Difference between revisions of "Fall 2018 and Spring 2019 Analysis Seminars"
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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
Contents
 1 Previous Analysis seminars
 2 Analysis Seminar Schedule
 3 Abstracts
 3.1 Simon Marshall
 3.2 Hong Wang
 3.3 Polona Durcik
 3.4 SongYing Li
 3.5 Hanlong Fan
 3.6 Kyle Hambrook
 3.7 Laurent Stolovitch
 3.8 Brian Cook
 3.9 Alexei Poltoratski
 3.10 Shaoming Guo
 3.11 Dean Baskin
 3.12 Lillian Pierce
 3.13 Loredana Lanzani
 3.14 Trevor Leslie
 3.15 Stefan Steinerberger
 3.16 Franc Forstnerič
 3.17 Andrew Zimmer
 3.18 Brian Street
 3.19 Zhen Zeng
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 BrascampLieb inequalities and extended boxes in R^n  Joris 
Oct 23  SongYing Li  UC Irvine  Estimates for the first positive eigenvalue of Kohn Laplacian on a pseudoHermitian 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 CauchyRiemann 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: BeurlingMalliavin 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 CauchySzegő projection in several complex variables  Xianghong 
Mar 12  Trevor Leslie  UW Madison  Energy Equality for the NavierStokes 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 WisconsinMadison  Maximal Hypoellipticity  Street 
Apr 30  Zhen Zeng  UPenn  Decay property of multilinear oscillatory integrals  Shaoming 
*Madison Lectures in Fourier Analysis  
Summer  
Sept 10  Jose Madrid  UCLA  Andreas, David  
Oct 15  Bassam Shayya  American University of Beirut  Andreas, Betsy 
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 BrascampLieb inequalities and extended boxes in R^n
BrascampLieb inequalities are L^p estimates for certain multilinear forms on functions on Euclidean spaces. In this talk we consider singular BrascampLieb inequalities, which arise when one of the functions is replaced by a CalderonZygmund 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.
SongYing Li
Estimates for the first positive eigenvalue of Kohn Laplacian on a pseudoHermitian 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 subLaplacian on a strictly pseudoconvex pseudoHermitian CR manifold, which include CR LichnerowiczObata 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 oneform 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 nonK\"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 CauchyRiemann manifolds and multisummability theory
We apply the multisummability theory from Dynamical Systems to CRgeometry. As the main result, we show that two realanalytic hypersurfaces in $\mathbb C^2$ are formally equivalent, if and only if they are $C^\infty$ CRequivalent at the respective point. As a corollary, we prove that all formal equivalences between realalgebraic Levinonflat 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: BeurlingMalliavin 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^2space. The BerulingMalliavin 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 socalled character sum. For example, both understanding the Riemann zeta function or Dirichlet Lfunctions 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 CauchySzegő 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 pseudoconvex, the CauchySzegő 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 pseudoconvex. Our starting point are the ideas of KiselmanBarrett introduced more than 30 years ago in the analysis of the Bergman projection. However the study of the CauchySzegő 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 CauchyLeray integral, where however the relevant counterexample is of much simpler nature.
Trevor Leslie
Energy Equality for the NavierStokes Equations at the First Possible Blowup Time
In this talk, we discuss the problem of energy equality for strong solutions of the NavierStokes 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 realvalued 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 codimension 1 (this is already interesting for trigonometric polynomials on the 2torus, 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 CalabiYau 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 KaehlerEinstein metric constructed by ChengYau. When the boundary of the domain is strongly pseudoconvex, ChengYau 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.