Difference between revisions of "Analysis Seminar"

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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]]===
  
= 2017-2018 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 (Colloquium)
+
|Sept 11
| Tess Anderson
+
| Simon Marshall
 
| UW Madison
 
| UW Madison
|[[#linktoabstract |   A Spherical Maximal Function along the Primes]]
+
|[[#Simon Marshall | Integrals of eigenfunctions on hyperbolic manifolds ]]
|Tonghai
+
|  
 
|-
 
|-
|September 19
+
|'''Wednesday, Sept 12'''
| Brian Street
+
| Gunther Uhlmann 
| UW Madison
+
| University of Washington
|[[#Brian Street  |  Convenient Coordinates ]]
+
| Distinguished Lecture Series
| Betsy
+
| See colloquium website for location
 
|-
 
|-
|September 26
+
|'''Friday, Sept 14'''
| Hiroyoshi Mitake
+
| Gunther Uhlmann 
| Hiroshima University
+
| University of Washington
|[[#Hiroyoshi Mitake  |  Derivation of multi-layered interface system and its application ]]
+
| Distinguished Lecture Series
| Hung
+
| See colloquium website for location
 
|-
 
|-
|October 3
+
|Sept 18
| Joris Roos
+
| Grad Student Seminar
| UW Madison
+
|  
|[[#Joris Roos  |  A polynomial Roth theorem on the real line ]]
+
|
| Betsy
+
|
 
|-
 
|-
|October 10
+
|Sept 25
| Michael Greenblatt
+
| Grad Student Seminar
| UI Chicago
+
|
|[[#Michael Greenblatt  |  Maximal averages and Radon transforms for two-dimensional hypersurfaces ]]
+
|
| Andreas
+
|
 
|-
 
|-
|October 17
+
|Oct 9
| David Beltran
+
| Hong Wang
| Basque Center of Applied Mathematics
+
| MIT
|[[#David Beltran Fefferman-Stein inequalities ]]
+
|[[#Hong Wang About Falconer distance problem in the plane ]]
| Andreas
+
| Ruixiang
 
|-
 
|-
|Wednesday, October 18, 4:00 p.m.  in B131
+
|Oct 16
|Jonathan Hickman
+
| Polona Durcik
|University of Chicago
+
| Caltech
|[[#Jonathan Hickman | Factorising X^n ]]
+
|[[#Polona Durcik |   Singular Brascamp-Lieb inequalities and extended boxes in R^n ]]
|Andreas
+
| Joris
 
|-
 
|-
|October 24
+
|Oct 23
| Xiaochun Li
+
| Song-Ying Li
| UIUC
+
| UC Irvine
|[[#Xiaochun Li  |  Recent progress on the pointwise convergence problems of Schroedinger equations ]]
+
|[[#Song-Ying Li  |  Estimates for the first positive eigenvalue of Kohn Laplacian on a pseudo-Hermitian manifold ]]
| Betsy
+
| Xianghong
 
|-
 
|-
|Thursday, October 26, 4:30 p.m. in B139
+
|Oct 30
| Fedor Nazarov
+
|Grad student seminar
| 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
+
|Nov 6
| Stefanie Petermichl
+
| Hanlong Fang
| University of Toulouse
+
| UW Madison
|[[#Stefanie Petermichl | Higher order Journé commutators   ]]
+
|[[#Hanlong Fang A generalization of the theorem of Weil and Kodaira on prescribing residues ]]
| Betsy, Andreas
+
| Brian
 
|-
 
|-
|Wednesday, November 1, 4:00 p.m. in B239 (Colloquium)
+
||'''Monday, Nov. 12, B139'''
| Shaoming Guo
+
| Kyle Hambrook
| Indiana University
+
| San Jose State University
|[[#Shaoming Guo Parsell-Vinogradov systems in higher dimensions ]]
+
|[[#Kyle Hambrook Fourier Decay and Fourier Restriction for Fractal Measures on Curves ]]
| Andreas
+
| Andreas
 +
|-
 +
|Nov 13
 +
| Laurent Stolovitch
 +
| Université de Nice - Sophia Antipolis
 +
|[[#Laurent Stolovitch  |  Equivalence of Cauchy-Riemann manifolds and multisummability theory ]]
 +
|Xianghong
 
|-
 
|-
|November 14
+
|Nov 20
| Naser Talebizadeh Sardari
+
| Grad Student Seminar
| UW Madison
+
|  
|[[#Naser Talebizadeh Sardari |   Quadratic forms and the semiclassical eigenfunction hypothesis ]]
+
|[[#linktoabstract |   ]]
| Betsy
+
|  
 
|-
 
|-
|November 28
+
|Nov 27
| Xianghong Chen
+
| No Seminar
| UW Milwaukee
+
|  
|[[#Xianghong Chen |   Some transfer operators on the circle with trigonometric weights ]]
+
|[[#linktoabstract |   ]]
| Betsy
+
|  
 
|-
 
|-
|Monday, December 4, 4:00, B139
+
|Dec 4
|  Bartosz Langowski and Tomasz Szarek
+
| No Seminar
| Institute of Mathematics, Polish Academy of Sciences
+
|[[#linktoabstract |   ]]
|[[#Bartosz Langowski and Tomasz Szarek |   Discrete Harmonic Analysis in the Non-Commutative Setting ]]
+
|  
| Betsy
 
 
|-
 
|-
|Wednesday, December 13, 4:00, B239 (Colloquium)
+
|Jan 22
|Bobby Wilson
+
| Brian Cook
|MIT
+
| Kent
|[[#Bobby Wilson | Projections in Banach Spaces and Harmonic Analysis ]]
+
|[[#Brian Cook  |   Equidistribution results for integral points on affine homogenous algebraic varieties ]]
| Andreas
+
| Street
 
|-
 
|-
| Monday, February 5, 3:00-3:50, B341  (PDE-GA seminar)
+
|Jan 29
| Andreas Seeger
+
| No Seminar
| UW
+
|  
|[[#Andreas Seeger | Singular integrals and a problem on mixing flows]]  
+
|[[#linktoabstract  |   ]]
 
|
 
|
 
|-
 
|-
|February 6
+
|Feb 5, '''B239'''
| Dong Dong
+
| Alexei Poltoratski
| UIUC
+
| Texas A&M
| [[#Dong Dong | Hibert transforms in a 3 by 3 matrix and applications in number theory]]
+
|[[#Alexei Poltoratski  |   Completeness of exponentials: Beurling-Malliavin and type problems ]]
|Betsy
+
| Denisov
 +
|-
 +
|'''Friday, Feb 8'''
 +
| Aaron Naber
 +
| Northwestern University
 +
|[[#linktoabstract  |  A structure theory for spaces with lower Ricci curvature bounds ]]
 +
| See colloquium website for location
 
|-
 
|-
|February 13
+
|Feb 12
| Sergey  Denisov
+
| Shaoming Guo
 
| UW Madison
 
| UW Madison
| [[#linkofabstract | Title]]
+
|[[#Shaoming Guo | Polynomial Roth theorems in Salem sets  ]]
 
|  
 
|  
 
|-
 
|-
|February 20
+
|'''Wed, Feb 13, B239'''
| Ruixiang Zhang
+
| Dean Baskin
| IAS (Princeton)
+
| TAMU
| [[#Ruixiang Zhang | The (Euclidean) Fractal Uncertainty Principle]]
+
|[[# Dean Baskin  |   Radiation fields for wave  equations ]]
| Betsy, Jordan, Andreas
+
| Colloquium
 +
|-
 +
|'''Friday, Feb 15'''
 +
| Lillian Pierce
 +
| Duke
 +
|[[#Lillian Pierce  |  Short character sums ]]
 +
|  Colloquium
 
|-
 
|-
|February 27
+
|'''Monday,  Feb 18, 3:30 p.m, B239.'''
|Detlef Müller
+
| Daniel Tataru
|University of Kiel
+
| UC Berkeley
| [[#Detlef Müller | On Fourier restriction for a non-quadratic hyperbolic surface]]
+
|[[#Daniel Tataru  |   A Morawetz inequality for water waves ]]
|Betsy, Andreas
+
| PDE Seminar
 
|-
 
|-
|Wednesday, March 7, 4:00 p.m.
+
|Feb 19
| Winfried Sickel
+
| Wenjia Jing
|Friedrich-Schiller-Universität Jena
+
|Tsinghua University
| [[#linkofabstract | Title]]
+
|Periodic  homogenization of Dirichlet problems in perforated domains: a unified proof
|Andreas
+
| PDE Seminar
 
|-
 
|-
|March 13
+
|Feb 26
 +
| No Seminar
 
|
 
|
|
 
| [[#linkofabstract | Title]]
 
 
|
 
|
 
|-
 
|-
|March 20
+
|Mar 5
|  
+
| Loredana Lanzani
|  
+
| Syracuse University
| [[#linkofabstract | Title]]
+
|[[#Loredana Lanzani  |   On regularity and irregularity of the Cauchy-Szegő projection in several complex variables ]]
 +
| Xianghong
 +
|-
 +
|Mar 12
 +
| Trevor Leslie
 +
| UW Madison
 +
|[[#Trevor Leslie  |  Energy Equality for the Navier-Stokes Equations at the First Possible Blowup Time ]]
 
|
 
|
 
|-
 
|-
|April 3
+
|Mar 19
 +
|Spring Break!
 
|  
 
|  
|  
+
|
| [[#linkofabstract | Title]]
 
 
|
 
|
 
|-
 
|-
|April 10
+
|Mar 26
 +
| No seminar
 
|  
 
|  
 +
|[[#linktoabstract  |    ]]
 
|  
 
|  
| [[#linkofabstract | Title]]
 
|
 
 
|-
 
|-
|Friday, April 13, 4:00 p.m. (Colloquium)
+
|Apr 2
|Jill Pipher
+
| Stefan Steinerberger
|Brown
+
| Yale
| [[#linkofabstract | Title]]
+
|[[#Stefan Steinerberger  |   Wasserstein Distance as a Tool in Analysis ]]
|WIMAW
+
| Shaoming, Andreas
 
|-
 
|-
|April 17
+
 
| TBA
+
|Apr 9
|  
+
| Franc Forstnerič
| [[#linkofabstract | Title]]
+
| Unversity of Ljubljana
|Andreas
+
|[[#Franc Forstnerič  |   Minimal surfaces by way of complex analysis ]]
 +
| Xianghong, Andreas
 
|-
 
|-
|April 24
+
|Apr 16
| Lenka Slavíková
+
| Andrew Zimmer
| University of Missouri
+
| Louisiana State University
| [[#linkofabstract | TBA]]
+
|[[#Andrew Zimmer  |   The geometry of domains with negatively pinched Kaehler metrics ]]
|Betsy, Andreas
+
| Xianghong
 
|-
 
|-
|May 1
+
|Apr 23
| Xianghong Gong
+
| Brian Street
| UW
+
| University of Wisconsin-Madison
| [[#linkofabstract | Title]]
+
|[[#Brian Street  |   Maximal Hypoellipticity ]]
|
+
| Street
|-
 
|May 15
 
|Gennady Uraltsev
 
|Cornell University
 
| [[#linkofabstract | TBA]]
 
|Betsy, Andreas
 
 
|-
 
|-
| May 16-18, [http://www.math.wisc.edu/~stovall/FA2018/ Workshop in Fourier Analysis]
+
|Apr 30
|
+
| Zhen Zeng
|
+
| UPenn
|
+
|[[#Zhen Zeng  |  Decay property of multilinear oscillatory integrals ]]
|Betsy, Andreas
+
| Shaoming
 
|-
 
|-
 
|}
 
|}
  
 
=Abstracts=
 
=Abstracts=
===Brian Street===
+
===Simon Marshall===
 +
 
 +
''Integrals of eigenfunctions on hyperbolic manifolds''
 +
 
 +
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.
 +
 
 +
 
 +
===Hong Wang===
 +
 
 +
''About Falconer distance problem in the plane''
 +
 
 +
If E is a compact set of Hausdorff dimension greater than 5/4 on the plane, we prove that there is a point x\in E such that the set of distances between x and E has positive Lebesgue measure. Our result improves upon Wolff's theorem for dim E> 4/3. This is joint work with Larry Guth, Alex Iosevich and Yumeng Ou.
 +
 
 +
===Polona Durcik===
 +
 
 +
''Singular Brascamp-Lieb inequalities and extended boxes in R^n''
 +
 
 +
Brascamp-Lieb inequalities are L^p estimates for certain multilinear forms on functions on Euclidean spaces. In this talk we consider singular Brascamp-Lieb inequalities, which arise when one of the functions is replaced by a Calderon-Zygmund kernel. We focus on a family of multilinear forms in R^n with a certain cubical structure and discuss their connection to some patterns in positive density subsets in R^n. Based on joint works with V. Kovac and C. Thiele.
 +
 
 +
 
 +
===Song-Ying Li===
 +
 
 +
''Estimates for the first positive eigenvalue of Kohn Laplacian on a pseudo-Hermitian manifold''
 +
 
 +
In this talk, I will present my recent works with my collaborators on the lower bound and upper bounds estimates
 +
for the first positive eigenvalues of Kohn Laplacian and sub-Laplacian on a strictly pseudoconvex pseudo-Hermitian CR manifold,
 +
which include CR Lichnerowicz-Obata theorem for the lower and upper bounds for the first positive eigenvalue for the
 +
Kohn Laplacian on strictly pseudoconvex hypersurfaces.
 +
 
 +
 
 +
===Hanlong Fan===
 +
 
 +
''A generalization of the theorem of Weil and Kodaira on prescribing residues''
 +
 
 +
An old theorem of Weil and Kodaira says that: For a K\"ahler manifold X, there exists a closed meromorphic one-form with residue divisor D if and only if D is homologous to zero. In this talk, I will generalize Weil and Kodaira's criterion to non-K\"ahler manifolds.
  
Title:  Convenient Coordinates
+
===Kyle Hambrook===
  
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".
+
''Fourier Decay and Fourier Restriction for Fractal Measures on Curves''
  
===Hiroyoshi Mitake===
+
I will discuss my recent work on some problems concerning
 +
Fourier decay and Fourier restriction for fractal measures on curves.
  
Title:  Derivation of multi-layered interface system and its application
+
===Laurent Stolovitch===
  
Abstract:  In this talk, I will propose a multi-layered interface system which can
+
''Equivalence of Cauchy-Riemann manifolds and multisummability theory''
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===
+
We apply the multisummability theory from Dynamical Systems to CR-geometry. As the main result, we show that two real-analytic hypersurfaces in $\mathbb C^2$ are formally equivalent, if and only if they are $C^\infty$ CR-equivalent at the respective point. As a corollary, we prove that all formal equivalences between real-algebraic Levi-nonflat hypersurfaces in $\mathbb C^2$  are algebraic (and in particular convergent). This is a joint work with I. Kossovskiy and B. Lamel.
  
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.
+
===Brian Cook===
  
===Michael Greenblatt===
+
''Equidistribution results for integral points on affine homogenous algebraic varieties''
  
Title:  Maximal averages and Radon transforms for two-dimensional hypersurfaces
+
Let Q be a homogenous integral polynomial of degree at least two. We consider certain results and questions concerning the distribution of the integral points on the level sets of Q.
  
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.
+
===Alexei Poltoratski===
  
===David Beltran===
+
''Completeness of exponentials: Beurling-Malliavin and type problems''
  
Title:  Fefferman Stein Inequalities
+
This talk is devoted to two old problems of harmonic analysis mentioned in the title. Both problems ask when a family of complex exponentials is complete (spans) an L^2-space. The Beruling-Malliavin problem was solved in the early 1960s and I will present its classical solution along with modern generalizations and applications. I will then discuss history and recent progress in the type problem, which stood open for more than 70 years.
  
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===
+
===Shaoming Guo===
  
Title: Factorising X^n.
+
''Polynomial Roth theorems in Salem sets''
  
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.
+
Let P(t) be a polynomial of one real variable. I will report a result on searching for patterns of the form (x, x+t, x+P(t)) within Salem sets, whose Hausdorff dimension is sufficiently close to one. Joint work with Fraser and Pramanik.  
  
===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===
+
===Dean Baskin===
  
Title: The Lerner-Ombrosi-Perez bound in the Muckenhoupt-Wheeden
+
''Radiation fields for wave equations''
conjecture is sharp.
 
  
Abstract: We show that the upper bound $[w]_{A_1}\log (e+[w]_{A_1})$ for
+
Radiation fields are rescaled limits of solutions of wave equations near "null infinity" and capture the radiation pattern seen by a distant observer. They are intimately connected with the Fourier and Radon transforms and with scattering theory. In this talk, I will define and discuss radiation fields in a few contexts, with an emphasis on spacetimes that look flat near infinity. The main result is a connection between the asymptotic behavior of the radiation field and a family of quantum objects on an associated asymptotically hyperbolic space.
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===
+
===Lillian Pierce===
Title: Higher order Journé commutators
 
  
Abstract: We consider questions that stem from operator theory via Hankel and
+
''Short character sums''
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
+
A surprisingly diverse array of problems in analytic number theory have at their heart a problem of bounding (from above) an exponential sum, or its multiplicative cousin, a so-called character sum. For example, both understanding the Riemann zeta function or Dirichlet L-functions inside the critical strip, and also counting solutions to Diophantine equations via the circle method or power sieve methods, involve bounding such sums. In general, the sums of interest fall into one of two main regimes: complete sums or incomplete sums, with this latter regime including in particular “short sums.” Short sums are particularly useful, and particularly resistant to almost all known methods. In this talk, we will see what makes a sum “short,” sketch why it would be incredibly powerful to understand short sums, and discuss a curious proof from the 1950’s which is still the best way we know to bound short sums. We will end by describing new work which extends the ideas of this curious proof to bound short sums in much more general situations.
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
+
===Loredana Lanzani===
analysis, while preserving the Ansatz from classical operator theory.
 
  
===Shaoming Guo ===
+
''On regularity and irregularity of the Cauchy-Szegő projection in several complex variables''
Title: Parsell-Vinogradov systems in higher dimensions
 
  
Abstract:
+
This talk is a survey of my latest, and now final, collaboration with Eli Stein.
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===
+
It is known that for bounded domains $D$ in $\mathbb C^n$ that are of class $C^2$ and are strongly pseudo-convex, the Cauchy-Szegő projection is bounded in $L^p(\text{b}D, d\Sigma)$ for $1<p<\infty$. (Here $d\Sigma$ is induced Lebesgue measure.)  We show, using appropriate worm domains, that this fails for any $p\neq 2$, when we assume that the domain in question is only weakly pseudo-convex. Our starting point are the ideas of Kiselman-Barrett introduced more than 30 years ago in the analysis of the Bergman projection. However the study of the Cauchy-Szegő projection raises a number of new issues and obstacles that need to be overcome. We will also compare these results to the analogous problem for the Cauchy-Leray integral, where however the relevant counter-example is of much simpler nature.
  
Title: Quadratic forms and the semiclassical eigenfunction hypothesis
+
===Trevor Leslie===
  
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.
+
''Energy Equality for the Navier-Stokes Equations at the First Possible Blowup Time''
  
===Xianghong Chen===
+
In this talk, we discuss the problem of energy equality for strong solutions of the Navier-Stokes Equations (NSE) at the first time where such solutions may lose regularity.  Our approach is motivated by a famous theorem of Caffarelli, Kohn, and Nirenberg, which states that the set of singular points associated to a suitable weak solution of the NSE has parabolic Hausdorff dimension of at most 1.  In particular, we furnish sufficient conditions for energy equality which depend on the dimension of the singularity set in addition to time and space integrability assumptions; in doing so we improve upon the classical results when attention is restricted to the first blowup time.  When our method is inconclusive, we are able to quantify the possible failure of energy equality in terms of the lower local dimension and the ''concentration dimension'' of a certain measure associated to the solution.  The work described is joint with Roman Shvydkoy (UIC).
  
Title:  Some transfer operators on the circle with trigonometric weights
+
===Stefan Steinerberger===
  
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.
+
''Wasserstein Distance as a Tool in Analysis''
  
===Bobby Wilson===
+
Wasserstein Distance is a way of measuring the distance between two probability distributions (minimizing it is a main problem in Optimal Transport). We will give a gentle Introduction into what it means and then use it to prove (1) a completely elementary but possibly new and quite curious inequality for real-valued functions and (2) a statement along the following lines: linear combinations of eigenfunctions of elliptic operators corresponding to high frequencies oscillate a lot and vanish on a large set of co-dimension 1 (this is already interesting for trigonometric polynomials on the 2-torus, sums of finitely many sines and cosines, whose sum has to vanish on long lines) and (3) some statements in Basic Analytic Number Theory that drop out for free as a byproduct.
  
Title: Projections in Banach Spaces and Harmonic Analysis
+
===Franc Forstnerič===
  
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.
+
''Minimal surfaces by way of complex analysis''
  
===Andreas Seeger===
+
After a brief historical introduction, I will present some recent developments in the theory of minimal surfaces in Euclidean spaces which have been obtained by complex analytic methods. The emphasis will be on results pertaining to the global theory of minimal surfaces including Runge and Mergelyan approximation, the conformal Calabi-Yau problem, properly immersed and embedded minimal surfaces, and a new result on the Gauss map of minimal surfaces.
  
Title: Singular integrals and a problem on mixing flows
+
===Andrew Zimmer===
  
Abstract: The talk will be about  results related to Bressan's mixing problem. We present  an inequality for the change of a  Bianchini semi-norm 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.
+
''The geometry of domains with negatively pinched Kaehler metrics''
  
===Dong Dong===
+
Every bounded pseudoconvex domain in C^n has a natural complete metric: the Kaehler-Einstein metric constructed by Cheng-Yau. When the boundary of the domain is strongly pseudoconvex, Cheng-Yau showed that the holomorphic sectional curvature of this metric is asymptotically a negative constant. In this talk I will describe some partial converses to this result, including the following: if a smoothly bounded convex domain has a complete Kaehler metric with close to constant negative holomorphic sectional curvature near the boundary, then the domain is strongly pseudoconvex. This is joint work with F. Bracci and H. Gaussier.
  
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.
+
===Brian Street===
  
===Ruixiang Zhang===
+
''Maximal Hypoellipticity''
  
Title: The (Euclidean) Fractal Uncertainty Principle
+
In 1974, Folland and Stein introduced a generalization of ellipticity known as maximal hypoellipticity. This talk will be an introduction to this concept and some of the ways it generalizes ellipticity.
  
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 |A||B| 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 Beurling-Malliavin 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===
+
===Zhen Zeng===
  
Title: On Fourier restriction for a non-quadratic hyperbolic surface
+
''Decay property of multilinear oscillatory integrals''
  
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.
+
In this talk, I will be talking about the conditions of the phase function $P$ and the linear mappings $\{\pi_i\}_{i=1}^n$ to ensure the asymptotic power decay properties of the following trilinear oscillatory integrals
 +
\[
 +
I_{\lambda}(f_1,f_2,f_3)=\int_{\mathbb{R}^m}e^{i\lambda P(x)}\prod_{j=1}^3 f_j(\pi_j(x))\eta(x)dx,  
 +
\]
 +
which falls into the broad goal in the previous work of Christ, Li, Tao and Thiele.
  
 
=Extras=
 
=Extras=
 
[[Blank Analysis Seminar Template]]
 
[[Blank Analysis Seminar Template]]

Latest revision as of 22:15, 23 April 2019

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 Brian at street(at)math

Previous Analysis seminars

Analysis Seminar Schedule

date speaker institution title host(s)
Sept 11 Simon Marshall UW Madison Integrals of eigenfunctions on hyperbolic manifolds
Wednesday, Sept 12 Gunther Uhlmann University of Washington Distinguished Lecture Series See colloquium website for location
Friday, Sept 14 Gunther Uhlmann University of Washington Distinguished Lecture Series See colloquium website for location
Sept 18 Grad Student Seminar
Sept 25 Grad Student Seminar
Oct 9 Hong Wang MIT About Falconer distance problem in the plane Ruixiang
Oct 16 Polona Durcik Caltech Singular Brascamp-Lieb inequalities and extended boxes in R^n Joris
Oct 23 Song-Ying Li UC Irvine Estimates for the first positive eigenvalue of Kohn Laplacian on a pseudo-Hermitian manifold Xianghong
Oct 30 Grad student seminar
Nov 6 Hanlong Fang UW Madison A generalization of the theorem of Weil and Kodaira on prescribing residues Brian
Monday, Nov. 12, B139 Kyle Hambrook San Jose State University Fourier Decay and Fourier Restriction for Fractal Measures on Curves Andreas
Nov 13 Laurent Stolovitch Université de Nice - Sophia Antipolis Equivalence of Cauchy-Riemann manifolds and multisummability theory Xianghong
Nov 20 Grad Student Seminar
Nov 27 No Seminar
Dec 4 No Seminar
Jan 22 Brian Cook Kent Equidistribution results for integral points on affine homogenous algebraic varieties Street
Jan 29 No Seminar
Feb 5, B239 Alexei Poltoratski Texas A&M Completeness of exponentials: Beurling-Malliavin and type problems Denisov
Friday, Feb 8 Aaron Naber Northwestern University A structure theory for spaces with lower Ricci curvature bounds See colloquium website for location
Feb 12 Shaoming Guo UW Madison Polynomial Roth theorems in Salem sets
Wed, Feb 13, B239 Dean Baskin TAMU Radiation fields for wave equations Colloquium
Friday, Feb 15 Lillian Pierce Duke Short character sums Colloquium
Monday, Feb 18, 3:30 p.m, B239. Daniel Tataru UC Berkeley A Morawetz inequality for water waves PDE Seminar
Feb 19 Wenjia Jing Tsinghua University Periodic homogenization of Dirichlet problems in perforated domains: a unified proof PDE Seminar
Feb 26 No Seminar
Mar 5 Loredana Lanzani Syracuse University On regularity and irregularity of the Cauchy-Szegő projection in several complex variables Xianghong
Mar 12 Trevor Leslie UW Madison Energy Equality for the Navier-Stokes Equations at the First Possible Blowup Time
Mar 19 Spring Break!
Mar 26 No seminar
Apr 2 Stefan Steinerberger Yale Wasserstein Distance as a Tool in Analysis Shaoming, Andreas
Apr 9 Franc Forstnerič Unversity of Ljubljana Minimal surfaces by way of complex analysis Xianghong, Andreas
Apr 16 Andrew Zimmer Louisiana State University The geometry of domains with negatively pinched Kaehler metrics Xianghong
Apr 23 Brian Street University of Wisconsin-Madison Maximal Hypoellipticity Street
Apr 30 Zhen Zeng UPenn Decay property of multilinear oscillatory integrals Shaoming

Abstracts

Simon Marshall

Integrals of eigenfunctions on hyperbolic manifolds

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.


Hong Wang

About Falconer distance problem in the plane

If E is a compact set of Hausdorff dimension greater than 5/4 on the plane, we prove that there is a point x\in E such that the set of distances between x and E has positive Lebesgue measure. Our result improves upon Wolff's theorem for dim E> 4/3. This is joint work with Larry Guth, Alex Iosevich and Yumeng Ou.

Polona Durcik

Singular Brascamp-Lieb inequalities and extended boxes in R^n

Brascamp-Lieb inequalities are L^p estimates for certain multilinear forms on functions on Euclidean spaces. In this talk we consider singular Brascamp-Lieb inequalities, which arise when one of the functions is replaced by a Calderon-Zygmund kernel. We focus on a family of multilinear forms in R^n with a certain cubical structure and discuss their connection to some patterns in positive density subsets in R^n. Based on joint works with V. Kovac and C. Thiele.


Song-Ying Li

Estimates for the first positive eigenvalue of Kohn Laplacian on a pseudo-Hermitian manifold

In this talk, I will present my recent works with my collaborators on the lower bound and upper bounds estimates for the first positive eigenvalues of Kohn Laplacian and sub-Laplacian on a strictly pseudoconvex pseudo-Hermitian CR manifold, which include CR Lichnerowicz-Obata theorem for the lower and upper bounds for the first positive eigenvalue for the Kohn Laplacian on strictly pseudoconvex hypersurfaces.


Hanlong Fan

A generalization of the theorem of Weil and Kodaira on prescribing residues

An old theorem of Weil and Kodaira says that: For a K\"ahler manifold X, there exists a closed meromorphic one-form with residue divisor D if and only if D is homologous to zero. In this talk, I will generalize Weil and Kodaira's criterion to non-K\"ahler manifolds.

Kyle Hambrook

Fourier Decay and Fourier Restriction for Fractal Measures on Curves

I will discuss my recent work on some problems concerning Fourier decay and Fourier restriction for fractal measures on curves.

Laurent Stolovitch

Equivalence of Cauchy-Riemann manifolds and multisummability theory

We apply the multisummability theory from Dynamical Systems to CR-geometry. As the main result, we show that two real-analytic hypersurfaces in $\mathbb C^2$ are formally equivalent, if and only if they are $C^\infty$ CR-equivalent at the respective point. As a corollary, we prove that all formal equivalences between real-algebraic Levi-nonflat hypersurfaces in $\mathbb C^2$ are algebraic (and in particular convergent). This is a joint work with I. Kossovskiy and B. Lamel.


Brian Cook

Equidistribution results for integral points on affine homogenous algebraic varieties

Let Q be a homogenous integral polynomial of degree at least two. We consider certain results and questions concerning the distribution of the integral points on the level sets of Q.

Alexei Poltoratski

Completeness of exponentials: Beurling-Malliavin and type problems

This talk is devoted to two old problems of harmonic analysis mentioned in the title. Both problems ask when a family of complex exponentials is complete (spans) an L^2-space. The Beruling-Malliavin problem was solved in the early 1960s and I will present its classical solution along with modern generalizations and applications. I will then discuss history and recent progress in the type problem, which stood open for more than 70 years.


Shaoming Guo

Polynomial Roth theorems in Salem sets

Let P(t) be a polynomial of one real variable. I will report a result on searching for patterns of the form (x, x+t, x+P(t)) within Salem sets, whose Hausdorff dimension is sufficiently close to one. Joint work with Fraser and Pramanik.



Dean Baskin

Radiation fields for wave equations

Radiation fields are rescaled limits of solutions of wave equations near "null infinity" and capture the radiation pattern seen by a distant observer. They are intimately connected with the Fourier and Radon transforms and with scattering theory. In this talk, I will define and discuss radiation fields in a few contexts, with an emphasis on spacetimes that look flat near infinity. The main result is a connection between the asymptotic behavior of the radiation field and a family of quantum objects on an associated asymptotically hyperbolic space.

Lillian Pierce

Short character sums

A surprisingly diverse array of problems in analytic number theory have at their heart a problem of bounding (from above) an exponential sum, or its multiplicative cousin, a so-called character sum. For example, both understanding the Riemann zeta function or Dirichlet L-functions inside the critical strip, and also counting solutions to Diophantine equations via the circle method or power sieve methods, involve bounding such sums. In general, the sums of interest fall into one of two main regimes: complete sums or incomplete sums, with this latter regime including in particular “short sums.” Short sums are particularly useful, and particularly resistant to almost all known methods. In this talk, we will see what makes a sum “short,” sketch why it would be incredibly powerful to understand short sums, and discuss a curious proof from the 1950’s which is still the best way we know to bound short sums. We will end by describing new work which extends the ideas of this curious proof to bound short sums in much more general situations.

Loredana Lanzani

On regularity and irregularity of the Cauchy-Szegő projection in several complex variables

This talk is a survey of my latest, and now final, collaboration with Eli Stein.

It is known that for bounded domains $D$ in $\mathbb C^n$ that are of class $C^2$ and are strongly pseudo-convex, the Cauchy-Szegő projection is bounded in $L^p(\text{b}D, d\Sigma)$ for $1<p<\infty$. (Here $d\Sigma$ is induced Lebesgue measure.) We show, using appropriate worm domains, that this fails for any $p\neq 2$, when we assume that the domain in question is only weakly pseudo-convex. Our starting point are the ideas of Kiselman-Barrett introduced more than 30 years ago in the analysis of the Bergman projection. However the study of the Cauchy-Szegő projection raises a number of new issues and obstacles that need to be overcome. We will also compare these results to the analogous problem for the Cauchy-Leray integral, where however the relevant counter-example is of much simpler nature.

Trevor Leslie

Energy Equality for the Navier-Stokes Equations at the First Possible Blowup Time

In this talk, we discuss the problem of energy equality for strong solutions of the Navier-Stokes Equations (NSE) at the first time where such solutions may lose regularity. Our approach is motivated by a famous theorem of Caffarelli, Kohn, and Nirenberg, which states that the set of singular points associated to a suitable weak solution of the NSE has parabolic Hausdorff dimension of at most 1. In particular, we furnish sufficient conditions for energy equality which depend on the dimension of the singularity set in addition to time and space integrability assumptions; in doing so we improve upon the classical results when attention is restricted to the first blowup time. When our method is inconclusive, we are able to quantify the possible failure of energy equality in terms of the lower local dimension and the concentration dimension of a certain measure associated to the solution. The work described is joint with Roman Shvydkoy (UIC).

Stefan Steinerberger

Wasserstein Distance as a Tool in Analysis

Wasserstein Distance is a way of measuring the distance between two probability distributions (minimizing it is a main problem in Optimal Transport). We will give a gentle Introduction into what it means and then use it to prove (1) a completely elementary but possibly new and quite curious inequality for real-valued functions and (2) a statement along the following lines: linear combinations of eigenfunctions of elliptic operators corresponding to high frequencies oscillate a lot and vanish on a large set of co-dimension 1 (this is already interesting for trigonometric polynomials on the 2-torus, sums of finitely many sines and cosines, whose sum has to vanish on long lines) and (3) some statements in Basic Analytic Number Theory that drop out for free as a byproduct.

Franc Forstnerič

Minimal surfaces by way of complex analysis

After a brief historical introduction, I will present some recent developments in the theory of minimal surfaces in Euclidean spaces which have been obtained by complex analytic methods. The emphasis will be on results pertaining to the global theory of minimal surfaces including Runge and Mergelyan approximation, the conformal Calabi-Yau problem, properly immersed and embedded minimal surfaces, and a new result on the Gauss map of minimal surfaces.

Andrew Zimmer

The geometry of domains with negatively pinched Kaehler metrics

Every bounded pseudoconvex domain in C^n has a natural complete metric: the Kaehler-Einstein metric constructed by Cheng-Yau. When the boundary of the domain is strongly pseudoconvex, Cheng-Yau showed that the holomorphic sectional curvature of this metric is asymptotically a negative constant. In this talk I will describe some partial converses to this result, including the following: if a smoothly bounded convex domain has a complete Kaehler metric with close to constant negative holomorphic sectional curvature near the boundary, then the domain is strongly pseudoconvex. This is joint work with F. Bracci and H. Gaussier.


Brian Street

Maximal Hypoellipticity

In 1974, Folland and Stein introduced a generalization of ellipticity known as maximal hypoellipticity. This talk will be an introduction to this concept and some of the ways it generalizes ellipticity.


Zhen Zeng

Decay property of multilinear oscillatory integrals

In this talk, I will be talking about the conditions of the phase function $P$ and the linear mappings $\{\pi_i\}_{i=1}^n$ to ensure the asymptotic power decay properties of the following trilinear oscillatory integrals \[ I_{\lambda}(f_1,f_2,f_3)=\int_{\mathbb{R}^m}e^{i\lambda P(x)}\prod_{j=1}^3 f_j(\pi_j(x))\eta(x)dx, \] which falls into the broad goal in the previous work of Christ, Li, Tao and Thiele.

Extras

Blank Analysis Seminar Template