November 19, 2012
The String Topology Category of Special Unitary Groups
Dr. Shoham Shamir
Abstract
The space of free loops on a compact manifold M, denoted LM, has been of major interest in topology for the last decade.
One reason for this interest is the discovery, by Chas and Sullivan, of a rich structure on the homology of LM - which is sometimes called the string homology of M. In particular, the string homology of M is a commutative ring, and the space of paths between any two submanifolds of M gives rise to a module over this ring. All this information can be neatly packaged into a construct called: the string topology category of M. Blumberg, Cohen and Teleman, who invented the string topology category, posed the following question: for which submanifolds N of M can we reconstruct the entire string homology of M using only the paths from N to itself? I will give an answer to this question when M is a special unitary group. Along the road we will meet novel techniques from algebra, and also a little bit of algebraic geometry. I will try to keep the talk as elementary as possible.