Fall 2021 and Spring 2022 Analysis Seminars: Difference between revisions

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'''Analysis Seminar
'''Analysis Seminar
'''
'''
[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  Betsy at stovall(at)math


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


= Analysis Seminar Schedule Spring 2017 =
= 2017-2018 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
|September 8 in B239 (Colloquium)
| Fabio Pusateri (Princeton)  
| Tess Anderson
|[[#Fabio Pusateri   |  The Water Waves Problem ]]
| UW Madison
| Sigurd Angenent
|[[#linktoabstract  |  A Spherical Maximal Function along the Primes]]
|Tonghai
|-
|September 19
| Brian Street
| UW Madison
|[[#Brian Street  |  Convenient Coordinates ]]
| Betsy
|-
|September 26
| Hiroyoshi Mitake
| Hiroshima University
|[[#Hiroyoshi Mitake  |  Derivation of multi-layered interface system and its application ]]
| Hung
|-
|October 3
| Joris Roos
| UW Madison
|[[#Joris Roos  |  A polynomial Roth theorem on the real line ]]
| Betsy
|-
|October 10
| Michael Greenblatt
| UI Chicago
|[[#Michael Greenblatt  |  Maximal averages and Radon transforms for two-dimensional hypersurfaces ]]
| Andreas
|-
|October 17
| David Beltran
| Basque Center of Applied Mathematics
|[[#David Beltran  |  Fefferman-Stein inequalities ]]
| Andreas
|-
|Wednesday, October 18, 4:00 p.m. in B131
|Jonathan Hickman
|University of Chicago
|[[#Jonathan Hickman  |  Factorising X^n  ]]
|Andreas
|-
|October 24
| Xiaochun Li
| 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 Lerner-Ombrosi-Perez 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  |  Parsell-Vinogradov 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 Non-Commutative Setting ]]
| Betsy
|-
|Wednesday, December 13, 4:00, B239 (Colloquium)
|Bobby Wilson
|MIT
|[[#Bobby Wilson | Projections in Banach Spaces and Harmonic Analysis ]]
| Andreas
|-
|January 30
|
|
| [[#linkofabstract | Title]]
|
|
|-
|-
|January 24, Joint Analysis/Geometry Seminar
|February 6
| Tamás Darvas (Maryland)
| Dong Dong
|[[#Tamás Darvas  | Existence of constant scalar curvature Kähler metrics and properness of the K-energy ]]
| UIUC
| Jeff Viaclovsky
| [[#Dong Dong | Hibert transforms in a 3 by 3 matrix and applications in number theory]]
|
|Betsy
|-
|-
|Monday, January 30, 3:30, VV901 (PDE Seminar)
|February 13
| Serguei Denissov (UW Madison)
|
|[[#Serguei Denissov  | Instability in 2D Euler equation of incompressible inviscid fluid ]]
|  
| [[#linkofabstract | Title]]
|  
|  
|-
|-
|February 7
|February 20
| Andreas Seeger (UW Madison)
| Ruixiang Zhang (tent.)
|[[#Andreas Seeger|   The Haar system in Sobolev spaces]]
| IAS (Princeton)
|
| [[#linkofabstract | Title]]
| Betsy, Jordan, Andreas
|-
|-
|February 21
|February 27
| Jongchon Kim (UW Madison)
|Detlef Müller (tent.)
|[[#Jongchon Kim |   Some remarks on Fourier restriction estimates ]]
|University of Kiel
| Andreas Seeger
| [[#linkofabstract | Title]]
|Betsy, Andreas
|-
|-
|March 7, Mathematics Department Distinguished Lecture
|Wednesday, March 7, 4:00 p.m.
| Roger Temam (Indiana) 
| Winfried Sickel
|[[#Roger Temam (Colloquium)  | On the mathematical  modeling of the humid atmosphere  ]]
|Friedrich-Schiller-Universität Jena
| Leslie Smith
| [[#linkofabstract | Title]]
|Andreas
|-
|-
|Wednesday, March 8, Joint Applied Math/PDE/Analysis  Seminar
|March 13
| Roger Temam (Indiana) 
| TBA
|[[#Roger Temam (Seminar)  |   Weak solutions of the Shigesada-Kawasaki-Teramoto system]]
|  
| Leslie Smith
| [[#linkofabstract | Title]]
|
|-
|-
|March 14
|March 20
| Xianghong Chen (UW Milwaukee)
|
|[[#Xianghong Chen  |   Restricting the Fourier transform to some oscillating curves ]]
|  
| Andreas Seeger
| [[#linkofabstract | Title]]
|
|
|-
|-
|March 21
|April 3
| SPRING BREAK
|
|[[#linktoabstract | ]]
|  
 
| [[#linkofabstract | Title]]
 
|
|-
|-
|March 27 (joint PDE/Analysis Seminar), 3:30, VV901
|April 10
| Sylvia Serfaty
|
|[[#linktoabstract |TBA ]]
|  
| Hung Tran
| [[#linkofabstract | Title]]
|
|
|-
|-
|March 28
|April 17
| Brian Cook (Fields Institute)
|
|[[#Brian Cook |Twists on the twisted ergodic theorems ]]
|  
| Andreas Seeger
| [[#linkofabstract | Title]]
|
|
|-
|-
|April 4
|April 24
| Francesco Di Plinio (Virginia)
|
|[[#linktoabstract  |   TBA ]]
|  
| Andreas Seeger
| [[#linkofabstract | Title]]
|
|
|-
|-
|April 11
|May 1
| Xianghong Gong (UW Madison)
|  
|[[#linktoabstract  |  TBA ]]
|  
|  
| [[#linkofabstract | Title]]
|
|-
| May 16-18, [http://www.math.wisc.edu/~stovall/FA2018/ Workshop in Fourier Analysis]
|
|
|
|
|Betsy, Andreas
|-
|-
|April 25 (joint PDE/Analysis Seminar)
| Chris Henderson (University of Chicago)
|[[#linktoabstract  |  TBA ]]
| Jessica Lin
|}
|}


=Abstracts=
=Abstracts=
===Brian Street===
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 sub-Riemannian geometry initiated by Nagel, Stein, and Wainger.  When the vector fields are complex one obtains a geometry with more structure which can be thought of as "sub-Hermitian".
===Hiroyoshi Mitake===
Title:  Derivation of multi-layered interface system and its application
Abstract:  In this talk, I will propose a multi-layered interface system which can
be formally derived by the singular limit of the weakly coupled system of
the Allen-Cahn equation.  By using the level set approach, this system can be
written as a quasi-monotone degenerate parabolic system.
We give results of the well-posedness of viscosity solutions, and study the
singularity of each layers. This is a joint work with H. Ninomiya, K. Todoroki.
===Joris Roos===
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 two-dimensional 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 two-weighted inequalities in which the weights are related via certain maximal function. These inequalites, which originated in work of Fefferman and Stein, have been established in an optimal way for different classical operators in Harmonic Analysis. In this talk, we survey some classical results and we present some recent Fefferman-Stein inequalities for pseudodifferential operators and for the solution operators to dispersive equations.
===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 number-theoretic 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 two-dimensional case. We proved that the solution to free Schrodinger equation in R^2 converges to its initial data, provided the initial data belongs to H^s for s larger than 1/3.  This result is sharp, up to the end point, due to Bourgain's example. The proof relies on the polynomial partitioning method and  the decoupling method. In addition, the pointwise convergence problem is closely related to Fourier restriction conjecture.
===Fedor Nazarov=== 
Title: The Lerner-Ombrosi-Perez bound in the Muckenhoupt-Wheeden
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


===  Fabio Pusateri  ===
Abstract: We consider questions that stem from operator theory via Hankel and
''The Water Waves problem''
Toeplitz forms and target (weak) factorisation of Hardy spaces. In
more basic terms, let us consider a function on the unit circle in its
Fourier representation. Let P_+ denote the projection onto
non-negative and P_- onto negative frequencies. Let b denote
multiplication by the symbol function b. It is a classical theorem by
Nehari that the composed operator P_+ b P_- is bounded on L^2 if and
only if b is in an appropriate space of functions of bounded mean
oscillation. The necessity makes use of a classical factorisation
theorem of complex function theory on the disk. This type of question
can be reformulated in terms of commutators [b,H]=bH-Hb with the
Hilbert transform H=P_+ - P_- . Whenever factorisation is absent, such
as in the real variable setting, in the multi-parameter setting or
other, these classifications can be very difficult.


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.
Such lines were begun by Coifman, Rochberg, Weiss (real variables) and
by Cotlar, Ferguson, Sadosky (multi-parameter) 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.


===  Tamás Darvas ===
The tools used go deep into modern advances in dyadic harmonic
''Existence of constant scalar curvature Kähler metrics and properness of the K-energy''
analysis, while preserving the Ansatz from classical operator theory.


Given a compact Kähler manifold $(X,\omega)$, we show that if there exists a constant
===Shaoming Guo ===
scalar curvature Kähler metric  cohomologous to $\omega$ then Mabuchi's K-energy is J-proper in an
Title: Parsell-Vinogradov systems in higher dimensions
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:
''Instability in 2D Euler equation of incompressible inviscid fluid''
I will present a few results on counting the numbers of integer solutions of Parsell-Vinogradov systems in higher dimensions.
Applications to Waring’s problem and to the problem of counting rational linear subspaces lying on certain hyper-surface will be discussed.
Joint works with Jean Bourgain, Ciprian Demeter and Ruixiang Zhang.


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


=== Andreas Seeger  ===
Title: Quadratic forms and the semiclassical eigenfunction hypothesis
''The Haar system in Sobolev spaces''


We consider the Haar system on  Sobolev  spaces and ask:
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.
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  ===
===Xianghong Chen===
''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.
Title:  Some transfer operators on the circle with trigonometric weights
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) ===
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.
''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.
===Bobby Wilson===
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.
=== Roger Temam (Seminar) ===
''Weak solutions of the Shigesada-Kawasaki-Teramoto system''


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.
Title: Projections in Banach Spaces and Harmonic Analysis
Based on an article with Du Pham, to appear in Nonlinear Analysis.


===  Xianghong Chen  ===
Abstract: In this talk, we will discuss the measure theoretic principles of orthogonal projections that follow from the classical Besicovitch-Federer projection theorem. The Besicovitch-Federer projection theorem offers a characterization of rectifiability of one-dimensional sets in R^d by the size of their projections to lines. We will focus on the validity of analogues to the Besicovitch-Federer projection theorem with respect to such sets in general Banach spaces. In particular, we will show that the projection theorem is false when the Banach space is infinite-dimensional and discuss related applications to questions in Harmonic Analysis. This is joint work with Marianna Csornyei and David Bate.
''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.
===Dong Dong===


=== Brian Cook ===
Title: Hibert transforms in a 3 by 3 matrix and applications in number theory
"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.
Abstract:  This talk could interest both analysts and number theorists. I will first present 35 variants of Hilbert transforms, with a focus on their connections with ergodic theory, number theory, and combinatorics. Then I will show how to use Fourier analysis tools to reduce a number theory problem (Roth theorem) to an algebraic geometry problem: this joint work Li and Sawin fully answers a question of Bourgain and Chang about three-term polynomial progressions in subsets of finite fields. I guarantee that a second-year graduate student can understand at least 50% of the talk.


=Extras=
=Extras=
[[Blank Analysis Seminar Template]]
[[Blank Analysis Seminar Template]]

Revision as of 21:17, 12 January 2018

Analysis Seminar

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

Previous Analysis seminars

2017-2018 Analysis Seminar Schedule

date speaker institution title host(s)
September 8 in B239 (Colloquium) Tess Anderson UW Madison A Spherical Maximal Function along the Primes Tonghai
September 19 Brian Street UW Madison Convenient Coordinates Betsy
September 26 Hiroyoshi Mitake Hiroshima University Derivation of multi-layered interface system and its application Hung
October 3 Joris Roos UW Madison A polynomial Roth theorem on the real line Betsy
October 10 Michael Greenblatt UI Chicago Maximal averages and Radon transforms for two-dimensional hypersurfaces Andreas
October 17 David Beltran Basque Center of Applied Mathematics Fefferman-Stein inequalities Andreas
Wednesday, October 18, 4:00 p.m. in B131 Jonathan Hickman University of Chicago Factorising X^n Andreas
October 24 Xiaochun Li UIUC 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 The Lerner-Ombrosi-Perez bound in the Muckenhoupt Wheeden conjecture is sharp Sergey, Andreas
Friday, October 27, 4:00 p.m. in B239 Stefanie Petermichl University of Toulouse Higher order Journé commutators Betsy, Andreas
Wednesday, November 1, 4:00 p.m. in B239 (Colloquium) Shaoming Guo Indiana University Parsell-Vinogradov systems in higher dimensions Andreas
November 14 Naser Talebizadeh Sardari UW Madison Quadratic forms and the semiclassical eigenfunction hypothesis Betsy
November 28 Xianghong Chen UW Milwaukee 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 Discrete Harmonic Analysis in the Non-Commutative Setting Betsy
Wednesday, December 13, 4:00, B239 (Colloquium) Bobby Wilson MIT Projections in Banach Spaces and Harmonic Analysis Andreas
January 30 Title
February 6 Dong Dong UIUC Hibert transforms in a 3 by 3 matrix and applications in number theory Betsy
February 13 Title
February 20 Ruixiang Zhang (tent.) IAS (Princeton) Title Betsy, Jordan, Andreas
February 27 Detlef Müller (tent.) University of Kiel Title Betsy, Andreas
Wednesday, March 7, 4:00 p.m. Winfried Sickel Friedrich-Schiller-Universität Jena Title Andreas
March 13 TBA Title
March 20 Title
April 3 Title
April 10 Title
April 17 Title
April 24 Title
May 1 Title
May 16-18, Workshop in Fourier Analysis Betsy, Andreas

Abstracts

Brian Street

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 sub-Riemannian geometry initiated by Nagel, Stein, and Wainger. When the vector fields are complex one obtains a geometry with more structure which can be thought of as "sub-Hermitian".

Hiroyoshi Mitake

Title: Derivation of multi-layered interface system and its application

Abstract: In this talk, I will propose a multi-layered interface system which can be formally derived by the singular limit of the weakly coupled system of the Allen-Cahn equation. By using the level set approach, this system can be written as a quasi-monotone degenerate parabolic system. We give results of the well-posedness of viscosity solutions, and study the singularity of each layers. This is a joint work with H. Ninomiya, K. Todoroki.

Joris Roos

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 two-dimensional 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 two-weighted inequalities in which the weights are related via certain maximal function. These inequalites, which originated in work of Fefferman and Stein, have been established in an optimal way for different classical operators in Harmonic Analysis. In this talk, we survey some classical results and we present some recent Fefferman-Stein inequalities for pseudodifferential operators and for the solution operators to dispersive equations.

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 number-theoretic 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 two-dimensional case. We proved that the solution to free Schrodinger equation in R^2 converges to its initial data, provided the initial data belongs to H^s for s larger than 1/3. This result is sharp, up to the end point, due to Bourgain's example. The proof relies on the polynomial partitioning method and the decoupling method. In addition, the pointwise convergence problem is closely related to Fourier restriction conjecture.

Fedor Nazarov

Title: The Lerner-Ombrosi-Perez bound in the Muckenhoupt-Wheeden 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 non-negative and P_- onto negative frequencies. Let b denote multiplication by the symbol function b. It is a classical theorem by Nehari that the composed operator P_+ b P_- is bounded on L^2 if and only if b is in an appropriate space of functions of bounded mean oscillation. The necessity makes use of a classical factorisation theorem of complex function theory on the disk. This type of question can be reformulated in terms of commutators [b,H]=bH-Hb with the Hilbert transform H=P_+ - P_- . Whenever factorisation is absent, such as in the real variable setting, in the multi-parameter setting or other, these classifications can be very difficult.

Such lines were begun by Coifman, Rochberg, Weiss (real variables) and by Cotlar, Ferguson, Sadosky (multi-parameter) 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: Parsell-Vinogradov systems in higher dimensions

Abstract: I will present a few results on counting the numbers of integer solutions of Parsell-Vinogradov systems in higher dimensions. Applications to Waring’s problem and to the problem of counting rational linear subspaces lying on certain hyper-surface will be discussed. Joint works with Jean Bourgain, Ciprian Demeter and Ruixiang Zhang.

Naser Talebizadeh Sardari

Title: Quadratic forms and the semiclassical eigenfunction hypothesis

Abstract: Let [math]\displaystyle{ Q(X) }[/math] be any integral primitive positive definite quadratic form in [math]\displaystyle{ k }[/math] variables, where [math]\displaystyle{ k\geq4 }[/math], and discriminant [math]\displaystyle{ D }[/math]. For any integer [math]\displaystyle{ n }[/math], we give an upper bound on the number of integral solutions of [math]\displaystyle{ Q(X)=n }[/math] in terms of [math]\displaystyle{ n }[/math], [math]\displaystyle{ k }[/math], and [math]\displaystyle{ 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]\displaystyle{ \mathbb{T}^d }[/math] for [math]\displaystyle{ d\geq 5 }[/math]. This conjecture is motivated by the work of Berry\cite{Berry, Michael} on semiclassical eigenfunction hypothesis.

Xianghong Chen

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 Besicovitch-Federer projection theorem. The Besicovitch-Federer projection theorem offers a characterization of rectifiability of one-dimensional sets in R^d by the size of their projections to lines. We will focus on the validity of analogues to the Besicovitch-Federer projection theorem with respect to such sets in general Banach spaces. In particular, we will show that the projection theorem is false when the Banach space is infinite-dimensional and discuss related applications to questions in Harmonic Analysis. This is joint work with Marianna Csornyei and David Bate.

Dong Dong

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 three-term polynomial progressions in subsets of finite fields. I guarantee that a second-year graduate student can understand at least 50% of the talk.

Extras

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