 
!align="left" | host(s)
 
!align="left" | host(s)
 
|-
 
|-
|September 8 in B239 (Colloquium)
+
|Sept 10
| Tess Anderson
+
| José Madrid
 +
| UCLA
 +
|[[#José Madrid  |  On the regularity of maximal operators on Sobolev Spaces ]]
 +
| Andreas, David
 +
|-
 +
|Sept 13 (Friday, B139)
 +
| Yakun Xi
 +
| University of  Rochester
 +
|[[#Yakun Xi  |  Distance sets on Riemannian surfaces and microlocal decoupling inequalities ]]
 +
| Shaoming
 +
|-
 +
|Sept 17
 +
| Joris Roos
 
| UW Madison
 
| UW Madison
|[[#linktoabstract A Spherical Maximal Function along the Primes]]
+
|[[#Joris Roos L^p improving estimates for maximal spherical averages ]]
|Tonghai
+
| Brian
 
|-
 
|-
|September 19
+
|Sept 20 (2:25 PM Friday, Room B139 VV)
| Brian Street
+
| Xiaojun Huang
 +
| Rutgers University–New Brunswick
 +
|[[#linktoabstract  |  A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries ]]
 +
| Xianghong
 +
|-
 +
|Oct 1
 +
| Xiaocheng Li
 
| UW Madison
 
| UW Madison
|[[#Brian Street Convenient Coordinates ]]
+
|[[#Xiaocheng Li  |  An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$ ]]
 +
| Simon
 +
|-
 +
|Oct 8
 +
| Jeff Galkowski
 +
| Northeastern University
 +
|[[#Jeff Galkowski Concentration and Growth of Laplace Eigenfunctions ]]
 
| Betsy
 
| Betsy
 
|-
 
|-
|September 26
+
|Oct 15
| Hiroyoshi Mitake
+
| David Beltran
| Hiroshima University
+
| UW Madison
|[[#Hiroyoshi Mitake |   Derivation of multi-layered interface system and its application ]]
+
|[[#David Beltran  |  Regularity of the centered fractional maximal function ]]
| Hung
+
| Brian
 +
|-
 +
|Oct 22
 +
| Laurent Stolovitch
 +
| University of Côte d'Azur
 +
|[[#Laurent Stolovitch | Linearization of neighborhoods of embeddings of complex compact manifolds ]]
 +
| Xianghong
 +
|-
 +
|<b>Wednesday Oct 23 in B129</b>
 +
|Dominique Kemp
 +
|Indiana University
 +
|[[#Dominique Kemp | Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature]]
 +
|Betsy
 
|-
 
|-
|October 3
+
|Oct 29
| Joris Roos
+
| Bingyang Hu
 
| UW Madison
 
| UW Madison
|[[#Joris Roos |  A polynomial Roth theorem on the real line ]]
+
|[[#Bingyang Hu  |    Sparse bounds of singular Radon transforms]]
 +
| Brian
 +
|-
 +
|Nov 5
 +
| Kevin O'Neill
 +
| UC Davis
 +
|[[#Kevin O'Neill |  A Quantitative Stability Theorem for Convolution on the Heisenberg Group ]]
 
| Betsy
 
| Betsy
 
|-
 
|-
|October 10
+
|Nov 12
| Michael Greenblatt
+
| Francesco di Plinio
| UI Chicago
+
| Washington University in St. Louis
|[[#Michael Greenblatt |  Maximal averages and Radon transforms for two-dimensional hypersurfaces ]]
+
|[[#Francesco di Plinio |  Maximal directional integrals along algebraic and lacunary sets]]
| Andreas
+
| Shaoming
 
|-
 
|-
|October 17
+
|Nov 13 (Wednesday)
| David Beltran
+
| Xiaochun Li
| Basque Center of Applied Mathematics
+
| UIUC
|[[#David Beltran Fefferman-Stein inequalities ]]
+
|[[#Xiaochun Li Roth's type theorems on progressions]]
| Andreas
+
| Brian, Shaoming
 
|-
 
|-
|Wednesday, October 18, 4:00 p.m.  in B131
+
|Nov 19
|Jonathan Hickman
+
| Joao Ramos
|University of Chicago
+
| University of Bonn
|[[#Jonathan Hickman | Factorising X^n  ]]
+
|[[#Joao Ramos |   Fourier uncertainty principles, interpolation and uniqueness sets ]]
|Andreas
+
| Joris, Shaoming
 
|-
 
|-
|October 24
+
|Jan 21
| Xiaochun Li
+
| No Seminar
| UIUC
+
|  
|[[#Xiaochun Li  |  Recent progress on the pointwise convergence problems of Schroedinger equations ]]
+
|
| Betsy
+
|
 
|-
 
|-
|Thursday, October 26, 4:30 p.m. in B139
+
|Friday, Jan 31, 4 pm, B239, Colloquium
| Fedor Nazarov
+
| Lillian Pierce
| Kent State University
+
| Duke University
|[[#Fedor Nazarov | The Lerner-Ombrosi-Perez bound in the Muckenhoupt Wheeden conjecture is sharp  ]]
+
|[[#Lillian Pierce |   On Bourgain’s counterexample for the Schrödinger maximal function ]]
| Sergey, Andreas
+
| Andreas, Simon
 
|-
 
|-
|Friday, October 27, 4:00 p.m.  in B239
+
|Feb 4
| Stefanie Petermichl
+
| Ruixiang Zhang
| University of Toulouse
+
| UW Madison
|[[#Stefanie Petermichl | Higher order Journé commutators   ]]
+
|[[#Ruixiang Zhang Local smoothing for the wave equation in 2+1 dimensions ]]
| Betsy, Andreas
+
| Andreas
 
|-
 
|-
|Wednesday, November 1, 4:00 p.m. in B239 (Colloquium)
+
|Feb 11
| Shaoming Guo
+
| Zane Li
 
| Indiana University
 
| Indiana University
|[[#Shaoming Guo Parsell-Vinogradov systems in higher dimensions ]]
+
|[[#Zane Li A bilinear proof of decoupling for the moment curve ]]
| Andreas
+
| Betsy
 +
|-
 +
|Feb 18
 +
| Sergey Denisov
 +
| UW Madison
 +
|[[#linktoabstract  |  De Branges canonical systems with finite logarithmic integral ]]
 +
| Brian
 
|-
 
|-
|November 14
+
|Feb 25
| Naser Talebizadeh Sardari
+
| Michel Alexis
 
| UW Madison
 
| UW Madison
|[[#Naser Talebizadeh Sardari Quadratic forms and the semiclassical eigenfunction hypothesis ]]
+
|[[#Michel Alexis The Steklov problem for trigonometric polynomials orthogonal to a Muckenhoupt weight ]]
| Betsy
+
| Sergey
 
|-
 
|-
|November 28
+
|Friday, Feb 28 (Colloquium)
| Xianghong Chen
+
| Brett Wick
| UW Milwaukee
+
| Washington University - St. Louis
|[[#Xianghong Chen Some transfer operators on the circle with trigonometric weights ]]
+
|[[#MBrett Wick The Corona Theorem]]
| Betsy
+
| Andreas
 
|-
 
|-
|Monday, December 4, 4:00, B139
+
|Mar 3
| Bartosz Langowski and Tomasz Szarek
+
| William Green
| Institute of Mathematics, Polish Academy of Sciences
+
| Rose-Hulman Institute of Technology
|[[#Bartosz Langowski and Tomasz Szarek Discrete Harmonic Analysis in the Non-Commutative Setting ]]
+
|[[#William Green Dispersive estimates for the Dirac equation ]]
 
| Betsy
 
| Betsy
 
|-
 
|-
|Wednesday, December 13, 4:00, B239 (Colloquium)
+
|Mar 10
|Bobby Wilson
+
| Ziming Shi
|MIT
+
| UW Madison
|[[#Bobby Wilson | Projections in Banach Spaces and Harmonic Analysis ]]
+
|[[#linktoabstract  |On the Sobolev space property of logarithmic modulus of holomorphic functions in C^n]]
| Andreas
+
| Xianghong
 
|-
 
|-
|January 30
+
|Mar 17
|  
+
| Spring Break!
|  
+
|
| [[#linkofabstract | Title]]
 
 
|
 
|
|-
 
|February 6
 
| Dong Dong
 
| UIUC
 
| [[#Dong Dong | Hibert transforms in a 3 by 3 matrix and applications in number theory]]
 
|Betsy
 
|-
 
|February 13
 
| Serguei Denissov
 
| UW
 
| [[#linkofabstract | Title]]
 
 
|  
 
|  
 
|-
 
|-
|February 20
+
|Mar 24
| Ruixiang Zhang
+
| Oscar Dominguez
| IAS (Princeton)
+
| Universidad Complutense de Madrid
| [[#Ruixiang Zhang | The (Euclidean) Fractal Uncertainty Principle]]
+
|Canceled
| Betsy, Jordan, Andreas
+
| Andreas
 
|-
 
|-
|February 27
+
|Mar 31
|Detlef Müller (tent.)
+
| Brian Street
|University of Kiel
+
| University of Wisconsin-Madison
| [[#linkofabstract | Title]]
+
|Canceled
|Betsy, Andreas
+
| Local
 
|-
 
|-
|Wednesday, March 7, 4:00 p.m.
+
|Apr 7
| Winfried Sickel
+
| Hong Wang
|Friedrich-Schiller-Universität Jena
+
| Institution
| [[#linkofabstract | Title]]
+
|Canceled
|Andreas
+
| Street
 
|-
 
|-
|March 13
+
|<b>Monday, Apr 13</b>
| TBA
+
|Yumeng Ou
|  
+
|CUNY, Baruch College
| [[#linkofabstract | Title]]
+
|Canceled
|
+
|Ruixiang
 
|-
 
|-
|March 20
+
|Apr 14
|  
+
| Tamás Titkos
|  
+
| BBS University of Applied Sciences & Rényi Institute
| [[#linkofabstract | Title]]
+
|Canceled
|
+
| Brian
 
|-
 
|-
|April 3
+
|Apr 21
|  
+
| Diogo Oliveira e Silva
|  
+
| University of Birmingham
| [[#linkofabstract | Title]]
+
|Canceled
|
+
| Betsy
|-
 
|April 10
 
|
 
|
 
| [[#linkofabstract | Title]]
 
|
 
 
|-
 
|-
|April 17
+
|Apr 28
|
+
| No Seminar
|
 
| [[#linkofabstract | Title]]
 
|
 
 
|-
 
|-
|April 24
+
|May 5
|
+
|Jonathan Hickman
|
+
|University of Edinburgh
| [[#linkofabstract | Title]]
+
|Canceled
|
+
| Andreas
|-
 
|May 1
 
|  
 
|  
 
| [[#linkofabstract | Title]]
 
|
 
 
|-
 
|-
| May 16-18, [http://www.math.wisc.edu/~stovall/FA2018/ Workshop in Fourier Analysis]
+
|Nov 17, 2020
|
+
| Tamás Titkos
 +
| BBS University of Applied Sciences & Rényi Institute
 
|
 
|
|
+
| Brian
|Betsy, Andreas
 
 
|-
 
|-
 
|}
 
|}
  
 
=Abstracts=
 
=Abstracts=
===Brian Street===
+
===José Madrid===
  
Title: Convenient Coordinates
+
Title: On the regularity of maximal operators on Sobolev Spaces
  
Abstract:  We discuss the method of picking a convenient coordinate system adapted to vector fields.  Let X_1,...,X_q be either real or complex C^1 vector fields.  We discuss the question of when there is a coordinate system in which the vector fields are smoother (e.g., C^m, or C^\infty, or real analytic). By answering this in a quantitative way, we obtain coordinate charts which can be used as generalized scaling maps. When the vector fields are real this is joint work with Stovall, and continues in the line of quantitative sub-Riemannian geometry initiated by Nagel, Stein, and Wainger.  When the vector fields are complex one obtains a geometry with more structure which can be thought of as "sub-Hermitian".
+
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.
  
===Hiroyoshi Mitake===
+
===Yakun Xi===
  
Title: Derivation of multi-layered interface system and its application
+
Title: Distance sets on Riemannian surfaces and microlocal decoupling inequalities
  
Abstract:   In this talk, I will propose a multi-layered interface system which can
+
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.
be formally derived by the singular limit of the weakly coupled system of  
 
the Allen-Cahn equation.  By using the level set approach, this system can be
 
written as a quasi-monotone degenerate parabolic system.
 
We give results of the well-posedness of viscosity solutions, and study the
 
singularity of each layers. This is a joint work with H. Ninomiya, K. Todoroki.
 
  
 
===Joris Roos===
 
===Joris Roos===
  
Title: A polynomial Roth theorem on the real line
+
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: For a polynomial P of degree greater than one, we show the existence of patterns of the form (x,x+t,x+P(t)) with a gap estimate on t in positive density subsets of the reals. This is an extension of an earlier result of Bourgain. Our proof is a combination of Bourgain’s approach and more recent methods that were originally developed for the study of the bilinear Hilbert transform along curves. This talk is based on a joint work with Polona Durcik and Shaoming Guo.
+
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.
  
===Michael Greenblatt===
+
===Xiaocheng Li===
  
Title:  Maximal averages and Radon transforms for two-dimensional hypersurfaces
+
Title:  An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$
  
Abstract:  A general local result concerning L^p boundedness of maximal averages over 2D hypersurfaces is described, where p > 2. The surfaces are allowed to have either the traditional smooth density function or a singularity growing as |(x,y)|^{-t} for some 0 < t < 2. This result is a generalization of a theorem of Ikromov, Kempe, and Mueller. Similar methods can be used to show sharp L^p to L^p_a Sobolev estimates for associated Radon transform operators when p is in a certain interval containing 2.
+
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===
 +
 
 +
<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===
 
===David Beltran===
  
Title:  Fefferman Stein Inequalities
+
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===
 +
 
 +
<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.
 +
 
  
Abstract:  Given an operator T, we focus on obtaining two-weighted inequalities in which the weights are related via certain maximal function. These inequalites, which originated in work of Fefferman and Stein, have been established in an optimal way for different classical operators in Harmonic Analysis. In this talk, we survey some classical results and we present some recent Fefferman-Stein inequalities for pseudodifferential operators and for the solution operators to dispersive equations.
+
===Kevin O'Neill===
  
===Jonathan Hickman===
+
<b>A Quantitative Stability Theorem for Convolution on the Heisenberg Group </b>
  
Title: Factorising X^n.
+
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.
  
Question: how many ways can the polynomial $X^n$ be factorised as a product of linear factors? Answer: it depends on the ring... In this talk I will describe joint work with Jim Wright investigating certain exponential sum estimates over rings of integers modulo N. This theory serves as a discrete analogue of the (euclidean) Fourier restriction problem, a central question in contemporary harmonic analysis. In particular, as part of this study, the question of counting the number of factorisations of polynomials over such rings naturally arises. I will describe how these number-theoretic considerations can themselves be approached via methods from harmonic analysis.
+
===Francesco di Plinio===
  
===Xiaochun Li ===
+
<b>Maximal directional integrals along algebraic and lacunary sets </b>
  
Title:  Recent progress on the pointwise convergence problems of Schrodinger equations
+
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.
  
Abstract:  Recently, Guth, Du and I solved the pointwise convergence problem of Schrodinger equations in two-dimensional case. We proved that the solution to free Schrodinger equation in R^2 converges to its initial data, provided the initial data belongs to H^s for s larger than 1/3.  This result is sharp, up to the end point, due to Bourgain's example. The proof relies on the polynomial partitioning method and  the decoupling method. In addition, the pointwise convergence problem is closely related to Fourier restriction conjecture.
+
===Laurent Stolovitch===
  
===Fedor Nazarov=== 
+
<b>Linearization of neighborhoods of embeddings of complex compact manifolds </b>
  
Title: The Lerner-Ombrosi-Perez bound in the Muckenhoupt-Wheeden
+
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.
conjecture is sharp.
 
  
Abstract: We show that the upper bound $[w]_{A_1}\log (e+[w]_{A_1})$ for
+
===Bingyang Hu===
the norm of the Hilbert transform on the line as an operator from $L^1(w)$
 
to $L^{1,\infty}(w)$ cannot be improved in general. This is a joint work
 
with Andrei Lerner and Sheldy Ombrosi.
 
  
===Stefanie Petermichl===
+
<b>Sparse bounds of singular Radon transforms</b>
Title: Higher order Journé commutators
 
  
Abstract: We consider questions that stem from operator theory via Hankel and
+
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.
Toeplitz forms and target (weak) factorisation of Hardy spaces. In
 
more basic terms, let us consider a function on the unit circle in its
 
Fourier representation. Let P_+ denote the projection onto
 
non-negative and P_- onto negative frequencies. Let b denote
 
multiplication by the symbol function b. It is a classical theorem by
 
Nehari that the composed operator P_+ b P_- is bounded on L^2 if and
 
only if b is in an appropriate space of functions of bounded mean
 
oscillation. The necessity makes use of a classical factorisation
 
theorem of complex function theory on the disk. This type of question
 
can be reformulated in terms of commutators [b,H]=bH-Hb with the
 
Hilbert transform H=P_+ - P_- . Whenever factorisation is absent, such
 
as in the real variable setting, in the multi-parameter setting or
 
other, these classifications can be very difficult.
 
  
Such lines were begun by Coifman, Rochberg, Weiss (real variables) and
+
===Lillian Pierce===
by Cotlar, Ferguson, Sadosky (multi-parameter) of characterisation of
+
<b> On Bourgain’s counterexample for the Schrödinger maximal function </b>
spaces of bounded mean oscillation via L^p boundedness of commutators.
 
We present here an endpoint to this theory, bringing all such
 
characterisation results under one roof.
 
  
The tools used go deep into modern advances in dyadic harmonic
+
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.
analysis, while preserving the Ansatz from classical operator theory.
+
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.”
  
===Shaoming Guo ===
+
===Ruixiang Zhang===
Title: Parsell-Vinogradov systems in higher dimensions
 
  
Abstract:
+
<b> Local smoothing for the wave equation in 2+1 dimensions </b>
I will present a few results on counting the numbers of integer solutions of Parsell-Vinogradov systems in higher dimensions.
 
Applications to Waring’s problem and to the problem of counting rational linear subspaces lying on certain hyper-surface will be discussed.
 
Joint works with Jean Bourgain, Ciprian Demeter and Ruixiang Zhang.
 
  
===Naser Talebizadeh Sardari===
+
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.
  
Title: Quadratic forms and the semiclassical eigenfunction hypothesis
+
===Zane Li===
  
Abstract:  Let <math>Q(X)</math> be any integral primitive positive definite quadratic form in <math>k</math> variables, where <math>k\geq4</math>,  and discriminant <math>D</math>. For any integer <math>n</math>, we give an upper bound on the number of integral solutions of <math>Q(X)=n</math>  in terms of <math>n</math>, <math>k</math>, and <math>D</math>. As a corollary, we prove  a conjecture of Lester and Rudnick on the small scale equidistribution of almost all functions belonging to any orthonormal basis of a given  eigenspace of the Laplacian on the flat torus <math>\mathbb{T}^d</math> for <math>d\geq 5</math>. This conjecture is motivated by the work of Berry\cite{Berry, Michael} on semiclassical eigenfunction hypothesis.
+
<b> A bilinear proof of decoupling for the moment curve</b>
  
===Xianghong Chen===
+
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.
  
Title:  Some transfer operators on the circle with trigonometric weights
 
  
Abstract:  A transfer operator is an averaging operator over the preimages of a given map. Certain dynamical properties of the map can be studied through its associated transfer operator. In this talk we will introduce a class of weighted transfer operators associated to the Bernoulli maps on the circle (i.e. multiplication by a given integer, mod 1). We will illustrate how the spectral properties of these operators may depend on the specific weight chosen and demonstrate multiple phase transitions. We also present some results on evaluating the spectral radii and corresponding eigenfunctions of these operators, as well as their connections to Fourier analysis. This is joint work with Hans Volkmer.
+
===Sergey Denisov===
  
===Bobby Wilson===
+
<b> De Branges canonical systems with finite logarithmic integral </b>
  
Title: Projections in Banach Spaces and Harmonic Analysis
+
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.
  
Abstract: In this talk, we will discuss the measure theoretic principles of orthogonal projections that follow from the classical Besicovitch-Federer projection theorem. The Besicovitch-Federer projection theorem offers a characterization of rectifiability of one-dimensional sets in R^d by the size of their projections to lines. We will focus on the validity of analogues to the Besicovitch-Federer projection theorem with respect to such sets in general Banach spaces. In particular, we will show that the projection theorem is false when the Banach space is infinite-dimensional and discuss related applications to questions in Harmonic Analysis. This is joint work with Marianna Csornyei and David Bate.
 
  
===Dong Dong===
+
===Michel Alexis===
  
Title: Hibert transforms in a 3 by 3 matrix and applications in number theory
+
<b>The Steklov problem for Trigonometric Polynomials orthogonal to a Muckenhoupt weight</b>
  
Abstract:  This talk could interest both analysts and number theorists. I will first present 35 variants of Hilbert transforms, with a focus on their connections with ergodic theory, number theory, and combinatorics. Then I will show how to use Fourier analysis tools to reduce a number theory problem (Roth theorem) to an algebraic geometry problem: this joint work Li and Sawin fully answers a question of Bourgain and Chang about three-term polynomial progressions in subsets of finite fields. I guarantee that a second-year graduate student can understand at least 50% of the talk.
+
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.
  
===Ruixiang Zhang===
+
===William Green===
 +
 
 +
<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.
  
Title:  The (Euclidean) Fractal Uncertainty Principle
+
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).
  
Abstract:  On the real line, a  version of the uncertainty principle says: If a nonzero function f has its Fourier support lying in B and |A||B| is much smaller than 1, then the L^2 norm of f on A cannot be close to the whole L^2 norm of f. Recently, Bourgain and Dyatlov proved a Fractal Uncertainty Principle (FUP) which has a similar statement. The difference is that in FUP the product of |A| and |B| can be much bigger, but A and B both have to be porous at many scales. We will introduce the theorem and then discuss some unusual features of its proof, most notably the application of the Beurling-Malliavin Theorem. In the original work the exponent was ineffective and we will also discuss why we can overcome this ineffectivity (joint work with Long Jin).
+
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