Georgia Institute of Technology College of Sciences

In the 1980’s Furuta and Fintushel-Stern applied the theory of instantons and Chern-Simons invariants to develop a criterion for a collection of Seifert fibred homology spheres to be independent in the homology cobordism group of oriented homology 3-spheres.

Solutions to the Yang-Baxter equation are one source of representations of the braid group. Solutions are difficult to find in general, but one systematic method to find some of them is via the theory of quantum groups. In this talk, we will introduce the Yang-Baxter equation, braided bialgebras, and the quantum group U_q(sl_2). Then we will see how to obtain the Burau and Lawrence-Krammer representations of the braid group as summands of natural representations of U_q(sl_2).

In 2007, Honda, Kazez, and Matic defined an invariant of contact 3-manifolds with convex boundaries using sutured Heegaard Floer homology (SHF). Last year, Steven Sivek and I defined an analogous contact invariant using sutured Monopole Floer homology (SMF). In this talk, I will describe work with Sivek to prove that these two contact invariants are identified by an isomorphism relating the two sutured theories. This has several interesting consequences.

In this talk I will finish the proof that braid groups are linear using the Lawrence-Krammer representation.

In this talk I will discuss the Lawrence-Krammer representation of the Braid Group and begin to sketch the the proof that braid groups are linear.

Most work on surgeries in contact manifolds has focused upon determining the situations where tightness is preserved. We will discuss an approach to this problem from the reverse angle: when negative surgery on a fibred knot in an overtwisted contact manifold produces a tight one. We will examine the various phenomena that occur, and discuss an approach to characterising them via Heegaard Floer homology.

How is the homological torsion of a hyperbolic 3-manifold related to its geometry? In this talk, I will explain some techniques to address this general question. In particular, I will discuss in detail the case of arithmetic manifolds, where the situation is presumably easier to understand.

In this talk we will begin by discussing the problem of understanding the topology of the space of Riemannian metrics of positive scalar curvature on a smooth manifold. Recently much progress has occurred in this topic. We will then look at an application of the theory of operads to this problem in the case when the underlying manifold is an n-sphere. In the case when n>2, this space is a homotopy commutative, homotopy associative H-space. In particular, we show that it admits an action of the little n-disks operad.