题目：Fractal uncertainty principle for discrete Fourier transform and random Cantor sets
主讲人: 韩晓龙（Han Xiaolong，California State University, Northridge）
时间: 9:00-10:00 am, 07/17/2023
The Fourier uncertainty principle describes a fundamental phenomenon that a function and its Fourier transform cannot simultaneously localize. Dyatlov and his collaborators (Zahl, Bourgain, Jin, Nonnenmacher) recently introduced a concept of Fractal Uncertainty Principle (FUP). It is a mathematical formulation concerning the limit of localization of a function and its Fourier transform on sets with certain fractal structure.
The FUP has quickly become an emerging topic in Fourier analysis and also has important applications to other fields such as scattering theory and quantum chaos. In this talk, we consider the discrete Fourier transform and the fractal sets are given by discrete Cantor sets. We present the FUP in this discrete setting with a much more favorable estimate than the one known before, when the Cantor sets are constructed by a random procedure. This is a joint work with Suresh Eswarathasan.
题目：Singular damped waves on manifolds
主讲人: 王若宇（Wang Ruoyu, University College London）
时间: 10:00-11:00 am, 07/17/2023
We will discuss a damped wave semigroup for damping exhibiting Hölder-type blowup near a hypersurface on a compact manifold. We will use vector field methods to prove a sharp energy decay result for singular damping on the torus, where the optimal rate of energy decay explicitly depends on the singularity of the damping. Dynamically, such fine rates result from that the transverse propagation of lower-order regularity fails to penetrate through the hypersurface where the damping potential is singular. This is a joint work with Perry Kleinhenz.
Title: Existence of Fourier series on Euclidean subsets
Time: 2023年4月17日 10：00-11：30
Fourier series is a very powerful tool in nature. In this talk we will introduce different types of Fourier basis, such as orthogonal basis, Riesz basis, frames, etc., and discuss about their existence on Euclidean subsets.