For more information, contact Peter Scott at pscott@umich.edu.

If you want to check the list of seminars held in the year 2000-1, click here.

Schedule of Talks

Abstracts

September 16

Thurston's Ending Lamination Conjecture provides a conjectural classification of all (marked) hyperbolic 3-manifolds homotopy equivalent to a fixed compact 3-manifold $M$. This classification is in terms of topological data, the marked homeomorphism type of the manifold, and geometric data, which encodes the asymptotic geometry of the ends of the manifold. Brock, Canary and Minsky recently established this conjecture in the case when $M$ has incompressible boundary. We will discuss the conjecture and some consequences of its proof.

September 23

This is a continuation of Dick Canary's talk from last week.

September 30

Let S be a closed, connected, orientable surface of genus at least 3, C(S) be the complex of curves on S and Mod_S^* be the extended mapping class group of S. We prove that a simplicial map, lambda : C(S) -> C(S), preserves nondisjointness (i.e. if alpha and beta are two vertices in C(S) such that i(alpha, beta) \noteq 0, then i(lambda(alpha), lambda(beta)) \noteq 0) iff it is induced by a homeomorphism of S. As a corollary, we prove that if K is a finite index subgroup of Mod_S^* and f : K -> Mod_S^* is an injective homomorphism, then f is induced by a homeomorphism of S and f has a unique extension to an automorphism of Mod_S^*.

October 21

Spectral theory is the subfield of differential geometry which provided the solution to Kac's famous question, "Can you hear the shape of a drum?" That is, can we use the Laplace spectrum of a manifold to draw conclusions about its geometry or topology? Here we ask this question in the context of Riemannian orbifolds. In an effort to "listen" to orbifold homotopy type, we will gain insight on the singular sets of isospectral orbifolds.

November 4

We obtain bounds on the total bending of the boundary of the convex core of a hyperbolic 3-manifold. These bounds will depend on the injectivity radius of the intrinsic geometry of the boundary of the convex hull of the limit set. This is joint work with Dick Canary.

November 11

We prove that, in fixed genus, the standard Heegaard splitting of a "generic" surface bundle is the unique splitting of minimal genus. The major tools used are Casson and Gordon's notion of strong irreducibility, the Rubinstein-Scharlemann graphic, and the curve complex. This is joint work with Dave Bachman.

November 18

In a recent preprint, Maher and Rubinstein prove that all free actions of Z_3 on the 3-sphere are conjugate to linear actions. I will discuss what is known about actions of finte groups on the 3-sphere in general, and then describe their arguments, but not in great detail.

November 25

By studying the space of projective structures on a surface S of genus at least two, we establish results on the topology of the space of hyperbolic metrics on SxR. This is joint work with Ken Bromberg.

December 2

In this talk, I'll address the simple question: which (Riemannian) metric on a given smooth manifold has the greatest amount of symmetry? I'll explain the notion of hidden symmetry, and show that on many manifolds there is a metric which has more hidden symmetry than any other metric. I will also pose a number of conjectures and questions concerning this phenomenon. This is joint work with Shmuel Weinberger.

This page last updated20 .