Over the past three decades, the ergodic theory for flows on homogeneous spaces has produced many beautiful applications in number theory, geometry and mathematical physics. In these lectures I will provide an introduction to equidistribution theorems for large spheres and horospheres, and explain their very recent application to directional statistics for lattices, quasicrystals and other aperiodic point sets. These lectures are introductory and aimed at a non-specialist mathematical audience. Much of the material is based on joint work with A. Strombergsson, D. El-Baz and I. Vinogradov.

Let f be the obvious covering map from Euclidean n-space to the n-torus. It is well known that if L is any straight line in n-space, then the closure of f(L) is a very nice submanifold of the n-torus. In 1990, Marina Ratner proved a beautiful generalization of this observation that replaces Euclidean space with any Lie group G, and allows L to be any subgroup of G that is ``unipotent.'' We will discuss the statement of this theorem and related results, some of the ideas in the proofs, and a few of the important consequences.

A classical problem in the theory of quadratic forms is to decide whether two given integral quadratic forms are equivalent. That is, whether their symmetric integral matrices A and B satisfy A=X’BX for some unimodular integral matrix X. For definite forms one can construct a simple decision procedure. Somewhat surprisingly, no such procedure was known for indefinite forms until the work of C. L. Siegel in the early 1970s. In the late 1990s, D. W. Masser made the following conjecture related to this problem: Let n be at least 3, and suppose A, B are equivalent. Then there exists a unimodular integral matrix X such that A=X’BX and ||X||< C(||A||+||B||)^k, where the constants C, k depend only on the dimension n. In this talk we shall discuss our recent resolution of this conjecture based on a joint work with Professor Gregory A. Margulis.

Over the past three decades, the ergodic theory for flows on homogeneous spaces has produced many beautiful applications in number theory, geometry and mathematical physics. In these lectures I will provide an introduction to equidistribution theorems for large spheres and horospheres, and explain their very recent application to directional statistics for lattices, quasicrystals and other aperiodic point sets. These lectures are introductory and aimed at a non-specialist mathematical audience. Much of the material is based on joint work with A. Strombergsson, D. El-Baz and I. Vinogradov.

For a simple Lie group G and a discrete subgroup Gamma of G, we will discuss  exponential decay of the matrix coefficients of G associated to functions on the quotient space Gamma\G. When G is of rank one, and Gamma is a lattice, this yields exponential mixing of the geodesic flow for  the associated Riemannian manifold.  We will also discuss a uniform  exponential decay for congruence subgroups of Gamma when Gamma is contained in an arithmetic subgroup of G.

We will first explain three properties of a Zariski dense subsemigroups of a semisimple real Lie group:

- the existence of loxodromic elements,

- the convexity and non degeneracy of the limit cone,

- the density of the group spanned by the Jordan projection.

We will then explain how to deduce from them three limit theorems for a random walk on this semigroup:

- the law of large numbers,

- the central limit theorem,

- the local limit theorem.

Let f be the obvious covering map from Euclidean n-space to the n-torus. It is well known that if L is any straight line in n-space, then the closure of f(L) is a very nice submanifold of the n-torus. In 1990, Marina Ratner proved a beautiful generalization of this observation that replaces Euclidean space with any Lie group G, and allows L to be any subgroup of G that is ``unipotent.'' We will discuss the statement of this theorem and related results, some of the ideas in the proofs, and a few of the important consequences.

We will discuss the action of unipotent subgroups of G=SL(2,R), SL(2,C) on homogeneous spaces G/\Gamma where \Gamma is a discrete, Zariski dense subgroup of G.  The focus will be on the case when \Gamma is not a lattice; some of the similarities  and contrasts between this case and the lattice case will be described.

The talk will be concerned with a property of lattices in the geometry of numbers which I refer to as "almost sure grid-DV with respect to a function F". The goal of the talk will be three-fold:

(1) Discuss a theorem relating this property to the dynamics of the lattice under the invariance group of the function.

(2) Apply the theorem to various classical discussions and see what it gives (it relates nicely to Minkowski's conjecture and to diophantine exponents).

(3) Discuss a mistake I have in the published paper and some open problems

In the past decade, substantial progress has been made in several beautiful arithmetic problems connected to thin groups: Apollonian packings, Zaremba's conjecture, and so on.  This progress was made possible by a series of developments, in particular the Affine Sieve of Bourgain-Gamburd-Sarnak from a decade ago.  In this talk we will focus on the arithmetic of Apollonian packings, and then move on to consider thin groups in their own right, apart from arithmetic applications.  Specifically, we will discuss how hyperbolic geometric methods have shed light on thinness of certain monodromy groups of hypergeometric equations, and how one might go about distinguishing thin groups from their non-thin counterparts in general.

Let $p(t,x,y)$ be the heat kernel on the universal cover $\widetilde{M}$ of a compact Riemannian manifold of negative curvature. We show that $$C(x,y) = \lim_{t \to \infty} e^{\lambda_0 t} t^{3/2} p(t,x,y) $$ is a positive function depending only on $x,y \in \widetilde{M}$, where $\lambda_0$ is the bottom of the spectrum. The function $C(x,y)$ can be described in terms of a Patterson-Sullivan density on $\partial \widetilde{M}$. We also show that $\lambda_0$-Martin boundary of $\widetilde{M}$ coincides with its topological boundary. We will explain how Margulis argument for counting geodesics as well as a uniform version of Dolgopyat's rapid-mixing of the geodesic flow are used to prove the results. This is a joint work with François Ledrappier.

I will discuss the problem of Diophantine approximation on group varieties and outline a strategy to establish analogues of classical results in Diophantine approximation in this setting. Specifically, I will talk about the problem of counting solutions to Diophantine inequalities on group varieties and its connection to spectral gap. This is joint work with Alexander Gorodnik and Amos Nevo.