Georgia Institute of Technology College of Sciences

Given any smooth manifold, there is a canonical symplectic structure on its cotangent bundle. A long standing idea of Arnol'd suggests that the symplectic topology of the cotangent bundle should contain a great deal of information about the smooth topology of its base. As a contrast, I show that when X is an open 4-manifold, this symplectic structure on T^*X does not depend on the choice of smooth structure on X. I will also discuss the particular cases of smooth structures on R^4 and once-punctured compact 4-manifolds.

In this series of talks I will begin by discussing the idea of studying smooth manifolds and their submanifolds using the symplectic (and contact) geometry of their cotangent bundles. I will then discuss Legendrian contact homology, a powerful invariant of Legendrian submanifolds of contact manifolds.

In this talk we are going to present a theorem that can be seen as related to S. Smale's theorem on the topology of the space of Legendrian loops. The framework will be slightly different and the space of Legendrian curves will be replaced by a smaller space $C_{\beta}$, that appears to be convenient in some variational problems in contact form geometry. We will also talk about the applications and the possible extensions of this result. This is a joint work with V. Martino.

We look at a paper of McMullen "Braid Groups and Hodge Theory" exploring representations of braid groups and their connections to arithemetic lattices.

We classify the Legendrian torus knots in S1XS2 with tight contact structure up to isotopy. This is a joint work with Feifei Chen and Fan Ding.

We discuss two concepts of low-dimensional topology in higher dimensions: near-symplectic manifolds and overtwisted contact structures. We present a generalization of near-symplectic 4-manifolds to dimension 6. By near-symplectic, we understand a closed 2-form that is symplectic outside a small submanifold where it degenerates. This approach uses some singular mappings called generalized broken Lefschetz fibrations. An application of this setting appears in contact topology.

The goal of this talk is to study geography and classification problem for Stein fillings of contact structures supported by planar open books. In the first part we will prove that for contact structures supported by planar open books Stein fillings have a finite geography. In the second part we will outline an approach to classify Stein fillings of manifolds supported by planar open books.

We study some finite quotients of the A_n Milnor fibre which coincide with the Stein surfaces that appear in Fintushel and Stern's rational blowdown construction. We show that these Stein surfaces have no exact Lagrangian submanifolds by using the already available and deep understanding of the Fukaya category of the A_n Milnor fibre coming from homological mirror symmetry. On the contrary, we find Floer theoretically essential monotone Lagrangian tori, finitely covered by the monotone tori that we studied in the A_n Milnor fibre.