Georgia Institute of Technology College of Sciences

An n-dimensional topological quantum field theory is a functor from the category of closed, oriented (n-1)-manifolds and n-dimensional cobordisms to the category of vector spaces and linear maps. Three and four dimensional TQFTs can be difficult to describe, but provide interesting invariants of n-manifolds and are the subjects of ongoing research. This talk focuses on the simpler case n=2, where TQFTs turn out to be equivalent, as categories, to Frobenius algebras. I'll introduce the two structures -- one topological, one algebraic -- explicitly describe the

I plan to discuss a method for defining Heegaard Floer invariants for 3-manifolds. The construction is inspired by contact geometry and has several interesting immediate applications to the study of tight contact structures on noncompact 3-manifolds. In this talk, I'll focus on one basic examples and indicate how one defines a contact invariant which can be used to give an alternate proof of James Tripp's classification of tight, minimally twisting contact structures on the open solid torus. This is joint work with John B. Etnyre and Rumen Zarev.

While orientable surfaces have been classified, the structure of their homeomorphism groups is not well understood. I will give a short introduction to mapping class groups, including a description of a crucial representation for these groups, the Magnus representation. In addition I will talk about some current work in which I use Johnson-type homomorphisms to define an infinite filtration of the kernel of the Magnus representation.

I'll present a new, simple proof that the Torelli group is generated by (infinitely many) bounding pair maps. At the end, I'll explain an application of this approach to the hyperelliptic Torelli group. The key is to take advantage of the "complex of minimizing cycles."

TBA

Torsion of a curve in Euclidean 3-space is a quantity which together with the curvature completely determines the curve up to a rigid motion. In this talk we use the curve shortening flow to show that the number of zero torsion points (or vertices) v a closed space curve c and the number p of the pair of parallel tangent lines of c satisfy the following sharp inequality: v + 2p > 5.

A theorem of Chris Wendl allows you to completely characterize symplectic fillings of certain open book decompositions by factorizations of their monodromy into Dehn twists. Olga Plamenevskaya and I use this to generalize results of Eliashberg, McDuff and Lisca to classify the fillings of certain Lens spaces. I'll discuss this and a newer version of Wendl's theorem, joint with Wendl and Sam Lisi, this time for spinal open books, and discuss a few more applications.

We will discuss the structure of the symmetric (or hyperelliptic) Torelli group. More specifically, we will investigatethe group generated by Dehn twists about symmetric separating curvesdenoted by H(S). We will show that Aut(H(S)) is isomorphic to the symmetricmapping class group up to the hyperelliptic involution. We will do this bylooking at the natural action of H(S) on the symmetric separating curvecomplex and by giving an algebraic characterization of Dehn twists aboutsymmetric separating curves.