Georgia Institute of Technology College of Sciences

After, briefly, recalling the definition of contact homology, a powerful but somewhat intractable and still largely unexplored invariant of Legendrian knots in contact structures, I will discuss various ways of constructing more tractable and computable invariants from it. In particular I will discuss linearizations, products, massy products, A_\infty structures and terms in a spectral sequence. I will also show examples that demonstrate some of these invariants are quite powerful.

Exact Topic TBA. Talk will be a general survery of branched covers, possibly including covers from the algebraic geometry perspective. In addition we will look at branched coveres in higher dimensions, in the contact world, and my current research interests. This talk will be a general survery, so very little background is assumed.

We prove that convex hypersurfaces M in R^n which are level sets of functions f: R^n --> R are C^1-regular if f has a nonzero partial derivative of some order at each point of M. Furthermore, applying this result, we show that if f is algebraic and M is homeomorphic to R^(n-1), then M is an entire graph, i.e., there exists a line L in R^n such that M intersects every line parallel L at precisely one point. Finally we will give a number of examples to show that these results are sharp.

We study the topology of the space bd K^n of complete convex hypersurfaces of R^n which are homeomorphic to R^{n-1}. In particular, using Minkowski sums, we construct a deformation retraction of bd K^n onto the Grassmannian space of hyperplanes. So every hypersurface in bd K^n may be flattened in a canonical way. Further, the total curvature of each hypersurface evolves continuously and monotonically under this deformation. We also show that, modulo proper rotations, the subspaces of bd K^n consisting of smooth, strictly convex, or positively curved

The Dehn function is a group invariant which connects geometric and combinatorial group theory; it measures both the difficulty of the word problem and the area necessary to fill a closed curve in an associated space with a disc. The behavior of the Dehn function for high-rank lattices in high-rank symmetric spaces has long been an openquestion; one particularly interesting case is SL(n,Z). Thurston conjectured that SL(n,Z) has a quadratic Dehn function when n>=4.

I will discuss a conjecture that relates the degree of the Jones polynomial of a knot and its parallels with the slopes of incompressible surfaces in the knot complement. I will present examples, as well as computational challenges.

In 3-dimensional contact topology one of the main problem is classifying Legendrian (transverse) knots in certain knot type up to Legendrian ( transverse) isotopy. In particular we want to decide if two (one in case of transverse knots) classical invariants of this knots are complete set of invariants. If it is, then we call this knot type Legendrian (transversely) simple knot type otherwise it is called Legendrian (transversely) non-simple.

The goal of this talk is to describe simple constructions by which we can construct any compact, orientable 3-manifold. It is well-known that every orientable 3-manifold has a Heegaard splitting. We will first define Heegaard splittings, see some examples, and go through a very geometric proof of this therem. We will then focus on the Dehn-Lickorish Theorem, which states that any orientation-preserving homeomorphism of an oriented 2-manifold without boundary can by presented as the composition of Dehn twists and homeomorphisms isotopic to the identity.

In this talk I will discuss a generalizations and/oo applications of bordered Floer homology. After reviewing the basic definitions and constructions, I will focus either on an application to sutured Floer homology developed by Rumen Zarev, or on applications of the theory to the knot Floer homology. (While it would be good to have attended the other two talks this week, this talk shoudl be independent of them.) This is a 2 hour talk.