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Friday, March 12, 2010 - 14:00 ,
Location: Skiles 269 ,
Alan Diaz ,
Georgia Tech ,
Organizer:

Khovanov homology is an invariant of oriented links, that is defined
as the cohomology of a chain complex built from the cube of
resolutions of a link diagram. Discovered in the late 90s, it is the
first of, and inspiration for, a series of "categorifications" of knot
invariants. In this first of two one-hour talks, I'll give some
background on categorification and the Jones polynomial,
defineKhovanov homology, work through some examples, and give a
portion of the proof of Reidemeister invariance.

Friday, March 5, 2010 - 14:00 ,
Location: Skiles 269 ,
Anh Tran ,
Georgia Tech ,
Organizer:

I will explain another approach to the conjecture and in particular,
study it for 2-bridge knots. I will give the proof of the conjecture
for a very large class of 2-bridge knots which includes twist knots and
many more (due to Le). Finally, I will mention a little bit about the
weak version of the conjecture as well as some relating problems.

Friday, February 19, 2010 - 14:00 ,
Location: Skiles 269 ,
Anh Tran ,
Georgia Tech ,
Organizer:

This is part 1 of a two part talk. The second part will continue next week.

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.

Friday, February 12, 2010 - 14:00 ,
Location: Skiles 269 ,
John Etnyre ,
Georgia Tech ,
Organizer:

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. I will also discuss what is known and not known about the relations between all of these invariants.

Friday, February 5, 2010 - 14:00 ,
Location: Skiles 269 ,
Meredith Casey ,
Georgia Tech ,
Organizer:

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.

Friday, January 29, 2010 - 14:00 ,
Location: Skiles 269 ,
Mohammad Ghomi ,
School of Mathematics, Georgia Tech ,
Organizer:

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.

Friday, January 15, 2010 - 14:00 ,
Location: Skiles 269 ,
Mohammad Ghomi ,
Georgia Tech ,
Organizer:

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.

Friday, December 4, 2009 - 14:00 ,
Location: Skiles 269 ,
Jim Krysiak ,
School of Mathematics, Georgia Tech ,
Organizer:

This talk will mostly be exposition on a result of M. Ghomi that
any C^2 knot in R^n can be C^1 perturbed into a knot of constant curvature
while preserving any smoothness properties.

Friday, November 6, 2009 - 15:00 ,
Location: Skiles 269 ,
Meredith Casey ,
Georgia Tech ,
Organizer:

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.

Friday, October 30, 2009 - 15:00 ,
Location: Skiles 269 ,
Shea Vela-Vick ,
Columbia University ,
Organizer: John Etnyre

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.