Georgia Institute of Technology College of Sciences

I will survey the program of realizing various versions of Floer homology as a theory of geometric cycles. This involves the description of infinite dimensional manifolds mapping to the relevant configuration spaces. This approach, which goes back to Atiyah's address at the Herman Weyl symposium, is in some ways technically simpler than the traditional construction based on Floer's version of Morse theory. In addition, it opens up the possibility of defining more refined invariants such as bordism andK-theory.

This talk is about the dilatations of pseudo-Anosov mapping classes obtained by pushing a marked point around a filling curve. After reviewing this "point-pushing" construction, I will give both upper and lower bounds on the dilatation in terms of the self-intersection number of the filling curve. I'll also give bounds on the least dilatation of any pseudo-Anosov in the point-pushing subgroup and describe the asymptotic dependence on self-intersection number.

The first hour of this talk gives a gentle introduction to yet another version of Heegaard Floer homology; Sutured Floer homology. This is the generalization of Heegaard Floer homology, for 3-manifolds with decorations (sutures) on their boundary. Sutures come naturally for contact 3-manifolds. Later we will concentrate on invariants for contact 3--manifolds in Heegaard Floer homology. This can be defined both for closed 3--manifolds, in this case they live in Heegaard Floer homology and for 3--manifolds with boundary, when the invariant is in sutured Floer homology.

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).