Quantum chaos and fractal uncertainty principle

Organizers:  Jean Bourgain, IAS and Semyon Dyatlov, MIT

Participants:  Alexis Drouot, Alexander Gamburd, Long Jin, Alex Kantorovich, Elon Lindenstrauss, Michael Magee, Frédéric Naud, Stéphane Nonnenmacher, Peter Sarnak, Alexander Sodin, Steve Zelditch, Ruixiang Zhang

Summary:  Fractal uncertainty principle (FUP) states that no function can be localized close to a fractal set in both position and frequency. It is a new method which can be applied to problems in quantum chaos, such as distribution of eigenvalues and eigenfunctions on compact manifolds and spectral gaps and resonance counting on noncompact manifolds. Despite recent progress by Bourgain, Dyatlov, Jin, and Zahl in the setting of hyperbolic surfaces, at the moment there are many open questions including:

  • how to prove FUP in higher dimensions,
  • what is the best exponent in FUP under suitable assumptions on the structure of the fractal sets involved,
  • how to adapt FUP techniques to more general hyperbolic systems such as manifolds of variable negative curvature.

Understanding these questions would likely require a combination of techniques in harmonic analysis, microlocal analysis, and hyperbolic dynamics. The goal of this workshop is to bring together experts from these three fields to advance our understanding of fractal uncertainty principle and its applications.

Working Group Report

S Nonnemacher Lecture An Introduction to Quantum Chaos

S Dyatlov Lecture Fractal Uncertainty Principle and its Applications

S Dyatlov Lecture Semiclassical analysis, chaotic dynamics, and fractal uncertainty principle

R Zhang Lecture Proof of Fractal Uncertainty Principle

L Jin Lecture Control of eigenfunctions on hyperbolic surfaces

F Naud Lecture An introduction to Dolgopyat's method

S Dyatlov Lecture Fractal uncertainty principle: improving over the volume bound