### Abstract:

Suppose MM is a closed Riemannian manifold with an orthonormal basis BB of L2(M)L2(M) consisting of Laplace eigenfunctions. A classical result of Shnirelman and others proves that if the geodesic flow on the cotangent bundle of MM is ergodic, then MM is quantum ergodic, in particular, on average, the probability measures defined by the functions ff in BB on MM tends on average towards the Riemannian measure on MM in the high energy limit (i.e, as the Laplace eigenvalues of f→∞f→∞).

We now want to look at a level aspect of this property, namely, instead of taking a fixed manifold and high energy eigenfunctions, we take a sequence of Benjamini-Schramm convergent compact Riemannian manifolds MjMj together with Laplace eigenfunctions ff whose eigenvalue varies in short intervals. This perspective has been recently studied in the context of graphs by Anantharaman and Le Masson, and for hyperbolic surfaces and manifolds by Abert, Bergeron, Le Masson, and Sahlsten. In my talk I want to discuss joint work with F. Brumley in which we study this question in higher rank, namely sequences of compact quotients of

SL(n,R)/SO(n)SL(n,R)/SO(n).

This talk is part of the online Number Theory Web Seminar, and will be streamed live on Zoom.

www.ntwebseminar.org/home

Registered users will receive an email before each talk with a link to the Zoom meeting.

Organisers:

Mike Bennett (University of British Columbia)

Philipp Habegger (University of Basel)

Alina Ostafe (UNSW Sydney)

Speaker

Jasmin Matz

Research Area

Number Theory Seminar

Affiliation

University of Copenhagen

Date

Tue, 24/11/2020 - 9:00pm

Venue

RC-4082, The Red Centre, UNSW