Berkeley Harmonic Analysis and Differential Equations Student Seminar
Spring
2019



Here is a list of links to material and conferences/workshops on PDEs and Harmonic Analysis.



January 29th
3:40 - 5:00 PM
740 Evans Hall
Eugenia Malinnikova
Remez inequality for solutions of elliptic PDEs. The Remez inequality for polynomials states that the maximum of the polynomial over an interval is controlled by its maximum over a subset of the interval of positive measure. The coefficient in the inequality depends on the degree of the polynomial and the result holds in higher dimensions.

We give a version of the Remez inequality for solutions of second order linear elliptic PDEs and their gradients. In this context, the degree of a polynomial is replaced by the Almgren frequency of the solution. We discuss other results on quantitative unique continuation for solutions of elliptic PDEs and their gradients and give some applications for the estimates of eigenfunctions for the Laplace-Beltrami operator. The talk is based on a joint work with A. Logunov.

February 5th
3:40 - 5:00 PM
740 Evans Hall
TBA
TBA  

February 12th
3:40 - 5:00 PM
740 Evans Hall
TBA
TBA  

February 19th
3:40 - 5:00 PM
740 Evans Hall
No HADES (Special Seminar on Wednesday)
Alexis Drouot on Wednesday, 4-5pm in 736 Evans

February 26th
3:40 - 5:00 PM
740 Evans Hall
James Rowan
Concentration Compactness Methods for Nonlinear Dispersive PDEs. Concentration compactness methods provide a powerful tool for proving global well-posedness and scattering for nonlinear dispersive equations. Once one has a small-data global well-posedness result, one knows that there is some minimal size of the initial data at which global well-posedness and scattering can fail. Then, using a profile decomposition, one can show that there is a minimal blowup solution that is almost periodic. One can then use tools like long-time Strichartz estimates and interaction Morawetz inequalities to rule out these "minimal enemies." I will illustrate this technique by presenting a proof, due to Killip and Visan, of the global well-posedness and scattering for the three-dimensional energy-critical defocusing NLS.

March 5th
3:40 - 5:00 PM
740 Evans Hall
TBA
TBA  

March 12th
3:40 - 5:00 PM
740 Evans Hall
Dongxiao Yu
Global Solutions of Quasilinear Wave Equations. In this talk, I will present a proof of global existence of solutions to certain quasilinear wave equations in three space dimensions with small initial data. The talk is based on a paper by Hans Lindblad published in 2008.

March 19th
3:40 - 5:00 PM
740 Evans Hall
No HADES (Prospective graduate student open house Monday)
Prospective Graduate student open house MONDAY March 18, 4-5 PM, 959 Evans  

March 26th
3:40 - 5:00 PM
740 Evans Hall
No HADES (Spring break)
 

April 2nd
3:40 - 5:00 PM
740 Evans Hall
Ruixiang Zhang
Real polynomials and the Fourier extension operator. The Fourier extension operator is a very interesting and difficult object to study in harmonic analysis. Stein conjectured that it is a bounded linear operator between some $L^p$ spaces. Recently people have found that auxiliary real polynomials can help one study Stein's above Restriction Conjecture. We will talk about a few interesting facts about zero sets of real polynomials, and why they can be useful in the study of the Restriction Conjecture.

April 9th
3:40 - 5:00 PM
740 Evans Hall
Semyon Dyatlov
Microlocal methods in hyperbolic dynamics. I will present a microlocal approach to some of the analytical problems in hyperbolic dynamics. Applications include exponential decay of correlations and a definition of Pollicott-Ruelle resonances for hyperbolic systems. The intuition comes from scattering theory, with scattering happening as frequency goes to infinity. Based on joint work with Maciej Zworski.

April 16th
3:40 - 5:00 PM
740 Evans Hall
Mitchell Taylor
Bases in Banach lattices. In this talk we will discuss bases in Banach lattices, and how they can be used to measure (non)-embeddability of a Banach space into a lattice. We will give several characterizations of basic sequences that "respect the lattice structure", and discuss some of the more unexpected corollaries. Time permitting, I will comment on existence of non-negative bases in Hilbert space.

April 23rd
3:40 - 5:00 PM
740 Evans Hall
Tim Laux
Weak-strong uniqueness for multiphase mean curvature flow. Multiphase mean curvature flow has, due to its importance in materials science, received a lot of attention over the last decades. On the one hand, there is substantial recent progress in the construction of weak solutions. On the other hand, strong solutions are - in particular in the planar case of networks - very well understood.

In this talk, after giving an overview of the topic, I will present a weak-strong uniqueness principle for multiphase mean curvature flow: as long as a strong solution to multiphase mean curvature flow exists, any distributional solution with optimal energy dissipation rate has to coincide with this solution.

In our proof we construct a suitable relative entropy functional, which in this geometric context may be viewed as a time-dependent variant of calibrations. Just like the existence of a calibration guarantees that one has found a global minimum, the existence of a "time-dependent calibration" ensures that the route of steepest descent in the energy landscape is unique and stable.

For the purpose of this talk, I will focus on two instructive model cases: a single smooth interface and a single triple junction.

This is a joint work (in progress) with Julian Fischer, Sebastian Hensel, and Thilo Simon.

April 30th
3:40 - 5:00 PM
740 Evans Hall
Xuwen Zhu
Self-adjoint extensions of Laplace operators on singular geometry. Spectral geometry aims at understanding how geometry influences the spectrum of geometrically related operators such as the Laplace operator. I will first talk about Von Neumann theory on classification of self-adjoint extensions of symmetric operators, and in particular focus on the Laplace operator on metrics with conical singularities. Then I will give a survey on how geometry (e.g. curvature and monodromy) and choice of self-adjoint extensions influence the spectrum, and discuss some results related to translation surfaces and spherical conical metrics.

May 7th
3:40 - 5:00 PM
740 Evans Hall
TBA
TBA




Upcoming Conferences and Summer Schools

Here is a link for those intersted in student support.

MathPrograms

Past:

Chicago Summer School in Analysis: 6/17/2017 - 6/30/2017 at U Chicago.

Hausdorff School: Dispersive Equations, Solitons, and Blow-up: 9/4/2017 - 9/8/2017 at University of Bonn.

Prairie Analysis Seminar 2017: 9/8/17 - 9/9/17 at Kansas State University.


Expository Articles and Info

Subcritical Scattering for Defocusing NLS, by Jason Murphy (UC Berkeley).

Generalizations of Fourier Analysis, and How to Apply Them, by W.T. Gowers.

A Study Guide for the l^2 Decoupling Theorem, by Jean Bourgain and Ciprian Demeter.

Notes on hyperbolic dynamics, by Semyon Dyatlov.


MSRI Links

Past:

Recent Developments in Harmonic Analysis, 5/15/2017 - 5/19/2017

Introductory Workshop: Harmonic Analysis, 1/23/2017 - 1/27/2017

Nonlinear dispersive PDE, quantum many particle systems and the world between, 7/17/2017 - 7/28/2017

New challenges in PDE: Deterministic dynamics and randomness, Fall 2015

Introductory Workshop: Randomness and long time dynamics in nonlinear evolution differential equations, Fall 2015


If you have more links to add to this list, please email me.