Georgia Institute of Technology College of Sciences

Topological quantum field theory associates to a surface a sequence of vector spaces and to curves on the surface, sequence of operators on that spaces. It is expected that these operators are Toeplitz although there is no general proof. I will state it in some particular cases and give applications to the asymptotics of quantum invariants like quantum 6-j symbols or quantum invariants of Dehn fillings of the figure eight knot. This is work in progress with (independently) L. Charles and T. Paul.

Given a knot, a simple Lie algebra L and an irreducible representation V of L one can construct a one-variable polynomial with integer coefficients. When L is the simplest simple Lie algebra (sl_2) this gives a sequence of polynomials, whose sequence of degrees is a quadratic quasi-polynomial. We will discuss a conjecture for the degree of the colored Jones polynomial for an arbitrary simple Lie algebra, and we will give evidence for sl_3. This is joint work with Thao Vuong.

In this talk, I'll focus on Seifert fibered spaces whose fiber structure is realized by the Reeb orbits of an appropriate contact form. I'll describe a rigorous combinatorial formulation of Legendrian contact homology for Legendrian knots in these manifolds. This work is joint with J. Sabloff.

In this talk we will give an introduction of Heegaard-Floer theory through examples. By exploring several explicit examples we hope to show that various aspects of the definitions that seem complicated, really aren't too bad and it really is possible to work with these fairly abstract things. While this is technically a continuation of last weeks talk, we will review enough material so that this talk should be self contained.

This will be an introduction to the basic aspects of Heegaard-Floer homology and knot Heegaard-Floer homology. After this talk (talks) we will be organizing a working group to go through various computations and results in knot Heegaard-Floer theory and invariants of Legendrian knots.

There are many conjectured connections between Heegaard Floer homology and the various homologies appearing in low dimensional topology and symplectic geometry. One of these conjectures states, roughly, that if \phi is a diffeomorphism of a closed Riemann surface, a certain portion of the Heegaard Floer homology of the mapping torus of \phi should be equal to the Symplectic Floer homology of \phi. I will discuss how this can be confirmed when \phi is periodic (i.e., when some iterate of \phi is the identity map).

A smooth knot in a contact 3-manifold is called Legendrian if it is always tangent to the contact planes. In this talk, I will discuss Legendrian knots in R^3 and the solid torus where knots can be conveniently viewed using their `front projections'. In particular, I will describe how certain decompositions of front projections known as `normal rulings' (introduced by Fuchs and Chekanov-Pushkar) can be used to give combinatorial descriptions for parts of the HOMFLY-PT and Kauffman polynomials.

 Deciding how to unknot a knotted piece of string (with its ends glued together) is not only a difficult problem in the real world, it is also a difficult and long studied problem in mathematics. (There are several notions of what one might mean by "unknotting" and I will leave the exact meaning a bit vague in this abstract.) In the past mathematicians have used a vast array of techniques --- from geometry to algebra, and even PDEs --- to study this question. I will discuss this question and (partially) recast it in terms of 4 dimensional topology.

We adapt techniques derived from the study of quasi-flats in Right Angled Artin Groups, and apply them to 2-dimensional Graph Braid Groups to show that the groups B_2(K_n) are quasi-isometrically distinct for all n.

We will prove that the mapping class group is finitely presented, using its action on the arc complex. We will also use the curve complex to show that the abstract commensurator of the mapping class group is the extended mapping class group. If time allows, we will introduce the complex of minimizing cycles for a surface, and use it to compute the cohomological dimension of the Torelli subgroup of the mapping class group. This is a followup to the previous talk, but will be logically independent.