Parabolicity of leaves, Mario Bonk.
Certain dynamical systems give rise to foliations where the leaves are quasiisometric to open simply connected surfaces. The question arises whether these leaves are parabolic or hyperbolic (equivalent to recurrence or transience of a random walk). This is related, for example, to Cannon's conjecture in geometric group theory or to Thurston's characterization of postcriticallyfinite rational maps.
I will discuss this in my talk and also mention some open problems in the area.

Zimmer's conjecture: subexponential growth, measure rigidity and strong property (T), David Fisher.
This talk is a sequel to the colloquium of the day before. I will try to make
it logically independent and selfcontained, but most of the history and motivation will occur in the colloquium talk and this talk will emphasize ideas of proofs of the following theorem. Let G be a cocompact lattice in SL(n,R) where n>2, M a compact manifold and a: G> Diff(M) a homomorphism. If dim(M)<n1, a(G) is finite. Furthermore if dim(M)=n1 and a(G) preserves a volume form, a(G) is finite. The proof has many surprising features, including that it uses hyperbolic dynamics to prove an essentially elliptic result and that it uses results on homogeneous dynamics, including Ratner's measure classification theorem, to prove results about inhomogeneous system. This is joint work with Aaron Brown and Sebastian
Hurtado.

The Geometric Burnside's Problem, Brandon Seward.
Burnside's Problem and the von Neumann Conjecture are classical problems from group theory which were long ago answered in the negative. In 1999, Kevin Whyte defined geometric analogs of these problems and proved the Geometric von Neumann Conjecture. In this talk, I will present a solution to the Geometric Burnside's Problem. I will also present a strengthening of Whyte's result and draw conclusions about the existence of regular spanning trees of Cayley graphs.

Hyperbolic rank rigidity for strict quarterpinching negatively curved manifold. Thang Nguyen.
A manifold is said to have higher hyperbolic rank if along every geodesic we can find a parallel vector field making curvature 1 with the geodesic. Examples of manifolds with higher hyperbolic rank are locally symmetric rank one manifolds. In this talk, I will present a rigidity statement: a compact higher hyperbolic rank manifold of curvature between [1, 1/4) is locally symmetric. This is a joint work with C. Connell and R. Spatzier.
