Berkeley Harmonic Analysis and Differential Equations Student Seminar

Fall 2019


We have moved to a new website for announcements and abstracts.
August 27th
3:40 - 5:00 PM
748 Evans Hall
Haoren Xiong
The method of complex scaling

Scattering resonances of Schrödinger operator with a compactly supported potential are defined as the poles of the meromorphic continuation of the resolvent. I will introduce the method of complex scaling which produces a natural family of non-self-adjoint operators whose discrete spectrum consists of resonances. Furthermore, we will see that similar results hold in the case of dilation analytic potentials.

September 3rd
3:40 - 5:00 PM
748 Evans Hall
No HADES
(MSRI workshop)  

September 10th
3:40 - 5:00 PM
748 Evans Hall
Stéphane Nonnenmacher
Random perturbations of nonselfadjoint operators, and the Gaussian Analytic Function  
The spectrum of a nonselfadjoint linear operator can be very unstable, that is sensitive to perturbations, an phenomenon usually referred to as the "pseudospectral effect". In order to quantify this phenomenon, we investigate a simple class of nonselfadjoint 1-dimensional semiclassical (pseudo-)differential operators, submitted to small random perturbations. The spectrum of this randomly perturbed operator is then viewed as a random point process on the complex plane, whose statistical properties we wish to analyze. Hager & Sjöstrand have shown that, in the semiclassical limit, the randomly perturbed eigenvalues satisfies a probabilistic form of Weyl's law, at the macroscopic scale. We in turn investigate the statistical distribution of the eigenvalues at the microscopic scale (scale of the distance between nearby eigenvalues). We show that at this scale, the spectral statistics satisfy a partial form of universality: spectral correlations can be expressed in terms of a universal object, the Gaussian Analytic Function (GAF), and a few parameters depending on the initial operator, and of the type of random disorder. A central tool in our analysis is a well-posed Grushin problem, which turns our spectral problem on L^2(R) into an effective nonlinear spectral problem on a finite dimensional subspace ("effective Hamiltonian"). This Grushin problem is set up by studying the "classical spectrum" of our initial semiclassical operator (a region in the complex plane), constructing quasimodes of this operator, and analyzing the (complex-)energy-dependence of these quasimodes. This is joint work with Matin Vogel.
September 17th
3:40 - 5:00 PM
740 Evans Hall
Alix Deleporte
Szegö kernels and Toeplitz operators  

Szegö kernels encode information on weighted holomorphic functions, or holomorphic sections. In an appropriate large curvature limit, they enjoy a semiclassical structure. Among other applications, these kernels are used to define an alternative quantization scheme : Berezin-Toeplitz quantization.

This talk will be an opportunity to further motivate attendance to M. Zworski's course.

September 24th
3:40 - 5:00 PM
740 Evans Hall
Mohandas Pillai
TBA  

October 1st
3:40 - 5:00 PM
740 Evans Hall
Katrina Morgan
TBA  

October 8th
3:40 - 5:00 PM
740 Evans Hall
Amir Vig
TBA  

October 15th
3:40 - 5:00 PM
740 Evans Hall
TBA
TBA  

October 22nd
3:40 - 5:00 PM
740 Evans Hall
TBA
TBA  

October 29th
3:40 - 5:00 PM
740 Evans Hall
TBA
TBA  

November 5th
3:40 - 5:00 PM
740 Evans Hall
Jacob Shapiro
TBA  

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

November 19th
3:40 - 5:00 PM
740 Evans Hall
TBA
TBA  

November 26th
3:40 - 5:00 PM
740 Evans Hall
TBA
TBA  

December 3rd
3:40 - 5:00 PM
740 Evans Hall
TBA
TBA  

December 10th
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

This semester:

Microlocal Analysis, 8/12/2019 - 12/13/2019

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 one of the organizers.