##### Department of Mathematics,

University of California San Diego

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### Math 211B - Group Actions Seminar

## Zhongkai Tao

#### UC Berkeley

## Fractal uncertainty principle via Dolgopyat's method in higher dimensions

##### Abstract:

The fractal uncertainty principle (FUP) was introduced by Dyatlov and Zahl which states that a function cannot be localized near a fractal set in both position and frequency spaces. It has rich applications in spectral gaps and quantum chaos on hyperbolic manifolds and has recently been an active area of research in harmonic analysis. I will talk about the history of the fractal uncertainty principle and explain its applications to spectral gaps. Then I will talk about our recent work, joint with Aidan Backus and James Leng, which proves a general fractal uncertainty principle for small fractal sets, improving the volume bound in higher dimensions. This generalizes the work of Dyatlov--Jin using Dolgopyat's method. As an application, we give effective essential spectral gaps for convex cocompact hyperbolic manifolds in higher dimensions with Zariski dense fundamental groups.

Host: Brandon Seward

### April 13, 2023

### 10:00 AM

APM 7218 and Zoom ID 967 4109 3409

Email an organizer for the password

Research Areas

Ergodic Theory and Dynamical Systems****************************