Difference between revisions of "Spring 2017 Analysis Seminars"
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Latest revision as of 14:36, 22 May 2017
Analysis Seminar Schedule Spring 2017
date  speaker  title  host(s)  

January 17, Math Department Colloquium  Fabio Pusateri (Princeton)  The Water Waves Problem  Sigurd Angenent  
January 24, Joint Analysis/Geometry Seminar  Tamás Darvas (Maryland)  Existence of constant scalar curvature Kähler metrics and properness of the Kenergy  Jeff Viaclovsky  
Monday, January 30, 3:30, VV901 (PDE Seminar)  Serguei Denissov (UW Madison)  Instability in 2D Euler equation of incompressible inviscid fluid  
February 7  Andreas Seeger (UW Madison)  The Haar system in Sobolev spaces  
February 21  Jongchon Kim (UW Madison)  Some remarks on Fourier restriction estimates  Andreas Seeger  
March 7, Mathematics Department Distinguished Lecture  Roger Temam (Indiana)  On the mathematical modeling of the humid atmosphere  Leslie Smith  
Wednesday, March 8, Joint Applied Math/PDE/Analysis Seminar  Roger Temam (Indiana)  Weak solutions of the ShigesadaKawasakiTeramoto system  Leslie Smith  
March 14  Xianghong Chen (UW Milwaukee)  Restricting the Fourier transform to some oscillating curves  Andreas Seeger  
March 21  SPRING BREAK 
 
Monday, March 27 (joint PDE/Analysis Seminar), 3:30, VV901  Sylvia Serfaty (NYU)  Mean Field Limits for Ginzburg Landau Vortices  Hung Tran  
March 28  Brian Cook (Fields Institute)  Twists on the twisted ergodic theorems  Andreas Seeger  
Friday, March 31, 4:00 p.m., B139  Laura Cladek (UBC)  Endpoint bounds for the lacunary spherical maximal operator  Andreas Seeger  
April 4  Francesco Di Plinio (Virginia)  Sparse domination of singular integral operators  Andreas Seeger  
April 11  Xianghong Gong (UW Madison)  Hoelder estimates for homotopy operators on strictly pseudoconvex domains with C^2 boundary  
April 25 (joint PDE/Analysis Seminar)  Chris Henderson (University of Chicago)  A localintime Harnack inequality and applications to reactiondiffusion equations  Jessica Lin 
Abstracts
Fabio Pusateri
The Water Waves problem
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 DengIonescuPausader  and sketch some of the main ideas.
Tamás Darvas
Existence of constant scalar curvature Kähler metrics and properness of the Kenergy
Given a compact Kähler manifold $(X,\omega)$, we show that if there exists a constant scalar curvature Kähler metric cohomologous to $\omega$ then Mabuchi's Kenergy is Jproper in an appropriate sense, confirming a conjecture of Tian from the nineties. The proof involves a careful study of weak minimizers of the Kenergy, and involves a surprising amount of analysis. This is joint work with Robert Berman and Chinh H. Lu.
Serguei Denissov
Instability in 2D Euler equation of incompressible inviscid fluid
We consider the patch evolution under the 2D Euler dynamics and study how the geometry of the boundary can deteriorate in time.
Andreas Seeger
The Haar system in Sobolev spaces
We consider the Haar system on Sobolev spaces and ask: 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
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. 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)
On the mathematical modeling of the humid atmosphere
The humid atmosphere is a multiphase system, made of air, water vapor, cloudcondensate, 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, noncontinuous (and nonmonotone) in the framework of nonlinear partial differential equations. 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 quasivariational inequalities.
Roger Temam (Seminar)
Weak solutions of the ShigesadaKawasakiTeramoto system
We will present a result of existence of weak solutions to the ShigesadaKawasakiTeramoto 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 selfcontained and does not rely on any earlier result. Based on an article with Du Pham, to appear in Nonlinear Analysis.
Xianghong Chen
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 monomiallike or finite type curves by work of BakOberlinSeeger, DendrinosMueller, and Stovall, where a geometric inequality (among others) plays an important role. Such an inequality fails to hold if the torsion demonstrates strong signchanging behavior, in which case endpoint restriction bounds may fail. In this talk I will present how one could obtain sharp nonendpoint 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.
Sylvia Serfaty
Mean Field Limits for Ginzburg Landau Vortices
GinzburgLandau type equations are models for superconductivity, superfluidity, BoseEinstein condensation. A crucial feature is the presence of quantized vortices, which are topological zeroes of the complexvalued solutions. This talk will review some results on the derivation of effective models to describe the statics and dynamics of these vortices, with particular attention to the situation where the number of vortices blows up with the parameters of the problem. In particular we will present new results on the derivation of mean field limits for the dynamics of many vortices starting from the parabolic GinzburgLandau equation or the GrossPitaevskii (=Schrodinger GinzburgLandau) equation.
Brian Cook
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 WienerWintner 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.
Laura Cladek
Endpoint bounds for the lacunary spherical maximal operator
Define the lacunary spherical maximal operator as the maximal operator corresponding to averages over spheres of radius 2^k for k an integer. This operator may be viewed as a model case for studying more general classes of singular maximal operators and Radon transforms. It is a classical result in harmonic analysis that this operator is bounded on L^p for p>1, but the question of weaktype (1, 1) boundedness (which would correspond to pointwise convergence of lacunary spherical averages for functions in L^1 has remained open. Although this question still remains open, we discuss some new endpoint bounds for the operator near L^1 that allows us to conclude almost everywhere pointwise convergence of lacunary spherical means for functions in a slightly smaller space than L\log\log\log L. This is based on joint work with Ben Krause.
Francesco di Plinio
Sparse domination of singular integral operators
Singular integral operators, which are a priori signed and nonlocal, can be dominated in norm, pointwise, or dually, by sparse averaging operators, which are in contrast positive and localized. The most striking consequence is that weighted norm inequalities for the singular integral follow from the corresponding, rather immediate estimates for the averaging operators. In this talk, we present several positive sparse domination results of singular integrals falling beyond the scope of classical CalderónZygmund theory; notably, modulation invariant multilinear singular integrals including the bilinear Hilbert transforms, variation norm Carleson operators, matrixvalued kernels, rough homogeneous singular integrals and critical BochnerRiesz means, and singular integrals along submanifolds with curvature. Collaborators: Amalia Culiuc, Laura Cladek, Jose Manuel CondeAlonso, Yen Do, Yumeng Ou and Gennady Uraltsev.
Xianghong Gong
Hoelder estimates for homotopy operators on strictly pseudoconvex domains with C^2 boundary
Abstract: We derive a new homotopy formula for a bounded strictly pseudoconvex domain of C^2 boundary by using a method of Lieb and Range, and we obtain estimates for its homotopy operator. We show that the dbar equation on the domain admits a solution gaining halfderivative in the HoelderZygmund spaces. The estimates are also applied to obtain a boundary regularity for Dsolutions on a suitable product domain in the Leviflat Euclidean spaces.
Chris Henderson
A localintime Harnack inequality and applications to reactiondiffusion equations
Abstract: The classical Harnack inequality requires one to look back in time to obtain a uniform lower bound on the solution to a parabolic equation. In this talk, I will introduce a Harnacktype inequality that allows us to remove this restriction at the expense of a slightly weaker bound. I will then discuss applications of this bound to (time permitting) three nonlocal reactiondiffusion equations arising in biology. In particular, in each case, this inequality allows us to show that solutions to these equations, which do not enjoy a maximum principle, may be compared with solutions to a related local equation, which does enjoy a maximum principle. Precise estimates of the propagation speed follow from this.