Geometric Analysis and Topology Seminar

Spring 2017

The seminar's usual time is Wednesday at 11:00am in 1314 Warren Weaver Hall (Directions). Special times and dates are marked in red. Click on the title of a talk for the abstract (if available).

Jan 25,  11am
1314 WWH
Mario Bonk
Parabolicity of leaves
Feb 21,  10am
805 WWH
David Fisher
(Indiana University)
Zimmer's conjecture: subexponential growth, measure rigidity and strong property (T)
Feb 22,  11am
1314 WWH
Brandon Seward
The Geometric Burnside's Problem
Apr 19,  11am
1314 WWH
Thang Nguyen
Hyperbolic rank rigidity for strict quarter-pinching negatively curved manifold.

Organizers: Sylvain Cappell, Jeff Cheeger, Bruce Kleiner, and Robert Young.


Parabolicity of leaves, Mario Bonk.  Certain dynamical systems give rise to foliations where the leaves are quasi-isometric 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 postcritically-finite 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 self-contained, 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)<n-1, a(G) is finite. Furthermore if dim(M)=n-1 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 quarter-pinching 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.

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