Harmonic Analysis and Differential Equations Student Seminar (HADES), Summer 2019

Talks are from 12:00 - 1:00 PM (Berkeley time) in 939 Evans hall, followed by lunch somewhere on Northside.

This summer iteration of HADES is meant to be a little more informal and less research-focused than HADES during the sememster.

July 16th
12:00-1:00 PM
939 Evans Hall
James Rowan
Paradifferential operators and nonlinear PDEs

Paradifferential operators and paraproducts give a way to use microlocal analysis to describe nonlinear operations like function composition and multiplication. The essential idea is that nonlinear interactions can be broken down into interactions of different frequency regimes. In this talk, I will introduce these tools and illustrate their usefulness by proving a commutator estimate of Kato and Ponce.

July 23rd
12:00 - 1:00 PM
939 Evans Hall
Jian Wang
Spectral theory for elliptic operators

I will establish a theorem of Hörmander on the asymptotic behaviour of the eigenvalues of an elliptic operator on a compact manifold. I will first use eigenvalues to construct solutions to a hyperbolic Cauchy problem and then construct asymptotic solutions. Then we will derive an asymptotic formula for the counting function of the eigenvalues from a trace formula.

July 30th
12:00 - 1:00 PM
939 Evans Hall
N/A
No HADES (Northwestern summer school)  

August 6th
12:00 - 1:00 PM
939 Evans Hall
N/A
No HADES (Northwestern summer school)  

August 13th
12:00 - 1:00 PM
939 Evans Hall
N/A
No HADES (Northwestern summer school)  

August 20st
12:00 - 1:00 PM
939 Evans Hall
Zirui Zhou
Prime gaps and Fourier optimization

In this talk I will present the main theorem in a paper by Carneiro, Milinovich, and Soundararajan. The authors establish a connection between a particular extremal problem in Fourier analysis and the problem of bounding the largest possible gap between consecutive primes assuming the Riemann hypothesis. In particular, this theorem implies that the universal constant $c$ in the classical result $\lim sup_{n\rightarrow \infty}\frac{p_{n+1}-p_n}{\sqrt{p_n}\log(p_n)}\leq c$ ($p_n$ denotes the nth prime) can get smaller than 1, which was the best result before this paper.

August 27th
12:00 - 1:00 PM
939 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.