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Abstract: I will review the definition of the universal Toeplitz C*-algebra Tu(P) associated to a submonoid P of a group. Tu(P) is given by a presentation based on Xin Li’s semilattice of constructible ideals with an extra condition. After briefly mentioning faithful representations and a uniqueness theorem for the reduced Toeplitz algebra Tλ(P), I will concentrate on the associated universal boundary quotient ∂Tu(P) and its reduced version ∂Tλ(P), and give sufficient conditions for the latter to be purely infinite simple. I intend to discuss some new applications, mainly to monoids arising from non-maximal orders in number fields but also to right LCM monoids with nontrivial units. This is joint work with Camila F. Sehnem.

Abstract: I will review the definition of the universal Toeplitz C*-algebra Tu(P) associated to a submonoid P of a group. Tu(P) is given by a presentation based on Xin Li’s semilattice of constructible ideals with an extra condition. After briefly mentioning faithful representations and a uniqueness theorem for the reduced Toeplitz algebra Tλ(P), I will concentrate on the associated universal boundary quotient ∂Tu(P) and its reduced version ∂Tλ(P), and give sufficient conditions for the latter to be purely infinite simple. I intend to discuss some new applications, mainly to monoids arising from non-maximal orders in number fields but also to right LCM monoids with nontrivial units. This is joint work with Camila F. Sehnem.

Abstract: In 1967, Haag, Hugenholtz, and Winnink introduced the KMS condition in a seminal paper as a condition for equilibrium in quantum statistical mechanics, since it survives, in some sense, to the procedure known as the thermodynamic limit. In the following years, many results were proved that vindicated this point of view, such as the variational principle. Some years later, Lanford and Ruelle, and Dobrushin independently, introduced another condition for equilibrium states better suited for classical systems, known today as the DLR equations, that nowadays are ubiquitous in classical statistical mechanics. In these two talks, I will briefly introduce both notions, discuss the apparent differences between them, and will sketch a proof that they are equivalent in some cases where both can be defined. I will also review some properties of the KMS states for the Ising model.

Abstract: In 1967, Haag, Hugenholtz, and Winnink introduced the KMS condition in a seminal paper as a condition for equilibrium in quantum statistical mechanics, since it survives, in some sense, to the procedure known as the thermodynamic limit. In the following years, many results were proved that vindicated this point of view, such as the variational principle. Some years later, Lanford and Ruelle, and Dobrushin independently, introduced another condition for equilibrium states better suited for classical systems, known today as the DLR equations, that nowadays are ubiquitous in classical statistical mechanics. In these two talks, I will briefly introduce both notions, discuss the apparent differences between them, and will sketch a proof that they are equivalent in some cases where both can be defined. I will also review some properties of the KMS states for the Ising model.

Abstract: In these two talks I will give a brief introduction to \'etale groupoids and their C*-algebras, and present examples of factor groupoids which arise from quotients of path spaces of Bratteli diagrams, and examples that arise from certain extensions of Cantor minimal systems constructed by Robin Deeley, Ian Putnam, and Karen Strung. I will also describe how to compute the K-theory of these algebras, and how they fit into the Elliott classification scheme.

Abstract: In these two talks I will give a brief introduction to \'etale groupoids and their C*-algebras, and present examples of factor groupoids which arise from quotients of path spaces of Bratteli diagrams, and examples that arise from certain extensions of Cantor minimal systems constructed by Robin Deeley, Ian Putnam, and Karen Strung. I will also describe how to compute the K-theory of these algebras, and how they fit into the Elliott classification scheme.

Abstract: In 2007 Kechris and Rosendal showed that there is a unique dense $G_\delta$ conjugacy class in the group of homeomorphisms of the Cantor set. An explicit construction of a member of this class, along with a characterization of members of the class, was given by Akin, Glasner, and Weiss shortly thereafter. A homeomorphism in this conjugacy class is knowns as a "generic" homeomorphism of the Cantor set. In this talk I will describe another context, arising from model theory, in which one can make sense of the notion of a "generic" structure quite generally, and in particular of a generic homeomorphism. I will explain why in this model-theoretic context there is a generic homeomorphism of the Cantor set, but the homeomorphism constructed by Akin, Glasner, and Weiss is not generic in this sense. No prior knowledge of model theory will be necessary.

Abstract: We discuss a construction applying to any smooth flow on a smooth compact manifold M yielding an \(\R\)-equivariant spectral triple over the *-algebra of smooth functions on M. The case of Krönecker flow on the 2-torus determines a spectral triple over C^\infty (T^2)\rtimes \Gamma for an appropriate dense countable subgroup of the real line, inducing a Poincaré duality for the irrational rotation algebra. If time allows we discuss the problem of extending this result to higher genus surfaces.

Abstract: We discuss a construction applying to any smooth flow on a smooth compact manifold M yielding an \(\R\)-equivariant spectral triple over the *-algebra of smooth functions on M. The case of Krönecker flow on the 2-torus determines a spectral triple over C^\infty (T^2)\rtimes \Gamma for an appropriate dense countable subgroup of the real line, inducing a Poincaré duality for the irrational rotation algebra. If time allows we discuss the problem of extending this result to higher genus surfaces.