题目: Analysis of gravitational instantons
主讲人: Zhu Xuwen (Northeastern University)
时间: 2:00-3:00 pm, 07/17/2025
地点: 双清综合楼C548
摘要: Gravitational instantons are non-compact Calabi--Yau metrics with L^2 bounded curvature and are categorized into six types. I will describe three projects on gravitational instantons including: (a) Fredholm theory and deformation of the ALH* type; (b) non-collapsing degeneration limits of ALH* and ALH types; (c) existence of stable non-holomorphic minimal spheres in some ALF types. These three projects utilize geometric microlocal analysis in different singular settings. Based on works joint with Rafe Mazzeo, Yu-Shen Lin and Sidharth Soundararajan.
题目: Lossless Strichartz and spectral projection estimates on manifolds with trapping
主讲人: Tao Zhongkai (IHES)
时间: 3:00-4:00 pm, 07/17/2025
地点: 双清综合楼C548
摘要: The Strichartz estimate is an important estimate in proving the well-posedness of dispersive PDEs. People believed that the lossless Strichartz estimate could not hold on manifolds with trapping (for example, the local smoothing estimate always comes with a logarithmic loss in the presence of trapping). Surprisingly, in 2009, Burq, Guillarmou, and Hassell proved a lossless Strichartz estimate for manifolds with trapping under the pressure condition. I will talk about their result as well as our recent work with Xiaoqi Huang, Christopher Sogge, and Zhexing Zhang, which goes beyond the pressure condition using the fractal uncertainty principle and a logarithmic short-time Strichartz estimate. If time permits, I will also talk about the lossless spectral projection estimate in the same setting.
题目: Control of eigenfunctions on negatively curved manifolds
主讲人: Semyon Dyatlov (MIT)
时间: 4:00-5:00 pm, 07/17/2025
地点: 双清综合楼C548
摘要: Semiclassical measures are a standard object studied in quantum chaos, capturing macroscopic behavior of sequences of eigenfunctions in the high energy limit. They have a long history of study going back to the Quantum Ergodicity theorem and the Quantum Unique Ergodicity conjecture. I will speak about the work with Jin and Nonnenmacher, proving that on a negatively curved surface, every semiclassical measure has full support. I will also discuss applications of this work to control for the Schrödinger equation and decay for the damped wave equation.
Our theorem was restricted to dimension 2 because the key new ingredient, the fractal uncertainty principle (proved by Bourgain and myself), was only known for subsets of the real line. I will talk about more recent joint work with Athreya and Miller in the setting of complex hyperbolic quotients and the work in progress by Kim and Miller in the setting of real hyperbolic quotients of any dimension. In these works there are potential obstructions to the full support property which can be classified by Ratner theory and geometrically described in terms of certain totally geodesic submanifolds. Time permitting, I will also mention a recent counterexample to Quantum Unique Ergodicity for higher-dimensional quantum cat maps, due to Kim and building on the previous counterexample of Faure-Nonnenmacher-De Bièvre
Past Talks：
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.
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.
