Georgia Institute of Technology College of Sciences

When do commuting homeomorphisms of S^2 have a common fixed point? Christian Bonatti gave the first sufficient condition: Commuting diffeomorphisms sufficiently close to the identity in Diff^+(S^2) always admit a common fixed point. In this talk we present a result of Michael Handel that extends Bonatti's condition to a much larger class of commuting homeomorphisms. If time permits, we survey results for higher genus surfaces due to Michael Handel and Morris Hirsch, and connections to certain compact foliated 4-manifolds.

The braid group has many applications throughout the world of math due to its simple yet rich structure. In this talk we will focus on the Burau representation of the braid group, which has important implications in knot theory. Most notably, the open problem of faithfulness of the Burau representation of the braid group on 4 strands is equivalent to whether or not the Jones polynomial can detect the unknot. The Burau representation has a topological interpretation that uses the mapping class definition of the braid group. We'll introduce the braid group first and then discuss the Burau representation. We will go through examples for small n and discuss the proof of nonfaithfulness for n > 4. Time permitting, we may introduce the Gassner representation.

Abstract: How big is a group?  One possible notion of the size of the group is the cohomological dimension, which is the largest n for which a group G can have non—trivial cohomology in degree n, possibly with twisted coefficients.  Following the work of Bestvina, Bux and Margalit, we compute the cohomological dimension of the terms Johnson filtration of a closed surface.  No background is required for this talk.