Friday, September 3, 2010 - 13:00 , Location: Skiles 114 , Dan Margalit , Georgia Tech , Organizer: Dan Margalit
The mapping class group is the group of symmetries of a surface (modulo homotopy). One way to study the mapping class group of a surface S is to understand its action on the set of simple closed curves in S (up to homotopy). The set of homotopy classes of simple closed curves can be organized into a simplicial complex called the complex of curves. This complex has some amazing features, and we will use it to prove a variety of theorems about the mapping class group. We will also state some open questions. This talk will be accessible to second year graduate students.
Friday, April 23, 2010 - 14:00 , Location: Skiles 269 , Thao Vuong , Georgia Tech , Organizer:
We will give definitions and then review a result by Floyd and Oertel that in a Haken 3-manifold M, there are a finite number of branched surfaces whose fibered neighborhoods contain all the incompressible, boundary-incompressible surfaces in M, up to isotopy. A corollary of this is that the set of boundary slopes of a knot K in S^3 is finite.
Friday, April 16, 2010 - 14:00 , Location: Skiles 269 , Igor Belegradek , Georgia Tech , Organizer:
I will discuss moduli spaces of Riemannian metrics with various curvature conditions, and then focus on the case of nonnegative sectional curvature.
Friday, April 9, 2010 - 14:00 , Location: Skiles 269 , Igor Belegradek , School of Mathematics, Georgia Tech , Organizer:
I will review results on the structure of open nonnegatively curved manifolds due to Cheeger-Gromoll, Perelman, and Wilking.
Friday, March 19, 2010 - 14:00 , Location: Skiles 269 , Alan Diaz , School of Mathematics, Georgia Tech , Organizer:
Last week we motivated and defined Khovanov homology, an invariant of oriented links whose graded Euler characteristic is the Jones polynomial. We'll discuss the proof of Reidemeister invariance, then survey some important applications and extensions, including Lee theory and Rasmussen's s-invariant, the connection to knot Floer homology, and how the latter was used by Hedden and Watson to show unknot detection for a large class of knots.
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.