

(92 intermediate revisions by 6 users not shown) 
Line 4: 
Line 4: 
 The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.   The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated. 
   
−  If you wish to invite a speaker please contact Betsy at stovall(at)math  +  If you wish to invite a speaker please contact Brian at street(at)math 
   
 ===[[Previous Analysis seminars]]===   ===[[Previous Analysis seminars]]=== 
   
−  = 20172018 Analysis Seminar Schedule =  +  = Analysis Seminar Schedule = 
 { cellpadding="8"   { cellpadding="8" 
 !align="left"  date   !align="left"  date 
Line 16: 
Line 16: 
 !align="left"  host(s)   !align="left"  host(s) 
     
−  September 8 in B239  +  Sept 11 
−   Tess Anderson  +   Simon Marshall 
−   UW Madison  +   Madison 
−  [[#linktoabstract  A Spherical Maximal Function along the Primes]]  +  [[#Simon Marshall  Integrals of eigenfunctions on hyperbolic manifolds ]] 
−  Tonghai  +   
     
−  September 19  +  Sept 18 
−   Brian Street  +   Person 
−   UW Madison  +   Institution 
−  [[#Brian Street  Convenient Coordinates ]]  +  [[#linktoabstract  Title ]] 
−   Betsy  +   Sponsor 
     
−  September 26  +  Sept 25 
−   Hiroyoshi Mitake  +   Gunther Uhlman 
−   Hiroshima University  +   University of Washington 
−  [[#Hiroyoshi Mitake  Derivation of multilayered interface system and its application ]]  +  [[#linktoabstract  Title ]] 
−   Hung  +   Qin Li 
     
−  October 3  +  Oct 2 
−   Joris Roos  +   Person 
−   UW Madison  +   Institution 
−  [[#Joris Roos  A polynomial Roth theorem on the real line ]]  +  [[#linktoabstract  Title ]] 
−   Betsy  +   Sponsor 
     
−  October 10  +  Oct 9 
−   Michael Greenblatt  +   Person 
−   UI Chicago  +   Institution 
−  [[#Michael Greenblatt  Maximal averages and Radon transforms for twodimensional hypersurfaces ]]  +  [[#linktoabstract  Title ]] 
−   Andreas  +   Sponsor 
     
−  October 17  +  Oct 16 
−   David Beltran  +   Person 
−   Basque Center of Applied Mathematics  +   Institution 
−  [[#David Beltran  FeffermanStein inequalities ]]  +  [[#linktoabstract  Title ]] 
−   Andreas  +   Sponsor 
     
−  Wednesday, October 18, 4:00 p.m. in B131  +  Oct 23 
−  Jonathan Hickman  +   Person 
−  University of Chicago  +   Institution 
−  [[#Jonathan Hickman  Factorising X^n ]]  +  [[#linktoabstract  Title ]] 
−  Andreas  +   Sponsor 
     
−  October 24  +  Oct 30 
−   Xiaochun Li  +   Person 
−   UIUC  +   Institution 
−  [[#Xiaochun Li  Recent progress on the pointwise convergence problems of Schroedinger equations ]]  +  [[#linktoabstract  Title ]] 
−   Betsy  +   Sponsor 
     
−  Thursday, October 26, 4:30 p.m. in B139  +  Nov 6 
−   Fedor Nazarov  +   Person 
−   Kent State University  +   Institution 
−  [[#Fedor Nazarov  The LernerOmbrosiPerez bound in the Muckenhoupt Wheeden conjecture is sharp ]]  +  [[#linktoabstract  Title ]] 
−   Sergey, Andreas  +   Sponsor 
     
−  Friday, October 27, 4:00 p.m. in B239  +  Nov 13 
−   Stefanie Petermichl  +   Person 
−   University of Toulouse  +   Institution 
−  [[#Stefanie Petermichl  Higher order Journé commutators ]]  +  [[#linktoabstract  Title ]] 
−   Betsy, Andreas  +   Sponsor 
     
−  Wednesday, November 1, 4:00 p.m. in B239 (Colloquium)  +  Nov 20 
−   Shaoming Guo  +   Thanksgiving! 
−   Indiana University  +   
−  [[#Shaoming Guo  ParsellVinogradov systems in higher dimensions ]]  +   
−   Andreas  +   
     
−  November 14  +  Dec 4 
−   Naser Talebizadeh Sardari  +   Person 
−   UW Madison  +   Institution 
−  [[#Naser Talebizadeh Sardari  Quadratic forms and the semiclassical eigenfunction hypothesis ]]  +  [[#linktoabstract  Title ]] 
−   Betsy  +   Sponsor 
     
−  November 28  +  Jan 22 
−   Xianghong Chen  +   Person 
−   UW Milwaukee  +   Institution 
−  [[#Xianghong Chen  Some transfer operators on the circle with trigonometric weights ]]  +  [[#linktoabstract  Title ]] 
−   Betsy  +   Sponsor 
     
−  Monday, December 4, 4:00, B139  +  Jan 29 
−   Bartosz Langowski and Tomasz Szarek  +   Person 
−   Institute of Mathematics, Polish Academy of Sciences  +   Institution 
−  [[#Bartosz Langowski and Tomasz Szarek  Discrete Harmonic Analysis in the NonCommutative Setting ]]  +  [[#linktoabstract  Title ]] 
−   Betsy  +   Sponsor 
     
−  December 12  +  Feb 5 
−   Alex Stokolos  +   Person 
−   GA Southern  +   Institution 
 [[#linktoabstract  Title ]]   [[#linktoabstract  Title ]] 
−   Andreas  +   Sponsor 
     
−  Wednesday, December 13, 4:00, B239 (Colloquium)  +  Feb 12 
−  Bobby Wilson (MIT)  +   Person 
−  [[#Bobby Wilson  Projections in Banach Spaces and Harmonic Analysis ]]  +   Institution 
−   Andreas  +  [[#linktoabstract  Title ]] 
 +   Sponsor 
     
−  January 30  +  Feb 19 
−    +   Person 
−    +   Institution 
−   [[#linkofabstract  Title]]  +  [[#linktoabstract  Title ]] 
−    +   Sponsor 
     
−  February 6  +  Feb 26 
−   Dong Dong  +   Person 
−   UIUC  +   Institution 
−   [[#Dong Dong  Hibert transforms in a 3 by 3 matrix and applications in number theory]]  +  [[#linktoabstract  Title ]] 
−  Betsy  +   Sponsor 
     
−  February 13  +  Mar 5 
−    +   Person 
−    +   Institution 
−   [[#linkofabstract  Title]]  +  [[#linktoabstract  Title ]] 
−    +   Sponsor 
     
−  February 20  +  Mar 12 
−    +   Person 
−    +   Institution 
−   [[#linkofabstract  Title]]  +  [[#linktoabstract  Title ]] 
−    +   Sponsor 
     
−  February 27  +  Mar 19 
 +  Spring Break!!! 
     
−  
 
−   [[#linkofabstract  Title]]
 
     
−  
 
−  Wednesday, March 7, 4:00 p.m.
 
−   Winfried Sickel
 
−  FriedrichSchillerUniversität Jena
 
−   [[#linkofabstract  Title]]
 
−  Andreas
 
−  
 
−  March 13
 
−  
 
−  
 
−   [[#linkofabstract  Title]]
 
     
     
−  March 20  +  Apr 2 
−    +   Person 
−    +   Institution 
−   [[#linkofabstract  Title]]  +  [[#linktoabstract  Title ]] 
−    +   Sponsor 
     
−  April 3  +  Apr 9 
−    +   Person 
−    +   Institution 
−   [[#linkofabstract  Title]]  +  [[#linktoabstract  Title ]] 
−    +   Sponsor 
     
−  April 10  +  Apr 9 
−    +   Person 
−    +   Institution 
−   [[#linkofabstract  Title]]  +  [[#linktoabstract  Title ]] 
−    +   Sponsor 
     
−  April 17  +  Apr 16 
−    +   Person 
−    +   Institution 
−   [[#linkofabstract  Title]]  +  [[#linktoabstract  Title ]] 
−    +   Sponsor 
     
−  April 24  +  Apr 23 
−    +   Person 
−    +   Institution 
−   [[#linkofabstract  Title]]  +  [[#linktoabstract  Title ]] 
−    +   Sponsor 
     
−  May 1  +  Apr 30 
−    +   Person 
−    +   Institution 
−   [[#linkofabstract  Title]]  +  [[#linktoabstract  Title ]] 
−    +   Sponsor 
     
 }   } 
   
 =Abstracts=   =Abstracts= 
−  ===Brian Street===  +  ===Simon Marshall=== 
−   +  
−  Title: Convenient Coordinates
 +  
−   +  
−  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".
 +  
−   +  
−  ===Hiroyoshi Mitake===
 +  
−   +  
−  Title: Derivation of multilayered interface system and its application
 +  
−   +  
−  Abstract: In this talk, I will propose a multilayered interface system which can
 +  
−  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===
 +  
−   +  
−  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.
 +  
−   +  
−  ===Michael Greenblatt===
 +  
−   +  
−  Title: Maximal averages and Radon transforms for twodimensional hypersurfaces
 +  
−   +  
−  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===
 +  
−   +  
−  Title: Fefferman Stein Inequalities
 +  
−   +  
−  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.
 +  
−   +  
−  ===Jonathan Hickman===
 +  
−   +  
−  Title: Factorising X^n.
 +  
−   +  
−  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.
 +  
−   +  
−  ===Xiaochun Li ===
 +  
−   +  
−  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.
 +  
−   +  
−  ===Fedor Nazarov===
 +  
   
−  Title: The LernerOmbrosiPerez bound in the MuckenhouptWheeden
 +  ''Integrals of eigenfunctions on hyperbolic manifolds'' 
−  conjecture is sharp.
 +  
   
−  Abstract: We show that the upper bound $[w]_{A_1}\log (e+[w]_{A_1})$ for
 +  Let X be a compact hyperbolic manifold, and let Y be a totally geodesic closed submanifold in X. I will discuss the problem of bounding the integral of a Laplace eigenfunction on X over Y, as the eigenvalue tends to infinity. I will present an upper bound for these integrals that is sharp on average, and briefly describe ongoing work with Farrell Brumley in which we attempt to produce eigenfunctions with very large periods. 
−  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===  +  ===Name=== 
−  Title: Higher order Journé commutators
 +  
   
−  Abstract: We consider questions that stem from operator theory via Hankel and
 +  Title 
−  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
 +  Abstract 
−  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 ===  +  ===Name=== 
−  Title: ParsellVinogradov systems in higher dimensions
 +  
   
−  Abstract:
 +  Title 
−  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===
 +  Abstract 
   
−  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.
 +  ===Name=== 
   
−  ===Xianghong Chen===
 +  Title 
   
−  Title: Some transfer operators on the circle with trigonometric weights
 +  Abstract 
   
−  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.
 
   
−  ===Dong Dong===  +  ===Name=== 
   
−  Title: Hibert transforms in a 3 by 3 matrix and applications in number theory  +  Title 
   
−  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.  +  Abstract 
   
 =Extras=   =Extras= 
 [[Blank Analysis Seminar Template]]   [[Blank Analysis Seminar Template]] 