Fall 2021 and Spring 2022 Analysis Seminars: Difference between revisions

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'''Analysis Seminar
'''Fall 2019 and Spring 2020 Analysis Seminar Series
'''
'''
[http://www.math.wisc.edu/~seeger/curr.html Current Semester]


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


===[[Previous Analysis seminars]]===
===[[Previous Analysis seminars]]===


= Analysis Seminar Schedule Spring 2017 =
= Analysis Seminar Schedule =
{| cellpadding="8"
{| cellpadding="8"
!align="left" | date   
!align="left" | date   
!align="left" | speaker
!align="left" | speaker
|align="left" | '''institution'''
!align="left" | title
!align="left" | title
!align="left" | host(s)
!align="left" | host(s)
|-
|-
|January 17, Math Department Colloquium
|Sept 10
| Fabio Pusateri (Princeton)  
| José Madrid
|[[#Fabio Pusateri   |  The Water Waves Problem ]]
| UCLA
| Sigurd Angenent
|[[#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
|[[#Joris Roos  |  L^p improving estimates for maximal spherical averages ]]
| Brian
|-
|Sept 20 (2:25 PM Friday, Room B139 VV)
| 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
|[[#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
|-
|Oct 15
| David Beltran
| UW Madison
|[[#David Beltran  |  Regularity of the centered fractional maximal function ]]
| 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
|-
|Oct 29
| Bingyang Hu
| UW Madison
|[[#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
|-
|Nov 12
| Francesco di Plinio
| Washington University in St. Louis
|[[#Francesco di Plinio  |  Maximal directional integrals along algebraic and lacunary sets]]
| Shaoming
|-
|Nov 13 (Wednesday)
| Xiaochun Li
| UIUC
|[[#Xiaochun Li  |   Roth's type theorems on progressions]]
| Brian, Shaoming
|-
|Nov 19
| Joao Ramos
| University of Bonn
|[[#Joao Ramos  Fourier uncertainty principles, interpolation and uniqueness sets ]]
| Joris, Shaoming
|-
|Jan 21
| No Seminar
|
|
|
|
|-
|-
|January 24, Joint Analysis/Geometry Seminar
|Friday, Jan 31, 4 pm, B239, Colloquium
| Tamás Darvas (Maryland)
| Lillian Pierce
|[[#Tamás Darvas Existence of constant scalar curvature Kähler metrics and properness of the K-energy ]]
| Duke University
| Jeff Viaclovsky
|[[#Lillian Pierce |   On Bourgain’s counterexample for the Schrödinger maximal function ]]
|
| Andreas, Simon
|-
|Feb 4
| Ruixiang Zhang
| UW Madison
|[[#Ruixiang Zhang |  Local smoothing for the wave equation in 2+1 dimensions ]]
| Andreas
|-
|Feb 11
| Zane Li
| Indiana University
|[[#Zane Li  |  A bilinear proof of decoupling for the moment curve ]]
| Betsy
|-
|-
|Monday, January 30, 3:30, VV901 (PDE Seminar)
|Feb 18
| Serguei Denissov (UW Madison)
| Sergey Denisov
|[[#Serguei Denissov | Instability in 2D Euler equation of incompressible inviscid fluid ]]
| UW Madison
|  
|[[#linktoabstract |   De Branges canonical systems with finite logarithmic integral ]]
| Brian
|-
|-
|February 7
|Feb 25
| Andreas Seeger (UW Madison)
|  Michel Alexis
|[[#Andreas Seeger|  The Haar system in Sobolev spaces]]
| UW Madison
|
|[[#Michel Alexis  |  The Steklov problem for trigonometric polynomials orthogonal to a Muckenhoupt weight ]]
| Sergey
|-
|-
|February 21
|Friday, Feb 28 (Colloquium)
| Jongchon Kim (UW Madison)
|  Brett Wick
|[[#Jongchon Kim Some remarks on Fourier restriction estimates ]]
| Washington University - St. Louis
| Andreas Seeger
|[[#MBrett Wick  The Corona Theorem]]
| Andreas
|-
|-
|March 7, Mathematics Department Distinguished Lecture
|Mar 3
| Roger Temam (Indiana) 
| William Green
|[[#Roger Temam (Colloquium) | On the mathematical  modeling of the humid atmosphere  ]]
| Rose-Hulman Institute of Technology
| Leslie Smith
|[[#William Green |   Dispersive estimates for the Dirac equation ]]
| Betsy
|-
|-
|Wednesday, March 8, Joint Applied Math/PDE/Analysis  Seminar
|Mar 10
| Roger Temam (Indiana) 
| Ziming Shi
|[[#Roger Temam (Seminar) |   Weak solutions of the Shigesada-Kawasaki-Teramoto system]]
| UW Madison
| Leslie Smith
|[[#linktoabstract |On the Sobolev space property of logarithmic modulus of holomorphic functions in C^n]]
| Xianghong
|-
|-
|March 14
|Mar 17
| Xianghong Chen (UW Milwaukee)
| Spring Break!
|[[#Xianghong Chen  |  Restricting the Fourier transform to some oscillating curves ]]
| Andreas Seeger
|
|
|
|
|-
|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
|-
|-
|March 21
|<b>Monday, Apr 13</b>
| SPRING BREAK
|Yumeng Ou
|[[#linktoabstract | ]]
|CUNY, Baruch College
 
|Canceled
 
|Ruixiang
|-
|-
|Monday, March 27 (joint PDE/Analysis Seminar), 3:30, VV901
|Apr 14
| Sylvia Serfaty (NYU)
| Tamás Titkos
|[[#Sylvia Serfaty |Mean Field Limits for Ginzburg Landau Vortices ]]
| BBS University of Applied Sciences & Rényi Institute
| Hung Tran
|Canceled
|
| Brian
|-
|-
|March 28
|Apr 21
| Brian Cook (Fields Institute)
| Diogo Oliveira e Silva
|[[#Brian Cook |Twists on the twisted ergodic theorems ]]
| University of Birmingham
| Andreas Seeger
|Canceled
|
| Betsy
|-
|-
|Friday, March 31, 4:00 p.m., B139
|Apr 28
| Laura Cladek (UBC)
| No Seminar
|[[#Laura Cladek | Endpoint bounds for the lacunary spherical maximal operator ]]
| Andreas Seeger
|
|-
|-
|April 4
|May 5
| Francesco Di Plinio (Virginia)
|Jonathan Hickman
|[[#Francesco di Plinio |  Sparse domination of singular integral operators ]]
|University of Edinburgh
| Andreas Seeger
|Canceled
|
| Andreas
|-
|-
|April 11
|Nov 17, 2020
| Xianghong Gong (UW Madison)
| Tamás Titkos
|[[#lXianghong Gong |  Hoelder estimates for homotopy operators on strictly pseudoconvex domains with C^2 boundary ]]
| BBS University of Applied Sciences & Rényi Institute
|  
|
|
| Brian
|-
|-
|April 25 (joint PDE/Analysis Seminar)
| Chris Henderson (University of Chicago)
|[[#|Chris Henderson  |  A local-in-time Harnack inequality and applications to reaction-diffusion equations]
| Jessica Lin
|}
|}


=Abstracts=
=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===


===  Fabio Pusateri  ===
Title: Distance sets on Riemannian surfaces and microlocal decoupling inequalities
''The Water Waves problem''


We will begin by introducing the free boundary Euler equations which are a system of nonlinear PDEs modeling the motion of fluids, such as waves on the surface of the ocean. We will discuss several works done on this system in recent years, and how they fit into the broader context of the study of nonlinear evolution problems. We will then focus on the question of global regularity for water waves, present some of our main results - obtained in collaboration with Ionescu and Deng-Ionescu-Pausader - and sketch some of the main ideas.
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.


===   Tamás Darvas ===
===Joris Roos===
''Existence of constant scalar curvature Kähler metrics and properness of the K-energy''


Given a compact Kähler manifold $(X,\omega)$, we show that if there exists a constant
Title: L^p improving estimates for maximal spherical averages
scalar curvature Kähler metric  cohomologous to $\omega$ then Mabuchi's K-energy is J-proper in an
appropriate sense, confirming a conjecture of Tian from the nineties. The proof involves a careful
study of weak minimizers of the K-energy, and involves a surprising amount of analysis. This is
joint work with Robert Berman and Chinh H. Lu.


=== Serguei Denissov  ===
Abstract: For a given compact set of radii $E$ we will discuss $L^p$ improving properties of maximal spherical averages with a supremum over $E$.
''Instability in 2D Euler equation of incompressible inviscid fluid''
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.


We consider the patch evolution under the 2D Euler dynamics and study how the geometry of the boundary can deteriorate in time.


=== Andreas Seeger  ===
''The Haar system in Sobolev spaces''


We consider the  Haar system on  Sobolev  spaces and ask:
===Joao Ramos===
When is it a Schauder basis?
When is it an unconditional  basis?
Some answers are given in recent joint work Tino Ullrich and Gustavo Garrigós.


=== Jongchon Kim  ===
Title: Fourier uncertainty principles, interpolation and uniqueness sets
''Some remarks on Fourier restriction estimates''


The Fourier restriction problem, raised by Stein in the 1960’s, is a hard open problem in harmonic analysis. Recently, Guth made some impressive progress on this problem using polynomial partitioning, a divide and conquer technique developed by Guth and Katz for some problems in incidence geometry.
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  
In this talk, I will introduce the restriction problem and the polynomial partitioning method. In addition, I will present some sharp L^p to L^q estimates for the Fourier extension operator that use an estimate of Guth as a black box.


=== Roger Temam (Colloquium) ===
$$ f(x) = \sum_{n \in \mathbb{Z}} f(n) \frac{\sin(\pi(x-n))}{\pi(x -n)}. $$
''On the mathematical  modeling of the humid atmosphere''


The humid atmosphere is a multi-phase system, made of air, water vapor, cloud-condensate, and rain water (and possibly ice / snow, aerosols and other components). The possible changes of phase due to evaporation and condensation make the equations nonlinear, non-continuous (and non-monotone) in the framework of nonlinear partial differential equations.
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 some modeling aspects, and some issues of existence, uniqueness and regularity for the solutions of the considered problems, making use of convex analysis, variational inequalities, and quasi-variational inequalities.
 
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 $f \equiv 0.$
=== Roger Temam (Seminar) ===
 
''Weak solutions of the Shigesada-Kawasaki-Teramoto system''
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===
 
<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
 
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.
 
 
===Kevin O'Neill===
 
<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.
 
===Francesco di Plinio===
 
<b>Maximal directional integrals along algebraic and lacunary sets </b>
 
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.
 
===Bingyang Hu===
 
<b>Sparse bounds of singular Radon transforms</b>
 
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===
<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===


We will present a result of existence of weak solutions to the Shigesada-Kawasaki-Teramoto system, in all dimensions. The method is based on new a priori estimates, the construction of approximate solutions and passage to the limit. The proof of existence is completely self-contained and does not rely on any earlier result.
<b> A bilinear proof of decoupling for the moment curve</b>
Based on an article with Du Pham, to appear in Nonlinear Analysis.


===  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.
''Restricting the Fourier transform to some oscillating curves''


I will talk about Fourier restriction to some compact smooth curves. The problem is relatively well understood for curves with nonvanishing torsion due to work of Drury from the 80's, but is less so for curves that contain 'flat' points (i.e. vanishing torsion). Sharp results are known for some monomial-like or finite type curves by work of Bak-Oberlin-Seeger, Dendrinos-Mueller, and Stovall, where a geometric inequality (among others) plays an important role. Such an inequality fails to hold if the torsion demonstrates strong sign-changing behavior, in which case endpoint restriction bounds may fail. In this talk I will present how one could obtain sharp non-endpoint results for certain space curves of this kind. Our approach uses a covering lemma for smooth functions that strengthens a variation bound of Sjolin, who used it to obtain a similar result for plane curves. This is joint work with Dashan Fan and Lifeng Wang.


===Sylvia Serfaty ===
===Sergey Denisov===


''Mean Field Limits for Ginzburg Landau Vortices''
<b> De Branges canonical systems with finite logarithmic integral </b>


Ginzburg-Landau type equations are models for superconductivity, superfluidity, Bose-Einstein condensation. A crucial feature is the presence of quantized vortices, which are topological zeroes of the complex-valued solutions. This talk will review some results on the derivation of effective models to describe the statics and dynamics of these vortices, with particular attention to the situation where the number of vortices blows up with the parameters of the problem. In particular we will present new results on the derivation of mean field limits for the dynamics of many vortices starting from the parabolic Ginzburg-Landau equation or the Gross-Pitaevskii (=Schrodinger Ginzburg-Landau) equation.
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===


=== Brian Cook ===
<b>The Steklov problem for Trigonometric Polynomials orthogonal to a Muckenhoupt weight</b>
''Twists on the twisted ergodic theorems''


The classical pointwise ergodic theorem has been adapted to include averages twisted by a phase polynomial, primary examples being the ergodic theorems of Wiener-Wintner and Lesigne. Certain uniform versions of these results are also known. Here uniformity refers to the collection of polynomials of degree less than some prescribed number. In this talk we wish to consider weakening the hypothesis in these latter results by considering uniformity over a smaller class of polynomials, which is naturally motivated when considering certain applications related to the circle method.
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===


=== Laura Cladek ===
<b> Dispersive estimates for the Dirac equation </b>
''Endpoint bounds for the lacunary spherical maximal operator''


Define the lacunary spherical maximal operator as the maximal operator corresponding to averages over spheres of radius 2^k for k an integer. This operator may be viewed as a model case for studying more general classes of singular maximal operators and Radon transforms. It is a classical result in harmonic analysis that this operator is bounded on L^p for p>1, but the question of weak-type (1, 1) boundedness (which would correspond to pointwise convergence of lacunary spherical averages for functions in L^1 has remained open. Although this question still remains open, we discuss some new endpoint bounds for the operator near L^1 that allows us to conclude almost everywhere pointwise convergence of lacunary spherical means for functions in a slightly smaller space than L\log\log\log L. This is based on joint work with Ben Krause.
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===


=== Francesco di Plinio ===
<b>On the Sobolev space property of logarithmic modulus of holomorphic functions in C^n</b>
''Sparse domination of singular integral operators''


Singular integral operators, which are a priori signed and non-local, can be dominated  in norm, pointwise, or dually, by sparse averaging operators, which are in contrast positive and localized. The most striking consequence is that weighted norm inequalities for the singular integral follow from the corresponding, rather immediate estimates for the averaging operators. In this talk, we present several positive sparse domination results of singular integrals falling beyond the scope of classical Calderón-Zygmund theory; notably, modulation invariant multilinear singular integrals including the bilinear Hilbert transforms, variation norm Carleson operators, matrix-valued kernels, rough homogeneous singular integrals and critical Bochner-Riesz means, and singular integrals along submanifolds with curvature.  Collaborators:  Amalia Culiuc, Laura Cladek, Jose Manuel Conde-Alonso, Yen Do, Yumeng Ou and Gennady Uraltsev.
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===


===Xianghong Gong===
<b>Isometries of Wasserstein spaces</b>
''Hoelder estimates for homotopy operators on strictly pseudoconvex domains with C^2 boundary''


Abstract: We derive a new homotopy formula for a bounded strictly pseudoconvex domain of C^2 boundary by using a method of Lieb and Range, and we obtain estimates for its homotopy operator. We show that the d-bar equation on the domain admits a solution gaining half-derivative in the Hoelder-Zygmund spaces. The estimates are also applied to obtain a boundary regularity for D-solutions on a suitable product domain in the Levi-flat Euclidean 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.


===Chris Henderson===
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).
''A local-in-time Harnack inequality and applications to reaction-diffusion equations''


Abstract: The classical Harnack inequality requires one to look back in time to obtain a uniform lower bound on the solution to a parabolic equation.  In this talk, I will introduce a Harnack-type inequality that allows us to remove this restriction at the expense of a slightly weaker bound.  I will then discuss applications of this bound to (time permitting) three non-local reaction-diffusion equations arising in biology.  In particular, in each case, this inequality allows us to show that solutions to these equations, which do not enjoy a maximum principle, may be compared with solutions to a related local equation, which does enjoy a maximum principle.  Precise estimates of the propagation speed follow from this.
Joint work with György Pál Gehér (University of Reading) and Dániel Virosztek (IST Austria).


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Revision as of 12:02, 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