**Yves Benoist (Université Paris-Sud): ***Arithmeticity of discrete subgroups*

The aim of this course will be to prove that any Zariski dense discrete subgroup of a higher rank simple Lie group which intersects cocompactly the unipotent radical of a proper parabolic subgroup is an arithmetic lattice. This work with Miquel extends previous work of Selberg and Hee Oh and answers an old question of Margulis.

The course will focus on concrete cases like SL(n,R) or SO(p,q) and will be a journey in the world of discrete subgroups of Lie groups. We will explain how classical tools and new techniques enter the proof: Zassenhauss neighborhoods, Auslander theorem, Margulis lemma, Bruhat decomposition, Borel-Harish-Chandra finiteness theorem, Borel density theorem, Raghunathan-Venkataramana congruence subgroup theorem, Weil rigidity theorem, Ratner topological theorem etc.

**Hee Oh (Yale University): ***Homogeneous Dynamics and Hyperbolic manifolds of infinite volume*

For a long time many questions have been answered for hyperbolic 3 manifolds of finite volume, but they are wide open for those of infinite volume. One such question is classification of possible closures of a geodesic plane in a hyperbolic 3-manifold. The main obstacle in the infinite volume case is scarcity of recurrence of unipotent flows, without which we cannot use unipotent dynamics.

In this course, we will discuss recent joint work with McMullen, Mohammadi, and in part with Benoist which establishes a closed or dense dichotomy for a geodesic plane in the interior of the convex core of M for any geometrically finite acylindrical hyperbolic 3-manifold. This discussion boils down to classifying the closures of SL(2, R) orbits in the quotient space Gamma\SL(2,C) where Gamma is a geometrically finite Zariski dense subgroup. When Gamma is not a lattice, no SL(2,R) orbit is dense in Gamma\SL(2,C). Description of possible SL(2,R)-orbit closures involves the canonical fractal set associated to Gamma, called the limit set of Gamma.

We will explain how some of the main techniques in unipotent dynamics can be adapted to the setting of infinite volume, and how the topological structure of the manifold Gamma\H^3 affects the recurrence property of unipotent flows.

**Giulio Tiozzo (University of Toronto): ***Random walks on weakly hyperbolic groups*

The distribution of sums of real-valued random variables is determined by the classical theorems of probability (law of large numbers, central limit theorem). Starting in the 1960’s Furstenberg, Oseledets and others have generalized such results tor noncommutative groups, e.g. groups of matrices.

In this course, we will consider random walks on groups of isometries of delta-hyperbolic spaces, and establish their asymptotic properties: for instance, sample paths almost surely converge to the boundary and have positive drift. In recent years, this has had many applications to low-dimensional topology, as e.g. the mapping class group and Out(F_n) act on certain (non-locally compact) hyperbolic spaces. We will discuss some such applications and their relations to Teichmuller theory.

*Lecture plan: *1) Introduction to delta-hyperbolic spaces and random walks

2) The horofunction boundary

3) Convergence to the boundary and positive drift

4) Genericity of hyperbolic elements

The course is mostly based on my joint work with J. Maher.

**Alex Wright (Stanford University):** *TBA*