Georgia Institute of Technology College of Sciences

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.

We will describe the Burau representation of the braid group and the related Gassner representation of the pure braid group. We will explore how the Burau representation is related to the Alexander polynomial.

I will discuss Eliashberg and Thurston's theorem that C^2 taut foliations can be approximated by tight contact structures. I will try to explain the importance of their work and why it is useful to weaken their smoothness assumption. This work is joint with Rachel Roberts.

Classically, there are two model category structures on coalgebras in the category of chain complexes over a field. In one, the weak equivalences are maps which induce an isomorphism on homology. In the other, the weak equivalences are maps which induce a weak equivalence of algebras under the cobar functor. We unify these two approaches, realizing them as the two extremes of a partially ordered set of model category structures on coalgebras over a cooperad satisfying mild conditions.

TBA

Under the operation of connected sum, the set of knots in the 3-sphere forms a monoid. Modulo an equivalence relation called concordance, this monoid becomes a group called the knot concordance group. We will consider various algebraic methods -- both classical and modern -- for better understanding the structure of this group.