# Difference between revisions of "Analysis Seminar"

Analysis Seminar Current Semester

The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.

# Analysis Seminar Schedule Spring 2017

date speaker title host(s)
January 17, Math Department Colloquium Fabio Pusateri (Princeton) The Water Waves Problem Angenent
January 24, Joint Analysis/Geometry Seminar Tamás Darvas (Maryland) Existence of constant scalar curvature Kähler metrics and properness of the K-energy 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
March 7, Mathematics Department Distinguished Lecture Roger Temam (Indiana) TBA Smith
Wednesday, March 8, Joint Applied Math/PDE/Analysis Seminar Roger Temam (Indiana) TBA Smith
March 14 Xianghong Chen (UW Milwaukee) Restricting the Fourier transform to some oscillating curves Seeger
March 21 SPRING BREAK

March 27 (joint PDE/Analysis seminar), 3:30, VV901 Sylvia Serfaty TBA Tran
March 28 Brian Cook (Fields Institute) TBA Seeger
April 4 Francesco Di Plinio (Virginia) TBA Seeger

# 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 Deng-Ionescu-Pausader - and sketch some of the main ideas.

### Tamás Darvas

Existence of constant scalar curvature Kähler metrics and properness of the K-energy

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 K-energy is J-proper in an 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

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.

### 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 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.