# Operator theory seminar

Title: Bost-Connes systems, Hecke algebras, and induction

Speaker: Marcelo Laca, University of Victoria

Date and time:
21 Jan 2015,
3:30pm -
4:30pm

Location: Clearihue C109

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A Bost-Connes type system is a C*-dynamical system, constructed from an algebraic number field, exhibiting a phase transition at low temperature with spontaneous symmetry-breaking that has a number theoretical interpretation in terms of class field theory. We show that this construction is functorial with respect to inclusion of number fields provided that, at the level of C*-dynamical systems, the arrows are taken to be equivariant C*-correspondences with normalized dynamics. As a second application of correspondences we discuss the relation of Hecke algebras of number fields to Bost-Connes systems.

This is joint work with Sergey Neshveyev and Mak Trifkovic.

Title: Index theory for manifolds with Baas-Sullivan singularities

Speaker: Robin Deeley

Date and time:
26 Nov 2014,
3:30pm -
4:30pm

Location: Clearihue C109

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We discuss index theory for manifolds with Baas-Sullivan singularities using the framework of geometric K-homology. The prototypical example of a manifold with Baas-Sullivan singularities is a z/k-manifold. Informally, a z/k-manifold is a pair of manifolds, one with boundary and one without boundary such that the boundary of the former decomposes into k-copies of the latter. The Freed-Melrose index theorem gives the natural analog of the Atiyah-Singer index theorem in the context of z/k-manifolds; the associated index is a z/k-valued index.

We will begin by introducing geometric (i.e., Baum-Douglas) K-homology and manifolds with Baas-Sullivan singularities. We then discuss "Baum-Douglas type" cycles which contain (smooth, compact, spin^c) Baas-Sullivan manifolds rather than ordinary (smooth, compact, spin^c) manifolds. This construction leads to a generalized homology theory, which serves as a framework for the study of index theory for Baas-Sullivan manifolds. In the case of z/k-manifolds, the generalized homology theory we obtain is K-homology with coefficients in z/k; this explains the z/k-valued index mentioned above.

Title: The Representation Theory of the Symmetric Groups

Speaker: Brittany Halverson-Duncan, University of Victoria

Date and time:
19 Nov 2014,
3:30pm -
4:30pm

Location: Clearihue C109

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In this talk I will introduce Young Tableaux, a method of describing the irreducible representations of the symmetric groups S_n. An irreducible representation of S_n restricts to a representation of S_{n-1}, which is not irreducible in general, but I should how to write such restricted representations as linear combinations of irreducibles. This computation gives structural information about the AF C*-algebra obtained by taking the group C*-algebra of the infinite permutation group S_{\infty}.