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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.

Abstract: It was known from work of an Huef, Laca, and Raeburn that the right Toeplitz system for the ax+b semigroup over the naturals exhibits more than one equilibrium state at it's critical temperature, but otherwise the structure of the supercritical equilibria was unknown. In recent work with Marcelo Laca, we obtained a classification of these supercritical equilibria. Using the algebraic structure of the diagonal, we rederived a result of Afsar, Larsen, and Neshveyev which provides a correspondence between the KMS beta states and a class of measures on the circle. We provide explicit formulas for these measures and their Fourier coefficients, and describe the phase transition using properties of the Lerch zeta function.