Difference between revisions of "Analysis Seminar"
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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. | In this talk, I will present a proof of the following Sobolev space property of logarithmic modulus of holomorphic functions in C^n. If f is a holomorphic function on the unit ball B(0,1) in C^n vanishing at the origin (i.e., f(0) = 0) but it is not identically zero, then log |f| ∈ W^{1,p}(B(0, r)) for any p < 2, but log |f| is not in W^{1,2}(B(0, r)) (r < 1). As you may see, this result is rather simple to prove in the complex plane due to the discreteness of zeros of holomorphic functions. In higher dimensions, we are going to apply Hironaka’s resolution of singularity and then Harvey- Polking removable singularity method to prove the existence of weak derivatives of log |f(z)|. This is part of a joint project with Ziming Shi at Madison. | ||
+ | |||
+ | |||
+ | ===Tamás Titkos=== | ||
+ | |||
+ | <b>Isometries of Wasserstein spaces</b> | ||
+ | |||
+ | Due to its nice theoretical properties and an astonishing number of applications via optimal transport problems, probably the most intensively studied metric nowadays is the $p$-Wasserstein metric. Given a complete and separable metric space $X$ and a real number $p\geq1$, one defines the $p$-Wasserstein space $\mathcal{W}_p(X)$ as the collection of Borel probability measures with finite $p$-th moment, endowed with a distance which is calculated by means of transport plans. | ||
+ | |||
+ | The main aim of our research project is to reveal the structure of the isometry group $\mathrm{Isom}(\mathcal{W}_p(X))$. Although $\mathrm{Isom}(X)$ embeds naturally into $\mathrm{Isom}(\mathcal{W}_p(X))$ by push-forward, and this embedding turned out to be surjective in many cases, these two groups are not isomorphic in general. Recently, Kloeckner described the isometry group of the quadratic Wasserstein space over the real line. It turned out that this group is extremely rich: it contains a flow of wild behaving isometries that distort the shape of measures. Following this line of investigation, we described $\mathrm{Isom}(\mathcal{W}_p(\mathbb{R}))$ and $\mathrm{Isom}(\mathcal{W}_p([0,1])$ for all $p\geq 1$. In this talk I will survey first some of the earlier results in the subject, and then I will present the key results of our recent manuscript \emph{"Isometric study of Wasserstein spaces -- The real line"} (to appear in Trans. Amer. Math. Soc., arXiv:2002.00859). | ||
+ | |||
+ | Joint work with György Pál Gehér (University of Reading) and Dániel Virosztek (IST Austria). | ||
=Extras= | =Extras= | ||
[[Blank Analysis Seminar Template]] | [[Blank Analysis Seminar Template]] |
Revision as of 10:59, 29 February 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
Contents
- 1 Previous Analysis seminars
- 2 Analysis Seminar Schedule
- 3 Abstracts
- 3.1 José Madrid
- 3.2 Yakun Xi
- 3.3 Joris Roos
- 3.4 Joao Ramos
- 3.5 Xiaojun Huang
- 3.6 Xiaocheng Li
- 3.7 Xiaochun Li
- 3.8 Jeff Galkowski
- 3.9 David Beltran
- 3.10 Dominique Kemp
- 3.11 Kevin O'Neill
- 3.12 Francesco di Plinio
- 3.13 Laurent Stolovitch
- 3.14 Bingyang Hu
- 3.15 Lillian Pierce
- 3.16 Ruixiang Zhang
- 3.17 Zane Li
- 3.18 Sergey Denisov
- 3.19 Michel Alexis
- 3.20 William Green
- 3.21 Yifei Pan
- 3.22 Tamás Titkos
- 4 Extras
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 | Yifei Pan | Purdue University Fort Wayne | 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 | Title | Andreas |
Mar 31 | Brian Street | University of Wisconsin-Madison | Title | Local |
Apr 7 | Hong Wang | Institution | Title | Street |
Monday, Apr 13 | Yumeng Ou | CUNY, Baruch College | TBA | Zhang |
Apr 14 | Tamás Titkos | BBS University of Applied Sciences & Rényi Institute | Isometries of Wasserstein spaces | Street |
Apr 21 | Diogo Oliveira e Silva | University of Birmingham | Title | Betsy |
Apr 28 | No Seminar | |||
May 5 | Jonathan Hickman | University of Edinburgh | Title | Andreas |
Abstracts
José Madrid
Title: On the regularity of maximal operators on Sobolev Spaces
Abstract: In this talk, we will discuss the regularity properties (boundedness and continuity) of the classical and fractional maximal operators when these act on the Sobolev space W^{1,p}(\R^n). We will focus on the endpoint case p=1. We will talk about some recent results and current open problems.
Yakun Xi
Title: Distance sets on Riemannian surfaces and microlocal decoupling inequalities
Abstract: In this talk, we discuss the generalization of the Falconer distance problem to the Riemannian setting. In particular, we extend the recent result of Guth-Iosevich-Ou-Wang for the distance set in the plane to general Riemannian surfaces. The key new ingredient is a family of refined decoupling inequalities associated with phase functions that satisfy Carleson-Sj\”olin condition. This is joint work with Iosevich and Liu.
Joris Roos
Title: L^p improving estimates for maximal spherical averages
Abstract: For a given compact set of radii $E$ we will discuss $L^p$ improving properties of maximal spherical averages with a supremum over $E$. Our results are sharp up to endpoints for a large class of $E$. A new feature is that the optimal exponents depend on both, the upper Minkowski dimension and the Assouad dimension of the set $E$. Joint work with Tess Anderson, Kevin Hughes and Andreas Seeger.
Joao Ramos
Title: Fourier uncertainty principles, interpolation and uniqueness sets
Abstract: A classical result in the theory of entire functions of exponential type, Shannon’s interpolation formula predicates that, given a function whose Fourier transform vanishes outside the interval $[-1/2,1/2]$, it is possible to recover it from its values at the integers. More specifically, it holds, in a suitable sense of convergence, that
$$ f(x) = \sum_{n \in \mathbb{Z}} f(n) \frac{\sin(\pi(x-n))}{\pi(x -n)}. $$
This formula is unfortunately unavailable for arbitrary Schwartz functions on the real line, but a recent result of Radchenko and Viazovska provides us with an explicit construction of an interpolation basis for even Schwartz functions. It states, in a nutshell, that we can recover explicitly the function given its values at the squares of roots of integers.
We will discuss a bit these two results, and explore, in connection to classical Fourier uncertainty results, the question of determining which pairs of sets $(A,B)$ satisfy that, if a Schwartz function $f$ vanishes on A and its Fourier transform vanishes on B, then $f \equiv 0.$
In particular, we will give sufficient conditions on $(\alpha,\beta)$ pairs of positive numbers so that, if $f$ vanishes at $\pm n^{\alpha}$ and its Fourier transform vanishes at $\pm n^{\beta}$, then $f$ is identically zero.
Xiaojun Huang
Title: A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries
Abstract: This is a joint work with Ming Xiao. We discuss how to construct a hyperbolic metric over a Stein space with spherical boundary. The technique we use is to employ holomorphic continuation along curves for multiple valued functions.
Xiaocheng Li
Title: An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$
Abstract: We prove an estimate for spherical functions $\phi_\lambda(a)$ on $\mathrm{SL}(3,\mathbb{R})$, establishing uniform decay in the spectral parameter $\lambda$ when the group parameter $a$ is restricted to a compact subset of the abelian subgroup $\mathrm{A}$. In the case of $\mathrm{SL}(3,\mathbb{R})$, it improves a result by J.J. Duistermaat, J.A.C. Kolk and V.S. Varadarajan by removing the limitation that $a$ should remain regular. As in their work, we estimate the oscillatory integral that appears in the integral formula for spherical functions by the method of stationary phase. However, the major difference is that we investigate the stability of the singularities arising from the linearized phase function by classifying their local normal forms when the parameters $\lambda$ and $a$ vary.
Xiaochun Li
Title: Roth’s type theorems on progressions
Abstract: The arithmetic progression problems were posed by Erd\”os-Turan, answered affirmatively by Semer\’edi. However, there are still many questions remained on precise quantitative description on how large a subset shall be in oredr to guarantee a progression in it. Involving with Fourier analysis, considerable work had been accomplished recently. We will give a survey on those progress, and report our recent progress on quantitative version of Roth’s type theorem on (polynomial) progressions of short length.
Jeff Galkowski
Concentration and Growth of Laplace Eigenfunctions
In this talk we will discuss a new approach to understanding eigenfunction concentration. We characterize the features that cause an eigenfunction to saturate the standard supremum bounds in terms of the distribution of L^2 mass along geodesic tubes emanating from a point. We also show that the phenomena behind extreme supremum norm growth is identical to that underlying extreme growth of eigenfunctions when averaged along submanifolds. Finally, we use these ideas to understand a variety of measures of concentration; in each case obtaining quantitative improvements over the known bounds.
David Beltran
Title: Regularity of the centered fractional maximal function
Abstract: I will report some recent progress regarding the boundedness of the map $f \mapsto |\nabla M_\beta f|$ from the endpoint space $W^{1,1}(\mathbb{R}^d)$ to $L^{d/(d-\beta)}(\mathbb{R}^d)$, where $M_\beta$ denotes the fractional version of the centered Hardy--Littlewood maximal function. A key step in our analysis is a relation between the centered and non-centered fractional maximal functions at the derivative level, which allows to exploit the known techniques in the non-centered case.
This is joint work with José Madrid.
Dominique Kemp
Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature
The celebrated l^2 decoupling theorem of Jean Bourgain and Ciprian Demeter presented a new perspective on a range of problems related to hypersurfaces with nonzero Gaussian curvature, such as exponential sum estimates, additive energy estimates, local smoothing, and counting solutions to Diophantine inequalities. The same authors also extended their theory to the n-dimensional cone. Following their steps, we prove optimal l^2 decoupling results for the remaining class of zero-curvature two-dimensional surfaces without umbilical points (the so-called tangent surfaces). We are also able to prove a decoupling theorem for the real analytic surfaces of revolution. These results should be viewed as partial progress toward the goal of proving a decoupling theorem for arbitrary real analytic hypersurfaces.
Kevin O'Neill
A Quantitative Stability Theorem for Convolution on the Heisenberg Group
Although convolution on Euclidean space and the Heisenberg group satisfy the same $L^p$ bounds with the same optimal constants, the former has maximizers while the latter does not. However, as work of Christ has shown, it is still possible to characterize near-maximizers. Specifically, any near-maximizing triple of the trilinear form for convolution on the Heisenberg group must be close to a particular type of triple of ordered Gaussians after adjusting by symmetry. In this talk, we will use the expansion method to prove a quantitative version of this characterization.
Francesco di Plinio
Maximal directional integrals along algebraic and lacunary sets
I will discuss two recent results obtained in collaboration with (partly) Natalia Accomazzo and Ioannis Parissis (U Basque Country). The first is a sharp $L^2$ estimate for the maximal averaging operator associated to sets of directions from algebraic sets in R^n of arbitrary codimension. The proof uses a new scheme of polynomial partitioning on manifolds which extends ideas by Larry Guth. The second result is a sharp estimate in all dimensions for the maximal directional singular integrals along lacunary directions. This settles a question of Parcet and Rogers. The proof uses a combination of two-dimensional and $n$-dimensional coverings combining seemingly contrasting ideas of Parcet-Rogers and of Nagel-Stein-Wainger.
Laurent Stolovitch
Linearization of neighborhoods of embeddings of complex compact manifolds
In this work, we address the following question due to Grauert: if a neighborhood M of a holomorphically embedded complex compact manifold C is formally equivalent to another one, are two neighborhoods biholomorphically equivalent? We shall present the case where the other neighborhood is the neighborhood of the zero section of the normal bundle of C in M. The solution to this problem involves "small divisors problems". This is joint work with X. Gong.
Bingyang Hu
Sparse bounds of singular Radon transforms
In this talk, we will first briefly talk about the general theory of sparse domination, and then talk about the sparse bounds of singular Radon transforms, which strengths the $L^p$ boundedness of such operators due to Christ, Nagel, Stein and Wainger in 1999.
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).