报告人 Speaker：Xiao Xiao (McGill University)
组织者 Organizer：金龙
时间 Time：3:30-5:00 pm, 08/27/2024
地点 Venue：Lecture Hall C548, Tsinghua University Shuangqing Complex Building A（清华大学双清综合楼A座C548报告厅)
Upcoming Talk:
题目: Complexified eigenfunctions and QER in complex setting
主讲人:
Xiao Xiao (McGill University)
时间:
3:30-5:00 pm, 08/27/2024
地点:
双清综合楼C548
摘要:
Quantum Ergodicity (QE) is the study of equidistribution property of a sequence of Laplace eigenfunctions in high-frequency limit. It is a curious question whether a QE sequence is still QE when restricted to a hypersurface. This is known as the Quantum Ergodic Restriction (QER) problem, which has found many important applications such as a lower bound on nodal counting. I will discuss famous known results on this problem by Toth-Zelditch and Christianson-Toth-Zelditch, as well as a new QER theorem in the setting of Grauert-tube complexification. The proof makes use of the complex structure in a crucial way that is not available in the real case. Time permitting, I will discuss some applications and possible future works.
Past Talks：
题目：Optimal Enhanced dissipation for contact Anosov flows.
主讲人: 陶中恺（Tao Zhongkai，University of California, Berkeley）
时间: 3:20-4:50 pm, 12/27/2023
地点: Lecture Hall C548, Tsinghua University Shuangqing Complex Building A（清华大学双清综合楼A座C548报告厅)
摘要:
The enhanced dissipation is motivated by the study of fluid mechanics, and has been studied by many people. I will talk about my recent work with Maciej Zworski, which shows a wide family of flows produces "optimal enhanced dissipation". I will also talk about background on normally hyperbolic trapping if time permits.
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
地点: 近春园西楼第一会议室
摘要:
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.
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
地点: 近春园西楼第一会议室
摘要:
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.
Title: Existence of Fourier series on Euclidean subsets
Speaker: 刘博辰（南方科技大学）
Time: 2023年4月17日 10：00-11：30
Venue: 近春园西楼第一会议室
Abstract:
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.
