Type: Seminar

Distribution: World

Expiry: 22 Aug 2011

Auth: athomas(.pmstaff;2039.2002)@p615.pc.maths.usyd.edu.au

There will be an Algebra Seminar by Jennifer Wilson on Monday 22 August. Please note the unusual day and venue. --------------------------------------------------------------------------- Speaker: Jennifer Wilson (University of Chicago) Date: Monday 22 August Time: 12.05-12.55pm Venue: Carslaw 375 Title: Representation stability for the cohomology of the groups of pure string motions Abstract: The string motion group Sigma_n, the group of motions of n disjoint, unlinked, unknotted circles in 3-space, is a generalization of the braid group. It can be identified with the symmetric automorphism group of the free group. The pure string motion group PSigma_n, the analogue of the pure braid group, admits an action by the hyperoctahedral group W_n. The rational cohomology of PSigma_n is not stable in the classical sense -- the dimension of the k^th cohomology group tends to infinity as n grows -- however, Church and Farb have recently developed a notion of stability for a sequence of vector spaces with a group action, which they call representation stability. Inspired by their recent work on the cohomology of the pure braid group, they conjectured that for each k>0, the k^th rational cohomology of PSigma_n is uniformly representation stable with respect to the induced action of W_n, that is, the description of the decomposition of the cohomology group into irreducible W_n-representations stabilizes for n >> k. In this talk, I will give an overview of the theory of representation stability, and outline a proof verifying this conjecture. This result has implications for the cohomology of the string motion group, and the permutation-braid group. ----------------------------------------------------------------------------