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Abstract: I will start by briefly recalling the construction and main properties of the ideal class group of a "number field". This is a finite group, whose order bears deep arithmetic information about the field. It has been known since the work of Dirichlet at the end of the Nineteenth Century that in some cases there are relationships among the orders of class groups in Galois extensions. Dirichlet's results have been put in a broader and conceptually clearer contest by Brauer's work on special values of Artin L functions at s=0 and I will discuss the interplay between analytic and representation-theoretic tools leading to these results. Finally, I will report on a joint work with L. Caputo focusing on the case of extensions whose Galois group is dihedral.

Abstract: The operators x, d/dx, and x+d/dx, figure in the basic set-up of quantum mechanics. They also can be used to prove a very powerful KK-theoretic version of the Bott Periodicity theorem. The operator x+d/dx, of most interest here, having discrete spectrum and being Fredholm, can also be used to build 2-summable spectral triples over certain crossed-product C*-algebras; we discuss work on their `Noncommutative Geometric' properties, work which is joint with Nathaniel Butler and Tyler Schultz.

Abstract. In a joint work with N. Brownlowe, J. Ramagge and M.F. Whittaker, we introduce the notion of a self-similar action of a groupoid $G$ on a finite higher-rank graph. To such an action, we associate two $C^*$-algebras: the Toeplitz algebra and the Cuntz--Pimsner algebra. We study the KMS states of these algebras in natural dynamics. We then illustrate our results by introducing the notion of a coloured-graph automaton, which we use to construct examples of self-similar actions. We compute the unique KMS states of the Cuntz--Pimsner algebra for some concrete examples.

Abstract: The irrational rotation algebra is known to be Poincaré self-dual in a KK-theoretic sense. The required K-homology fundamental class was constructed by Connes out of the Dolbeault operator on the 2-torus, but so far, there has not been an explicit description of the dual element. In this (and potentially next week's) talk, I will geometrically construct a finitely generated projective module representing said K-theory class, by using a pair of transverse Kronecker flows on the 2-torus. The results I will be presenting are a more refined version of two talks I gave in June 2018 at UVic.

Abstract: Given a C*-algebra, A, and a C*-subalgebra, A', Karoubi introduced the notion of a relative K-group, K(A';A). I won't actually give a definition (Mitch will sometime later this year), but I'll will describe some basic properties. Of particular interest will be excision results, which, in some sense, say that this group depends only on the set theoretic difference difference A-A'. That doesn't make much sense as written, as the set A-A' isn't a C*-algebra. But the talk will focus on some examples and techniques where we can make sense of it.

Abstract: In the previous talk we introduced a class of hyperbolic dynamical systems called Smale spaces . In this talk, following the philosophy of noncommutative topology, we will encode certain dynamical aspects of Smale spaces into topological groupoids and then construct their C*-algebras. A particularly nice class of such C*-algebras are the Ruelle algebras introduced by Ian Putnam. They can be thought as higher-dimensional analogues of Cuntz-Krieger algebras. Ruelle algebras have the following features:

• They are separable, simple, nuclear, purely infinite, stable and in the UCT class. Hence, they can be classified by their K-theory.
• They satisfy a noncommutative version of Poincare duality; i.e. there are natural isomorphisms between the K-theory and the K-homology of the Ruelle algebras.

We will discuss the above features with a focus on the second one.

Abstract: This will be part one of a series of two talks. In the first part we will discuss about a class of hyperbolic dynamical systems called Smale spaces. This part will not have a C*-algebraic flavour. However, we will give examples and try to convince the audience that they are interesting.

Abstract: I will explain how to compute the partition functions of C*-dynamical systems arising from actions of congruence monoids on rings of algebraic integers in number fields. Then I will show how to use these partition functions to recover information from the C*-dynamical system about the underlying number field and a certain class field that is naturally associated with the initial data. This talk is based on joint work with Marcelo Laca (University of Victoria) and Takuya Takeishi (Kyoto Institute of Technology), and it builds on Marcelo's talk from last week.

Abstract: The C*-dynamical systems from number theory introduced in joint work with Cuntz and Deninger have been generalized by Bruce by considering affine semigroups whose multiplicative part is a congruence monoid. In these systems, the phase transition of equlibrium states at low temperature does not respect the energy levels, and this calls for a closer look at what is really meant by `partition function' in the context of general C*-dynamical systems. The talk is about this closer look and what it reveals.

This research is part of current joint work with Chris Bruce (UVic) and Takuya Takeishi (Kyoto Inst. Tech.).

In a forthcoming talk, Chris will discuss the phase transition of systems from congruence monoids, and how the associated partition functions give rise to C*-dynamical invariants that can be used to retrieve information about the number field.