Georgia Institute of Technology College of Sciences

I will introduce the AJ conjecture (by Garoufalidis) which relates the A-polynomial and the colored Jones polynomial of a knot in the 3-sphere. Then I will verify it for the trefoil and the figure 8 knots (due to Garoufalidis) and torus knots (due to Hikami) by explicit calculations.

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 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 hypersurfaces are each contractible, which settles a question of H. Rosenberg.

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. We will prove this theorm, and then see some applications and examples. With both of these resutls together, we will have shown that using only handlebodies and Dehn twists one can construct any compact, oriented 3-manifold.