Schedule of Talks

Abstracts

Simplicial Spaces of Homomorphisms

In this talk we will discuss the construction of a family of spaces that filter tha classigying space of a group. This filtration is based on the descending central series of free groups. Cohomology calculations are provided for compact Lie groups and homotopy properties of these spaces are considered for finite groups, including their description as homotopy colimits. The first layer of this filtration (built out of commuting elements) is described in detail for transitively commutative groups. Stable homotopy decompositionsof these spaces are also given.
This is joint work with Alex Adem and Fred Cohen.

On Boundaries of CAT(0) Groups

The CAT(0) condition is a geometric notion of nonpositive curvature similar to Gromov's definition of hyperbolicity.  If a group G acts properly discontinuously and cocompactly on a CAT(0) space Xthen we call G a CAT(0) group.  In this setup the visual boundary of X is called a boundary of G.  In contrast to the hyperbolic setting, the boundary of a CAT(0) group need not be well-defined up to homeomorphism.  In this talk we present some results concerning boundaries of CAT(0) groups as well as motivating open questions.

Dehn surgeries that reduce the Thurston norm of a fibered manifold

Suppose K is a knot on the fiber of a surface bundle over the circle. If we do surgery on K with slope specified by the fiber, then the Thurston norm of the homology class of the fiber will decrease in the new manifold. We will show that the converse is also true. Namely, if a Dehn surgery on a winding number 0 knot in a fibered manifold reduces the Thurston norm of the homology class of the fiber, then the knot must lie on the fiber and
the slope is the natural one.

Supersymmetric Euclidean field theories and generalized cohomology

This is a report of ongoing joint work with Peter Teichner (Berkeley). Elaborating Segal's axiomatic approach to conformal field theories, we define supersymmetric Euclidean field theories over a manifold X. It turns out that the set of concordance classes of such field theories over X of dimension d is in bijective correspondence to the cohomology of X (with complex coefficients) for d=0 and to the K-theory of X for d=1. We speculate that for d=2 we obtain the "Topological Modular Form theory" of Hopkins-Miller. Evidence is provided by our result that the partition function of a supersymmetric Euclidean field theory of dimension 2 is a weakly holomorphic integral modular form.

Curvatures of surfaces in hyperbolic 3-manifolds

Given a surface in a hyperbolic 3-manifold, how much can it be flattened via isotopy? I will discuss universal bounds on the principal curvatures of incompressible surfaces and strongly irreducible Heegaard surfaces in hyperbolic 3-manifolds. I will also discuss bounds on the curvatures of codimension-1 laminations in closed hyperbolic 3-manifolds.

Growth rate of intersection numbers for free group automorphisms

P. Scott has defined a notion of intersection number between splittings of a group which generalizes the familiar notion of intersection number between curves on a surface. We investigate the aymptotic behavior of this intersection number between trees in Culler Vogtmann Outer Space when one of the trees is iterated by a fully irreducible free group automorphism. This is joint work with Jason Behrstock and Mladen Bestvina.

Cocycle categories

The set of maps between objects in a homotopy category can be identified with path components of a category of cocycles, in great generality. The
applications, to date, have predominantly consisted of homotopy classification results in non-abelian cohomology theory. Cocycle category methods lead to an expanded (and interesting) version of the Verdier hypercovering theorem. Cocycle categories might also be used to give a new description of etale homotopy theory, which would involve neither hypercovers nor pro-objects.

Volume and homology in the space of cycles

No abstract available...

A strong Tits alternative and arithmetic heights on character varieties

We prove an effective uniform version of the Tits alternative on an arbitrary field and derive several new results on the structure of linear
groups, mainly about their growth and number of relations. The main statement can be reformulated in terms of first order logic as the
equality between seemingly unrelated algebraic varieties. This allows to reduce mod p and get new applications such as bounds on the girth of
Cayley graphs of subgroups of $GL(n,F_q)$.

Proofs rely on the notion of "arithmetic spectral radius" of a finite family of matrices and key ingredients are some results from Diophantine
Geometry such as theorems of Bilu and Zhang on the set of points of small height on algebraic varieties.

We will also mention the Lehmer conjecture, in connection with the solvable case, which turns out to be surprisingly harder.

Homology of the automorphism group of a free group

Bounding the stable genus of Heegaard splittings from below

It is a classical result that given two Heegaard splittings of a 3-manifold, there is a third Heegaard splitting, called a common stabilization, that results from adding handles to each of the original two Heegaard surfaces. It is an open problem to determine how many handles are needed, in general, to form this common stabilization. I will describe a way to construct, for each integer k, a 3-manifold with Heegaard splittings of genus 2k and 2k-1 such that the smallest common stabilization has genus at 3k-1.

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.

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 December 27, 2008.