Georgia Institute of Technology College of Sciences

A contact structure on a 3-manifold is called overtwisted ifthere is a certain kind of embedded disk called an overtwisted disk; it istight if no such disk exists. A Legendrian knot in an overtwisted contact3-manifold is loose if its complement is overtwisted and non-loose if itscomplement is tight. We define and compare two geometric invariants, depthand tension, that measure how far from loose is a non-loose knot. This isjoint work with Sinem Onaran.

We will start by defining the Jones polynomial of a knot and talking about some of its classical applications to knot theory. We will then define a fancier version ("categorification") of the Jones polynomial, called Khovanov homology and mention some of its applications. We will conclude by talking about a further refinement, a Khovanov homotopy type, sketch some of the ideas behind its construction, and mention some applications.

Houghton's groups are a family of subgroups of infinite permutation groups known for their cohomological properties. Here, I describe some aspects of their geometry and metric properties including families of self-quasi-isomtries. This is joint work with Jose Burillo, Armando Martino and Claas Roever.

We discuss a simple construction a finite dimensional algebra("bipartite algebra") to a bipartite oriented graph, and explain how thestudy of the representation theory of these algebras produces acategorification of the cut and flow lattices of graphs. I'll also mentionwhy we suspect that bipartite algebras should arise naturally in severalother contexts. This is joint work with Anthony Licata.

We'll prove the simplest case of Hirzebruch's signature theorem, which relates the first Pontryagin number of a smooth 4-manifold to the signature of its intersection form. If time permits, we'll discuss the more general case of 4k-manifolds. The result is relevant to Prof. Margalit's ongoing course on characteristic classes of surface bundles.

For a fixed integer n, consider the nerve L_n of the topological poset of orthogonal decompositions of complex n-space into proper orthogonal subspaces. The space L_n has an action by the unitary group U(n), and we study the fixed points for subgroups of U(n). Given a prime p, we determine the relatively small class of p-toral subgroups of U(n) which have potentially non-empty fixed points.

We introduce a theory of virtual Legendrian knots. A virtual Legendrian knot is a cooriented wavefront on an oriented surface up to Legendrian isotopy of its lift to the unit cotangent bundle and stabilization and destablization of the surface away from the wavefront. We show that the groups of Vassiliev invariants of virtual Legendrian knots and of virtual framed knots are isomorphic. In particular, Vassiliev invariants cannot be used to distinguish virtual Legendrian knots that are isotopic as virtual framed knots and have equal virtual Maslov numbers.

We show that after stabilizations of opposite parity and braid isotopy, any twobraids in the same topological link type cobound embedded annuli. We use this to prove thegeneralized Jones conjecture relating the braid index and algebraic length of closed braidswithin a link type, following a reformulation of the problem by Kawamuro. This is joint workwith Doug Lafountain.