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Abstract: In the theory of harmonic analysis, one considers unitary representations of groups bearing some geometric significance. These may give rise to summable representations of an appropriate convolution algebra, and the trace can be computed either spectrally (in terms of irreducible factors) or geometrically, resulting in a "trace formula" that relates these two perspectives. Hallmark examples of this method include the Poisson summation formula and the Selberg trace formula. The work of Connes (1996, 1999) and Meyer (2004, 2007) considers geometric representations of groups arising from number theoretic data, and the trace formulas in their examples recover the explicit formulas of Weil relating prime numbers (or ideals) to the zeros of L-functions. In this talk, I will provide some background on trace formulas for groups, including one of the examples of Meyer, and introduce the group and representation that I am considering in my research.

Abstract: In noncommutative geometry, a spectral triple is a set of data which encodes a geometric phenomenon in an analytic way. The authors Butler, Emerson, and Schulz introduced a method of producing such spectral triples using the annihilation and creation operators of quantum mechanics. Their construction produced the Heisenberg cycle, a 2-dimensional spectral triple over the C*-algebra crossed-product of uniformly continuous, bounded functions on the real line crossed with the discretely topologized reals. In this talk, I will discuss a higher dimensional analogue of this Heisenberg cycle, properties of its zeta functions, as well as some promising examples of pullbacks for which the corresponding zeta functions can be meromorphically extended to the complex plane.

Abstract: Étale groupoid C*-algebras are ubiquitous and it is useful to have concrete methods for verifying that a given C*-algebra is isomorphic to a groupoid C*-algebra. It is a result, essentially due to Brown, Clark, Farthing, and Sims, that a Hausdorff second countable étale groupoid that is topologically principal and minimal has a simple reduced C*-algebra. This makes verifying faithfulness of its representations trivial. Removing the minimality condition one obtains an older result stating that faithfulness of a representation can be verified on the canonical commutative subalgebra of the groupoid. In this talk I will discuss the notion of an étale groupoid which is relatively topologically principal with respect to an open subgroupoid, as well as the consequences of this condition for representations of the groupoid C*-algebra.

Abstract: We address the question: when does a purely infinite, simple, separable classifiable C*-algebra have a groupoid model? The answer was given up to stable isomorphism by Jack Spielberg and later in the unital case by L.O. Clark, J. Fletcher and A. an Huef. We will describe an alternate approach. This first requires a construction of Deaconu and Renault which I will describe. We further need a construction of mine into dynamical systems which I call 'binary factors' and finally an idea of orbit-breaking for Deaconu-Renault groupoids. This is joint work with Robin Deeley and Karen Strung.

Abstract: We address the question: when does a purely infinite, simple, separable classifiable C*-algebra have a groupoid model? The answer was given up to stable isomorphism by Jack Spielberg and later in the unital case by L.O. Clark, J. Fletcher and A. an Huef. We will describe an alternate approach. This first requires a construction of Deaconu and Renault which I will describe. We further need a construction of mine into dynamical systems which I call 'binary factors' and finally an idea of orbit-breaking for Deaconu-Renault groupoids. This is joint work with Robin Deeley and Karen Strung.

Abstract: We consider a family of Pimsner algebras associated to correspondences constructed from self-similar group actions and we investigate the equilibrium states (the KMS states) for the natural time evolution on these algebras. For all inverse temperatures above a critical value, the KMS states on the Toeplitz algebra are given in a very concrete way by traces on the full group algebra of the group. At the critical inverse temperature, the KMS states factor through states of the Cuntz-Pimsner algebra; if the self-similar group is contracting, then the Cuntz-Pimsner algebra has only one KMS state. We apply these results to a number of examples, including self-similar actions associated to integer dilation matrices, and the canonical self-similar actions of the basilica group and the Grigorchuk group.

Abstract: C*-algebras are often said to resemble `noncommutative spaces.' One way of making this idea precise makes use of the idea of Morita Equivalence. This is an introductory talk on Morita equivalence and its role in K-theory, representation theory (of groups), and group actions.

Abstract: Imprimitivity theorems establish the Morita equivalence between C*-algebras arising from certain group/groupoid dynamics. For example, Green's imprimitivity theorem arises when two groups act on a space by commuting free and proper actions. Recently, we introduced the notion of self-similar actions (also known as Zappa-Szep product) of groupoids on other groupoids and Fell bundles. These self-similar actions encode two-way actions between the groupoids, in contrast to the one-way action encoded by the classical group/groupoid actions. We prove a generalized imprimitivity theorem arising from such self-similar actions, which establishes a new way of constructing Morita equivalent groupoid and Fell bundles. This is joint work with Anna Duwenig.