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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.

Abstract: An irrational rotation algebra is a C*-algebra defined as the crossed product of the continuous functions on the circle, T=R/Z, by the Z-action induced from translation on T by integer multiples of a real number ℏ (modulo Z). Such irrational rotation algebras serve as some of the first motivating examples in noncommutative geometry; In particular, the early 90’s saw Connes use an adaptation of the Dirac cycle on the 2-torus to obtain a K-homology class (and representative cycle) over the tensor product of two identical irrational rotation algebras which induces a self-dialuty in the sense of KK-theory. In this presentation, I will introduce higher dimensional analogues of irrational rotation algebras constructed by considering lattices in Euclidean space of dimension possibly greater than one. Until recently, the aim in considering such higher dimensional rotation algebras was to provide a class of geometrically defined C*-algebras exhibiting KK-duality, with the particular duality classes being constructed in a manner involving a pair of (suitably) transverse groupoids arising from appropriately chosen lattices. However, the current goal is now to construct interesting spectral triples using our notion of transverse groupoids and, as in the work of Butler, Emerson, and Schulz, determine when one can meromorphically extend the resulting zeta functions. After some basic setup and notation, I will discuss the new goal, and a spectral triple of interest. Time permitting, I may also touch on the previous goal and, in particular, our candidate unit and co-unit for a KK-duality between pairs of these higher dimensional rotation algebras.

Abstract: The relative Baum-Connes conjecture claims that a certain relative Baum-Connes assembly map is an isomorphism. It provides an algorithm of the computation of the K-theory of relative group C*-algebras. In my talk, I will present several cases when the relative assembly maps are isomorphic (or injective). The strong relative Novikov conjecture states that the relative assembly map is injective. I will also talk about the applications of the strong Novikov conjecture in geometry and topology, especially about the relative higher signatures of manifolds with boundary.

Abstract: We study actions of right LCM semigroups on C*-algebras by endomorphisms that respect the ideal structure of the semigroup, and prove an equivariant Stinespring dilation theorem that combines a Naimark dilation of a semigroup of contractions with a Stinespring dilation of a completely positive unital *-linear map. This requires that we identify a class of contractive covariant representations and show they can be dilated to isometric covariant representations if and only if the C*-algebra map is unital and completely positive. We also analyze what happens for the boundary quotient. This is joint work with Boyu Li.

Abstract: The right-regular and left-regular representations of the ax+b monoid of the integers gives KK-equivalent C*-algebras with very different KMS structures under the natural gauge actions. A new feature which appears in the right-regular system is phase transition at high temperatures, which makes the analysis of these states more nuanced. I will start with a discussion of KMS states and their structure. I will then introduce the right ax+b system and its presentations, show that its KMS states can be identified with subconformal measures on the circle with respect to the wrapping action, and sketch the computation of these measures using twisted zeta functions.

Abstract: A Gromov hyperbolic group is a group with a certain large-scale negative curvature property. Almost all groups are hyperbolic (e.g. fundamental groups of a compact Riemann surface of genus g is hyperbolic unless g=0 or 1. The theory of hyperbolicity is important in topology in the classification of manifolds. Hyperbolic groups also are interesting from the point of view of dynamical systems. Any hyperbolic group can be compactified by adding a boundary to it. The boundary is a compact metrizable space on which the group acts by homeomorphisms. These boundary actions of hyperbolic groups code in special cases, asymptotic behaviour of geodesics on negatively curved surfaces, and determine simple purely infinite C*-algebras with Type III von Neumann closures. This means that the traditional tools of Noncommutative Geometry cannot be used to endow the corresponding `noncommutative spaces' with geometric structure. In these talks we report on progress on developing a `twisted' NCG for them, building on previous work of the author and Bogdan Nica.

Abstract: The construction of C*-algebras from groupoids is a very general method for constructing C*-algebras, including many of great importance. I will give a short review the construction for etale groupoids. The Elliott classification program for C*-algebras has been a huge undertaking over the past three decades and has given many new insights which were unimagined thirty years ago. I will give a short overview of the subject (from a non-expert). The obvious question which links these topics is: Which C*-algebras which are classified by Elliott arise from groupoids? I will discuss various results to answer this. I will try to keep the two talks at a fairly elementary level, although this will involve avoiding a lot of technical issues.