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 Events
University of Chicago
Mon 2 Apr 2012, 2:00pm SPECIAL
Topology and related seminars
WMAX 110 (PIMS)
The cohomology groups of the pure string motion group are uniformly representation stable
WMAX 110 (PIMS)
Mon 2 Apr 2012, 2:00pm-3:00pm

Abstract

In the past two years, Church, Farb and others have developed the concept of 'representation stability', an analogue of homological stability for a sequence of groups or spaces admitting group actions. In this talk, I will give an overview of this new theory, using the pure string motion group P\Sigma_n as a motivating example. The pure string motion group, which is closely related to the pure braid group, is not cohomologically stable in the classical sense -- for each k>0, the dimension of the degree k rational cohomology of P\Sigma_n tends to infinity as n grows. The groups H^k(P\Sigma_n, \Q) are, however, representation stable with respect to a natural action of the hyperoctahedral group W_n -- that is, in some precise sense, the description of the decomposition of these cohomology groups into irreducible W_n-representations stabilizes for n>>k. I will outline a proof of this result, verifying a conjecture by Church and Farb.
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University of Tokyo
Wed 4 Apr 2012, 4:00pm SPECIAL
Topology and related seminars
WMAX 216
Self-linking number of transverse knots in general open books
WMAX 216
Wed 4 Apr 2012, 4:00pm-5:00pm

Abstract

Every transverse knot in a contact 3-manifold is represented as a closed braid in an open book. In this talk, based on a new technique called an open book foliation, we give a formula of self-linking number in terms of braids and open books. Surprisingly, our self-linking number formula essentially uses Johnson's homomorphism. This is a joint work with Keiko Kawamuro (Univ. Iowa).
 
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Université de Louvain
Thu 12 Apr 2012, 2:00pm SPECIAL
Topology and related seminars
WMAX 216 (PIMS)
From the eversion of the sphere to spaces of knots
WMAX 216 (PIMS)
Thu 12 Apr 2012, 2:00pm-3:00pm

Abstract

A famous result by Steven Smale states that we can turn the sphere inside-out through immersions: this is called the eversion of the sphere. We will explain this result and the strategy of its proof which is a "cut-and-paste" strategy quite standard in algebraic topology. This approach allows us to understand globally the space of all immersions of a given manifold in another one, like the space of all immersion of the sphere in R^3 in the case of Smale's eversion. This theory has been enhanced by Goodwillie in the 1990's to understand spaces of embeddings. We will explain how this can be applied to understand spaces of knots, that is the spaces of all embeddings of a circle into a fixed euclidean space.
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