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SUMMARY:Control of eigenfunctions on negatively curved manifolds - Semyon 
  Dyatlov (Massachusetts Institute of Technology)
DTSTART:20260115T101500Z
DTEND:20260115T111500Z
UID:TALK238180@talks.cam.ac.uk
DESCRIPTION:Semiclassical measures are a standard object studied in quantu
 m chaos\, capturing macroscopic behavior of sequences of eigenfunctions in
  the high energy limit. They have a long history of study going back to th
 e 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 Schr&oum
 l\;dinger equation and decay for the damped wave equation.\n&nbsp\;\nOur t
 heorem was restricted to dimension 2 because the key new ingredient\, the 
 fractal uncertainty principle (proved by Bourgain and myself)\, was only k
 nown for subsets of the real line. I will talk about more recent joint wor
 k with Athreya and Miller in the setting of complex hyperbolic quotients a
 nd the work by Kim and Miller in the setting of real hyperbolic quotients 
 of any dimension. In these works there are potential obstructions to the f
 ull support property which can be classified by Ratner theory and geometri
 cally described in terms of certain totally geodesic submanifolds. Time pe
 rmitting\, I will also mention a recent counterexample to Quantum Unique E
 rgodicity for higher-dimensional quantum cat maps\, due to Kim and buildin
 g on the previous counterexample of Faure-Nonnenmacher-De Bi&egrave\;vre.
LOCATION:Seminar Room 1\, Newton Institute
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