# Trimester Seminar Series

## Tuesday, March 10th, 10:30 a.m.

### Law of large numbers for the coefficients

Speaker: Yves Benoist (Université Paris-Sud)

Abstract:

I will recall Furstenberg's law of large numbers for the norm of products of independent equidistributed random matrices when the first moment is finite. I will then explain a law of large numbers for the matrix coefficients of these products. The main tool will be Kesten amenability criterion. Joint with JF. Quint.

## Tuesday, March 3rd, 3 p.m.

### Cycle integrals and traces of CM values of modular forms

Abstract:

In this talk we give an introduction to the study of generating series of the traces of the CM values and geodesic cycle integrals of different modular forms. First we define modular forms and their generalizations. Then we briefly discuss the theory of theta lifts that gives a conceptual framework for such generating series. We end with some applications of this theory.

## Thursday, February 20th, 10:30 a.m.

### Random walk on the torus by linear and affine transformations

Speaker: Elon Lindenstrauss (The Hebrew University of Jerusalem)

Abstract:

I will present several quantitative results regarding the distribution of individual trajectories of a random walk on the d-torus (R/Z)^d by linear and affine transformation, including some work in progress with Weikun He and Tsviqa Lakrec.

## Tuesday, February 11th, 10:30 a.m.

### Duke’s theorems and their variants

Speaker: Philippe Michel (Ecole Polytechnique Federale de Lausanne)

Abstract:

What is commonly refereed as "Duke’s theorems" are a collection of equidistribution results for representations of integers by ternary quadratic forms on the associated quadratic surfaces. By duality they are equivalent to equidistribution of torus orbits in homogeneous spaces attached to quaternions algebras. In this talk, I will review these theorems, the various methods available to approach these (automorphic forms and/or dynamics) and discuss some recent variants.

## Thursday, January 30rd, 10:30 a.m.

### Measure Rigidity for non-maximal diagonal subgroups

Speaker: Manfred Einsiedler (ETH Zürich)

Abstract:

Strong measure classification theorems are one of the engines behind many applications of homogeneous dynamics e.g. to number theory. Unlike the case of unipotent dynamics (resolved by M. Ratner) the case of diagonal subgroups is still unresolved. First of all all current techniques require a positive entropy assumption. However, even with this assumption we only understand such measures if the action is by a Cartan subgroup or if we are considering joinings. We will present a new method that can handle non-maximal cases in the case of homogeneous spaces defined by $SL_2^k$ or $SL_3^2$. This is joint work with Elon Lindenstrauss.

## Tuesday, January 28st, 10:30 a.m.

### On the sup-norm problem for arithmetic hyperbolic manifolds

Speaker: Djordje Milicevic (Bryn Mawr College and Max Planck Institute for Mathematics)

#### Abstract:

The correspondence principle of quantum mechanics predicts that the long-term dynamics of the geodesic flow on a compact Riemannian manifold are reflected in the limiting behavior of its high-energy Laplace-Beltrami eigenstates. On negatively curved manifolds, the expectation of a diffused quantum picture to reflect the strong classical mixing is quantified by statements such as the QUE and the sup-norm problem, which asks about the asymptotics of the extreme pointwise values of Laplace eigenfunctions and now occupies a prominent place at the intersection of spectral theory, geometry/dynamics, and analytic number theory.

The sup-norm is a very sensitive measurement that can often detect fine invariant features of the geodesic flow such as unstable embedded hypersurfaces. In this talk, we will discuss the state of the art of the sup-norm problem on arithmetic hyperbolic manifolds, motivate the arithmetic techniques that give access to some of the chaotic dynamics, and present several upper and lower bounds for the sup-norm of Hecke-Maass cusp forms of myself and joint with Blomer, Harcos, and Maga, obtained by combining spectral, diophantine, and geometric arguments.

## Tuesday, January 21st, 2:45 p.m.

### Welcome Meeting

with Christoph Thiele, Director of the HIM

## Tuesday, January 21st, 3 p.m.

### Quantum Unique Ergodicity

Speaker: Lior Silberman (The University of British Columbia)

#### Abstract:

In this colloquium-style talk I will discuss some aspects of the equidistribution problem for Laplace eigenfunctions on Riemannian manifolds.  In the latter part of the talk I will concentrate on the case of locally symmetric spaces, where I will discuss positive results for exact eigenfunctions (with and without reference to the number-theoretic symmetries of the manifold), and negative results for approximate eigenfunctions.

Most of the results are (independently) joint with A. Venkatesh, N. Anantharaman, and S. Eswarathasan.