Organizer
Jin Long 金龙
Speakers
Zhu Xuwen (Northeastern University)
Tao Zhongkai (IHES)
Semyon Dyatlov (MIT)
Time
Thur., 2:00-5:00 pm, July 17, 2025
Venue
C548, Shuangqing Complex Building A
Lecture 1
Speaker:
Zhu Xuwen (Northeastern University)
Time：
2:00-3:00 pm
Title：
Analysis of gravitational instantons
Abstract: 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.
Lecture 2
Speaker:
Tao Zhongkai (IHES)
Time：
3:00-4:00 pm
Title:
Lossless Strichartz and spectral projection estimates on manifolds with trapping
Abstract: 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.
Lecture 3
Speaker:
Semyon Dyatlov (MIT)
Time：
4:00-5:00 pm
Title:
Control of eigenfunctions on negatively curved manifolds
Abstract: 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.
