Georgia Institute of Technology College of Sciences

We will look at various instances of how working with skeins (diagrams) provides a way to describe the existence of various topological quantum invariants that were originally described using representation theory. This provides a very simple description of these invariants. Along the way we will look at how to describe the algebraic data (ribbon categories) topologically and also how one could observe instances of certain dualities that exist between certain categories using these diagrams. 

Giroux and Pardon conjectured that a Lagrangian L in a Weinstein manifold W is regular (that is, compatible with the Weinstein structure in a natural sense) if there is a Lefschetz fibration p: W \to \C such that p(L) is a ray. In this talk, I will discuss forthcoming joint work with A. Roy and L. Wang, which establishes this conjecture. As an application of the proof, we show how all fillings of the rainbow closures of a positive braid can be described by manipulations of arcs in the base of an appropriate Lefschetz fibration.

A knot in a contact manifold is Legendrian if it is everywhere tangent to the contact planes. The classification problem in Legendrian knot theory has always generated significant interest. The problem gets a lot more complicated when we consider links. In this talk, I'll survey some of the results in this area and then discuss the classification problem for cable links of uniformly thick knot type.  If time permits, I'll also mention the classification of links in the overtwisted setting. Part of this is joint work with John Etnyre, Hyunki Min, and Tom Rodewald. 

Dehn surgery is a fundamental construction in topology where one removes a neighborhood of a knot from the three-sphere and reglues to obtain a new three-manifold. The Cosmetic Surgery Conjecture predicts two different surgeries on the same non-trivial knot always gives different three-manifolds. We discuss how gauge theory, in particular, the Chern-Simons functional, can help approach this problem. This technique allows us to solve the conjecture in essentially all but one case. This is joint work with Ali Daemi and Mike Miller Eismeier.