Schedule of Talks

Abstracts

The Topology of Deformation Spaces of Hyperbolic 3-Manifolds

For any closed surface S, the deformation space AH(S) is the space of all marked hyperbolic 3-manifolds homotopy equivalent to S. After reviewing some of the classical results that describe topology of the interior of AH(S), we will show that there are certain points on the boundary where AH(S) is not locally connected. This is a generalization of Ken Bromberg’s result that the space of Kleinian punctured torus groups is not locally connected.

Uniform uniform exponential growth of subgroups of the mapping class group

Let Mod(S) denote the mapping class group of a compact, orientable surface S. Finitely generated subgroups of Mod(S) which are not virtually abelian have uniform
exponential growth with minimal growth rate bounded below by a constant depending only on S. We describe the proof of this and related results.

On the space of discrete linear groups

I will discuss a work in progress with Emmaneul Breuillard. Let Hom(F_k, G) be the set of homomorphisms from a free group F_k into a Lie group G. We study the subset of this space consisting of homomorphisms with discrete image and prove several results generalizing classical results of Jorgensen in the case where G is SL(2,R) or SL(2,C). As additional
motivation for this work, we use an idea of Lubotzky to produce an interesting new class of locally rigid subgroups of Lie groups.

Computations in motivic stable homotopy theory

The Morel-Voevodsky motivic homotopy category provides a framework in which homotopy-theoretic constructions and techniques can be applied to algebraic varieties. Until recently, though, few computations were known beyond those done by Voevodsky. After introducing motivic homotopy theory, I will present some motivic versions of spectral sequences from topology. In topology, these are used in calculations of stable homotopy groups, while their motivic analogues converge to motivic stable homotopy groups, which are already known to be deep objects related to the theory of quadratic forms and other arithmetic structures. I will give some examples providing new information about these groups.

Hyperbolic structures on compression bodies

We study Ford domains of geometrically finite structures on compression bodies obtained from adding a one-handle to a torus cross an interval. We find that in many cases, the topology of the compression body predicts geometric information on its geometry. Namely, the core of the attached one-handle is geodesic. These results have applications to
the geometry of single cusped tunnel number one manifolds. This is joint with Marc Lackenby.

A quasicategorical approach to Thom spectra and twisted Umkehr maps

Let R be an associative ring spectrum. I shall describe several constructions of the R-module Thom spectrum associated to a map f : X--> BGL1R. The space BGL1R classifies the twists of R-theory, and to a fibration of manifolds g : Y-->X I shall as sociated an Umkehr map g! from the (f g)-twisted R-theory of Y to the f-twisted R-theory of X. In the case of K-theory, this twisted Umkehr map appears in the study of D-brane charge. I shall review this story, and then discuss the analogous construction for TMF .

Rigidity of representations of 3-manifolds in SO(4,1)

We prove that infinitely many Dehn fillings in a 2-bridge knot are rigid in SO(4,1): namely its holonomy representation in SO(3,1) is the unique discrete and faithful representation in SO(4,1), up to conjugacy. This is based in previous work of M. Kapovich and K. Scannell about local rigidity, and it is joint with S. Francaviglia.

Geometric limits of knot complements

We prove that every one ended hyperbolic 3-manifold which embeds in the sphere and has finitely generated fundamental group is a geometric limit of knot complements.
This is a joint work with Jessica Purcell.

On completions of Verlinde rings

Let G be a simple, simply-connected, compact Lie group, and let m be a non-negative integer. The "level m Verlinde ring of G" (an object similar in spirit to an ordinary representation ring, related to certain representations of the loop group LG = Map(S^1, G)) may be realized as a quotient of the representation ring of G. In other words, V(G,m) = R(G)/I_m, where I_m is the "Verlinde ideal." By a result of Freed-Hopkins-Teleman and a twisted generalization of the Atiyah-Segal completion theorem, the (m+h(G))-twisted K-theory of LBG is isomorphic to the completion of V(G,m) at the augmentation ideal of R(G) (as an R(G)-module), where h(G) is the dual Coxeter number of G. Let V(G,m)^ denote this completion. After a brief review of relevant definitions, I will describe the additive structure of V(G,m)^. In particular, we will see that after further completing V(G,m)^ at a prime p, it splits as a finite sum of copies of the p-adic numbers.

Homological representation theory of the mapping class group

I seek to understand the algebraic structure of the mapping class group and the dynamical behavior of individual classes by studying the representation theory of the mapping class group on the homology of certain finite covers. I will explain how we can construct a faithful infinite-dimensional representation of the mapping class group and recover the Nielsen-Thurston classification of each class. I will also indicate connections with the representation theory of nilpotent Lie groups.

For more information, contact Enrique Torres-Giese.

If you want to check the list of seminars held in previous terms, click on the appropriate link below.

Fall 2008, Winter 2008, Fall 2007, Winter 2007, Fall 2006, Winter 2006, Fall 2005, Winter 2005, Fall 2004, Winter 2004, Fall 2003, Winter 2003, Fall 2002, the year 2000-1.

This page last updated on August 21, 2009.