

(313 intermediate revisions by 10 users not shown) 
Line 1: 
Line 1: 
−  '''Analysis Seminar
 
−  '''
 
   
−  The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.  +  The 20202021 Analysis Seminar will be organized by David Beltran and Andreas Seeger. 
 +  It will be online at least for the Fall semester, with details to be announced in September. 
 +  The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar earlier to accomodate speakers). 
   
−  If you wish to invite a speaker please contact Betsy at stovall(at)math
 +  Zoom links will be sent to those who have signed up for the Analysis Seminar List. For instructions how to sign up for seminar lists, see https://www.math.wisc.edu/node/230 
   
−  ===[[Previous Analysis seminars]]===
 +  If you'd like to suggest speakers for the spring semester please contact David and Andreas (dbeltran at math, seeger at math). 
   
−  = 20172018 Analysis Seminar Schedule =  +  
 +  
 +  =[[Previous_Analysis_seminars]]= 
 +  
 +  https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars 
 +  
 +  = Current Analysis Seminar Schedule = 
 { cellpadding="8"   { cellpadding="8" 
 !align="left"  date   !align="left"  date 
Line 16: 
Line 22: 
 !align="left"  host(s)   !align="left"  host(s) 
     
−  September 8 in B239 (Colloquium)  +  September 22 
−   Tess Anderson  +  Alexei Poltoratski 
−   UW Madison  +  UW Madison 
−  [[#linktoabstract  A Spherical Maximal Function along the Primes]]  +  [[#linktoabstract  Title ]] 
−  Tonghai  +   
     
−  September 19  +  September 29 
−   Brian Street  +  Polona Durcik 
−   UW Madison  +   Chapman University 
−  [[#Brian Street  Convenient Coordinates ]]  +  [[#linktoabstract  Title ]] 
−   Betsy  +   
     
−  September 26  +  September 30 
−   Hiroyoshi Mitake  +  Joris Roos 
−   Hiroshima University  +  University of Massachusetts  Lowell 
−  [[#Hiroyoshi Mitake  Derivation of multilayered interface system and its application ]]  +  [[#linktoabstract  Title ]] 
−   Hung  +   
     
−  October 3  +  October 6 
−   Joris Roos  +  Andrew Zimmer 
−   UW Madison  +  UW Madison 
−  [[#Joris Roos  A polynomial Roth theorem on the real line ]]  +  [[#linktoabstract  Title ]] 
−   Betsy  +   
     
−  October 10  +  October 13 
−   Michael Greenblatt  +  Hong Wang 
−   UI Chicago  +  Princeton/IAS 
−  [[#Michael Greenblatt  Maximal averages and Radon transforms for twodimensional hypersurfaces ]]  +  [[#linktoabstract  Title ]] 
−   Andreas  +   
     
−  October 17  +  October 20 
−   David Beltran  +  Kevin Luli 
−   Basque Center of Applied Mathematics  +  UC Davis 
−  [[#David Beltran  FeffermanStein inequalities ]]  +  [[#linktoabstract  Title ]] 
−   Andreas  +   
     
−  Wednesday, October 18, 4:00 p.m. in B131  +  October 27 
−  Jonathan Hickman  +  Terence Harris 
−  University of Chicago  +   Cornell University 
−  [[#Jonathan Hickman  Factorising X^n ]]  +  [[#linktoabstract  Title ]] 
−  Andreas  +   
     
−  October 24  +  November 3 
−   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
 
−   Fedor Nazarov
 
−   Kent State University
 
−  [[#Fedor Nazarov  The LernerOmbrosiPerez bound in the Muckenhoupt Wheeden conjecture is sharp ]]
 
−   Sergey, Andreas
 
−  
 
−  Friday, October 27, 4:00 p.m. in B239
 
−   Stefanie Petermichl
 
−   University of Toulouse
 
−  [[#Stefanie Petermichl  Higher order Journé commutators ]]
 
−   Betsy, Andreas
 
−  
 
−  Wednesday, November 1, 4:00 p.m. in B239 (Colloquium)
 
−   Shaoming Guo
 
−   Indiana University
 
−  [[#Shaoming Guo  ParsellVinogradov systems in higher dimensions ]]
 
−   Andreas
 
−  
 
−  November 14
 
−   Naser Talebizadeh Sardari
 
−   UW Madison
 
−  [[#Naser Talebizadeh Sardari  Quadratic forms and the semiclassical eigenfunction hypothesis ]]
 
−   Betsy
 
−  
 
−  November 28
 
−   Xianghong Chen
 
−   UW Milwaukee
 
−  [[#Xianghong Chen  Some transfer operators on the circle with trigonometric weights ]]
 
−   Betsy
 
−  
 
−  Monday, December 4, 4:00, B139
 
−   Bartosz Langowski and Tomasz Szarek
 
−   Institute of Mathematics, Polish Academy of Sciences
 
−  [[#Bartosz Langowski and Tomasz Szarek  Discrete Harmonic Analysis in the NonCommutative Setting ]]
 
−   Betsy
 
−  
 
−  Wednesday, December 13, 4:00, B239 (Colloquium)
 
−  Bobby Wilson
 
−  MIT
 
−  [[#Bobby Wilson  Projections in Banach Spaces and Harmonic Analysis ]]
 
−   Andreas
 
−  
 
−   Monday, February 5, 3:003:50, B341 (PDEGA seminar)  
−   Andreas Seeger
 
−   UW  
−  [[#Andreas Seeger  Singular integrals and a problem on mixing flows]]
 
     
 +   
     
−  February 6  +  November 10 
−   Dong Dong  +  Óscar Domínguez 
−   UIUC  +   Universidad Complutense de Madrid 
−   [[#Dong Dong  Hibert transforms in a 3 by 3 matrix and applications in number theory]]  +  [[#linktoabstract  Title ]] 
−  Betsy  +   
     
−  February 13  +  November 17 
−   Sergey Denisov  +  Tamas Titkos 
−   UW Madison  +  BBS U of Applied Sciences and Renyi Institute 
−   [[#Sergey Denisov  Spectral Szegő theorem on the real line]]  +  [[#linktoabstract  Title ]] 
     
     
−  February 20  +  November 24 
−   Ruixiang Zhang  +  Shukun Wu 
−   IAS (Princeton)  +  University of Illinois (UrbanaChampaign) 
−   [[#Ruixiang Zhang  The (Euclidean) Fractal Uncertainty Principle]]  +  [[#linktoabstract  Title ]] 
−   Betsy, Jordan, Andreas  +   
     
−  February 27  +  December 1 
−  Detlef Müller  +   Jonathan Hickman 
−  University of Kiel  +   The University of Edinburgh 
−   [[#Detlef Müller  On Fourier restriction for a nonquadratic hyperbolic surface]]  +  [[#linktoabstract  Title ]] 
−  Betsy, Andreas
 
−  
 
−  Wednesday, March 7, 4:00 p.m.
 
−   Winfried Sickel
 
−  FriedrichSchillerUniversität Jena
 
−   [[#Winfried Sickel  On the regularity of compositions of functions]]
 
−  Andreas
 
−  
 
−  March 13
 
−  
 
     
−   [[#linkofabstract  Title]]
 
−  
 
     
−  March 20  +  December 8 
−   Betsy Stovall  +  TBA 
−   UW
 
−   [[#linkofabstract  Two endpoint bounds via inverse problems]]
 
−  
 
−  
 
−  April 3
 
     
 +  [[#linktoabstract  Title ]] 
     
−   [[#linkofabstract  Title]]
 
−  
 
     
−  April 10  +  February 2 
−   Martina Neuman  +  Jongchon Kim 
−   UC Berkeley  +   UBC 
−   [[#Martina Neuman  GowersHostKra norms and Gowers structure on Euclidean spaces]]  +  [[#linktoabstract  Title ]] 
−   Betsy  +   
     
−  Friday, April 13, 4:00 p.m. (Colloquium)  +  February 9 
−  Jill Pipher  +  Bingyang Hu 
−  Brown  +  Purdue University 
−   [[#Jill Pipher  Mathematical ideas in cryptography]]  +  [[#linktoabstract  Title ]] 
−  WIMAW
 
−  
 
−  April 17
 
     
−  
 
−   [[#linkofabstract  Title]]
 
−  
 
−  
 
−  April 24
 
−   Lenka Slavíková
 
−   University of Missouri
 
−   [[#Lenka Slavíková  <math>L^2 \times L^2 \to L^1</math> boundedness criteria]]
 
−  Betsy, Andreas
 
−  
 
−  May 1
 
−   Xianghong Gong
 
−   UW
 
−   [[#linkofabstract  Title]]
 
−  
 
−  
 
−   '''May 7'''
 
−   Ebru Toprak
 
−   UIUC
 
−   [[#linkofabstract  TBA]]
 
−  Betsy
 
−  
 
−   '''May 15'''
 
−   Gennady Uraltsev
 
−   Cornell
 
−   [[#linkofabstract  TBA]]
 
−   Andreas, Betsy
 
−  
 
−   May 1618, [http://www.math.wisc.edu/~stovall/FA2018/ Workshop in Fourier Analysis]
 
−  
 
−  
 
−  
 
−  Betsy, Andreas
 
     
 +  February 16 
 +  David Beltran 
 +  UW  Madison 
 +  [[#linktoabstract  Title ]] 
 }   } 
   
 =Abstracts=   =Abstracts= 
−  ===Brian Street===  +  ===Name=== 
   
−  Title: Convenient Coordinates  +  Title 
   
−  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 subRiemannian 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 "subHermitian".  +  Abstract 
   
−  ===Hiroyoshi Mitake===
 
   
−  Title: Derivation of multilayered interface system and its application
 +  ===Name=== 
   
−  Abstract: In this talk, I will propose a multilayered interface system which can
 +  Title 
−  be formally derived by the singular limit of the weakly coupled system of
 
−  the AllenCahn equation. By using the level set approach, this system can be
 
−  written as a quasimonotone degenerate parabolic system.
 
−  We give results of the wellposedness of viscosity solutions, and study the
 
−  singularity of each layers. This is a joint work with H. Ninomiya, K. Todoroki.
 
   
−  ===Joris Roos===
 +  Abstract 
   
−  Title: A polynomial Roth theorem on the real line
 
   
−  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.
 +  ===Name=== 
   
−  ===Michael Greenblatt===
 +  Title 
   
−  Title: Maximal averages and Radon transforms for twodimensional hypersurfaces
 +  Abstract 
   
−  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.
 
   
−  ===David Beltran===  +  ===Name=== 
   
−  Title: Fefferman Stein Inequalities  +  Title 
   
−  Abstract: Given an operator T, we focus on obtaining twoweighted 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 FeffermanStein inequalities for pseudodifferential operators and for the solution operators to dispersive equations.  +  Abstract 
   
−  ===Jonathan Hickman===
 
   
−  Title: Factorising X^n.
 +  ===Name=== 
   
−  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 numbertheoretic considerations can themselves be approached via methods from harmonic analysis.
 +  Title 
   
−  ===Xiaochun Li ===
 +  Abstract 
   
−  Title: Recent progress on the pointwise convergence problems of Schrodinger equations
 
   
−  Abstract: Recently, Guth, Du and I solved the pointwise convergence problem of Schrodinger equations in twodimensional 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.
 +  =Extras= 
−   +  [[Blank Analysis Seminar Template]] 
−  ===Fedor Nazarov===  
−   
−  Title: The LernerOmbrosiPerez bound in the MuckenhouptWheeden
 
−  conjecture is sharp.
 
−   
−  Abstract: We show that the upper bound $[w]_{A_1}\log (e+[w]_{A_1})$ for
 
−  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===
 
−  Title: Higher order Journé commutators
 
−   
−  Abstract: We consider questions that stem from operator theory via Hankel and
 
−  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
 
−  nonnegative 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]=bHHb with the
 
−  Hilbert transform H=P_+  P_ . Whenever factorisation is absent, such
 
−  as in the real variable setting, in the multiparameter setting or
 
−  other, these classifications can be very difficult.
 
−   
−  Such lines were begun by Coifman, Rochberg, Weiss (real variables) and
 
−  by Cotlar, Ferguson, Sadosky (multiparameter) of characterisation of
 
−  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
 
−  analysis, while preserving the Ansatz from classical operator theory.
 
−   
−  ===Shaoming Guo ===
 
−  Title: ParsellVinogradov systems in higher dimensions
 
−   
−  Abstract:
 
−  I will present a few results on counting the numbers of integer solutions of ParsellVinogradov systems in higher dimensions.
 
−  Applications to Waring’s problem and to the problem of counting rational linear subspaces lying on certain hypersurface will be discussed.
 
−  Joint works with Jean Bourgain, Ciprian Demeter and Ruixiang Zhang.
 
−   
−  ===Naser Talebizadeh Sardari===
 
   
−  Title: Quadratic forms and the semiclassical eigenfunction hypothesis
 
   
−  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.
 +  Graduate Student Seminar: 
   
−  ===Xianghong Chen===
 +  https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html 
−   
−  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.
 
−   
−  ===Bobby Wilson===
 
−   
−  Title: Projections in Banach Spaces and Harmonic Analysis
 
−   
−  Abstract: In this talk, we will discuss the measure theoretic principles of orthogonal projections that follow from the classical BesicovitchFederer projection theorem. The BesicovitchFederer projection theorem offers a characterization of rectifiability of onedimensional sets in R^d by the size of their projections to lines. We will focus on the validity of analogues to the BesicovitchFederer 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 infinitedimensional and discuss related applications to questions in Harmonic Analysis. This is joint work with Marianna Csornyei and David Bate.
 
−   
−  ===Andreas Seeger===
 
−   
−  Title: Singular integrals and a problem on mixing flows
 
−   
−  Abstract: The talk will be about results related to Bressan's mixing problem. We present an inequality for the change of a Bianchini seminorm of characteristic functions under the flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator for which one proves bounds on Hardy spaces. This is joint work with Mahir Hadžić, Charles Smart and Brian Street.
 
−   
−  ===Dong Dong===
 
−   
−  Title: Hibert transforms in a 3 by 3 matrix and applications in number theory
 
−   
−  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 threeterm polynomial progressions in subsets of finite fields. I guarantee that a secondyear graduate student can understand at least 50% of the talk.
 
−   
−  ===Sergey Denisov===
 
−   
−  Title: Spectral Szegő theorem on the real line
 
−   
−  Abstract: For even measures on the real line, we give the criterion for the logarithmic integral to converge in terms of the corresponding DeBranges system (or Krein's string). The applications to probability (linear prediction for stationary Gaussian processes) will be explained. This is the joint result with R. Bessonov.
 
−   
−  ===Ruixiang Zhang===
 
−   
−  Title: The (Euclidean) Fractal Uncertainty Principle
 
−   
−  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 AB 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 BeurlingMalliavin Theorem. In the original work the dependence on the dimensions of both fractals was ineffective. We will also discuss why we can overcome this ineffectivity (joint work with Long Jin).
 
−   
−  ===Detlef Müller===
 
−   
−  Title: On Fourier restriction for a nonquadratic hyperbolic surface
 
−   
−  Abstract: In contrast to what is known about Fourier restriction for elliptic surfaces, rather little is known about hyperbolic surfaces. Hitherto, basically only the quadric $z=xy$ had been studied successfully. In my talk, after giving some background on Fourier restriction, I shall report on recent joint work with S. Buschenhenke and A. Vargas on a cubic perturbation of this quadric. Our analysis reveals that the geometry of the problem changes drastically in the presence of a perturbation term, and that new techniques, compared to the elliptic case, are required to handle more general hyperbolic surfaces.
 
−   
−  ===Winfried Sickel===
 
−   
−  Title: On the regularity of compositions of functions
 
−   
−  Abstract: Let <math>E</math> denote a Banach space of locally integrable functions on <math>\mathbb{R}</math>. To each continuous function <math>f:\mathbb{R} \to \mathbb{R}</math>
 
−  we associate the composition operator
 
−  <math>T_f(g):= f\circ g</math>, <math>g\in E</math>.
 
−  The properties of <math>T_f</math> strongly depend on the chosen function space <math>E</math>.
 
−  In my talk I will concentrate on Sobolev spaces <math>W^m_p</math> and Slobodeckij spaces <math>W^s_p</math>.
 
−  The main aim will consist in giving a survey on necessary and sufficient conditions on <math>f</math>
 
−  such that the composition operator maps such a space <math>E</math> into itself.
 
−   
−  ===Martina Neuman===
 
−   
−  Title: GowersHostKra norms and Gowers structure on Euclidean spaces
 
−   
−  Abstract: The investigation on BrascampLieb data  their structure, their extremizability, their stability and regularity of their constants  has been an active one in Harmonic Analysis. In this talk, I'll present an example of a BrascampLieb structure: a socalled Gowers structure on Euclidean spaces, together with the related GowersHostKra norms  these were originally tools in additive combinatorics context. I'll dissertate on what happens when a function nearly achieves its GowersHostKra norm in a Euclidean context  this can be seen as continuation of the work of EisnerTao  and a related stability result of the Gowers structure on Euclidean spaces.
 
−   
−  ===Jill Pipher===
 
−   
−  Title: Mathematical ideas in cryptography
 
−   
−  Abstract: This talk does not assume prior knowledge of public key crypto (PKC). I'll talk about the history of the subject and some current areas of research,
 
−  including homomorphic encryption.
 
−   
−  ===Lenka Slavíková===
 
−   
−  Title: <math>L^2 \times L^2 \to L^1</math> boundedness criteria
 
−   
−  Abstract: It is a consequence of Plancherel's identity that a linear multiplier operator associated with a function <math>m</math> is bounded from <math>L^2</math> to itself if and only if <math>m</math> belongs to the space <math>L^\infty</math>. In this talk we will investigate the <math>L^2 \times L^2 \to L^1</math> boundedness of bilinear multiplier operators which is as central in the bilinear theory as the <math>L^2</math> boundedness is in the linear multiplier theory. We will present a sharp <math>L^2 \times L^2 \to L^1</math> boundedness criterion for a class of bilinear operators associated with a multiplier given by a signed sum of dyadic dilations of a given function, in terms of the <math>L^q</math> integrability of this function; precisely we will show that boundedness holds if and only if <math>q<4</math>. We will then discuss applications of this result concerning bilinear rough singular integrals and bilinear dyadic spherical maximal functions. This is a joint work with L. Grafakos and D. He.
 
−   
−  =Extras=
 
−  [[Blank Analysis Seminar Template]]
 