Geometry seminar:    Fall 2004

Tuesday 9/21

Brian Munson (Stanford)

Layers of the embedding tower and homotopy invariants

Thursday 9/30 (Topology/Geometry seminar)

Sa'ar Hersonsky (Ben-Gurion University)

Approximation by maximal cusps in boundaries of deformation spaces of Kleinian groups

Tuesday 10/19

Wilbur Whitten

Knot group epimorphisms

Thursday 10/21 (Geometry/Topology seminar)

Joanna Kania-Bartoszynska (Boise State and NSF)

Turaev-Viro invariants of 3-manifolds and normal surfaces.

Abstract: The formula for the Turaev-Viro invariant of a 3-manifold depends
on a complex parameter t. When t is not a root of unity, the formula
becomes an infinite sum. We analyze convergence of this sum
when t does not lie on the unit circle, in the presence of an efficient
triangulation of the three-manifold. The terms of the sum can be indexed by
surfaces lying in the three-manifold. The contribution of a surface is
largest when the surface is normal and when its genus is the lowest.
This is joint work with Charles Frohman, University of Iowa.

Tuesday 10/26

Chris Herald (University of Nevada)

SU(n) flat connections and knots

Abstract: If X is a closed 3-manifold with the homology of S^3, then

the Casson invariant is a count (with sign) of the representations of

\pi_1 X into SU(2) modulo conjugation, or equivalently a count of

the flat SU(2) connections on X modulo gauge equivalence.
For knot complements Y=S^3 -K, there is a related family invariants,
indexed by \theta \in [0,2\pi] which count representations of \pi_1 Y
sending the knot meridians to matrices with trace 2 cos(\theta). These
invariants have been related equivariant knot signature.
In this talk I will discuss these results, and then outline some
work in progress to generalize the knot invariants to SU(n) by analyzing
the flat moduli space of SU(n) connections on the knot complement.

Thursday 10/28 (Geometry/Topology seminar)

Hans Boden (McMaster University)

Calculations of the Casson-Curtis SL(2,C) invariant

Abstract: This talk, which presents joint work with Cynthia Curtis, will

focus on the SL(2,C) analogue of Casson's invariant. This 3-manifold

invariant was defined earlier by Curtis, and we give a simple closed formula

for the invariant for Seifert-fibered homology 3-spheres. One way to establish

this formula is to use the correspondence between the SL(2,C) character

varieties and the moduli spaces of parabolic Higgs bundles of rank two.

These results can then be utilized to provide computations for families of

3-manifolds arising as Dehn surgeries on knots with Seifert slopes. For

example, we describe computations of the Casson-Curtis for surgeries

on twist knots. These computations employ Curtis's surgery formula

together with information about the Culler-Shalen seminorms.

Tuesday 11/23

Ismar Volic (UVa)

Introduction to embedding calculus

Thursday 12/2 (Geometry/Topology seminar)

Yongwu Rong (George Washington University)

A Khovanov type categorification for the chromatic polynomial

Thursday 12/9 (Geometry/Topology seminar)

Frank Quinn (Virginia Tech)

Are topological 4-manifolds high-dimensional?

Abstract: The development of 4-dimensional topology due largely to Freedman

showed that many 4-manifolds share the structural patterns of higher dimensional

manifolds. These methods failed to work for manifolds with "large" fundamental

groups, and Freedman conjectured that there is some residual "low dimensional"

behavior in these manifolds. However there is now reason to suspect that they are in

fact high dimensional. I will outline a still-provisional proof of the topological surgery

conjecture. The approach is quite indirect and "epistemically impure" (goes far outside

the setting of the data and the desired answer). Is there a deep reason an indirect proof is

needed, or are we just missing a trick? Insight into this would throw light on remaining questions.

Thursday 12/9 (Colloquium)

Dan Freed (University of Texas – Austin)

Representations of loop groups and topology

Abstract: A loop group is the infinite dimensional space of maps from a
circle into a compact Lie group, with pointwise multiplication.
These groups have a distinguished class of representations, termed
`positive energy', which arise in many contexts. A recent joint
result with Michael Hopkins and Constantin Teleman locates the
group of isomorphism classes of these representations (sometimes
called the Verlinde algebra) in the topology of the Lie group.
The basic construction has ramifications in finite dimensions as
well, where it relates to Kirillov's ideas on coadjoint orbits.
I will begin with that more elementary case and some specific
examples.

Friday 12/10

Jack Morava (Johns Hopkins University)

Topological gravity and 4D h-cobordisms