Algebra Seminar: Wilson -- Representation stability for the cohomology of the groups of pure string motions

There will be an Algebra Seminar by Jennifer Wilson on Monday 22 August.  Please note
the unusual day and venue.  

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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.  

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