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Abstract: One of the foundational results about C*-algebras is Gelfand's theorem establishing a duality between commutative unital C*-algebras and compact Hausdorff spaces. I will discuss the question of how effectively computable this duality is; specifically, is there an algorithm that takes as input a description of a commutative unital C*-algebra and produces as output a description of the spectrum of A? In the first talk I will provide all the necessary background from computability theory to be able to make this question precise, and also describe previous work on computable dualities. In the second talk I will outline a proof that the Gelfand duality is computable.

These talks are based on joint work with P. Burton, A. Fox, I. Goldbring, M. Harrison-Trainor, T. McNicholl, A. Melnikov, and T. Thewmorakot.

Abstract: One of the foundational results about C*-algebras is Gelfand's theorem establishing a duality between commutative unital C*-algebras and compact Hausdorff spaces. I will discuss the question of how effectively computable this duality is; specifically, is there an algorithm that takes as input a description of a commutative unital C*-algebra and produces as output a description of the spectrum of A? In the first talk I will provide all the necessary background from computability theory to be able to make this question precise, and also describe previous work on computable dualities. In the second talk I will outline a proof that the Gelfand duality is computable.

These talks are based on joint work with P. Burton, A. Fox, I. Goldbring, M. Harrison-Trainor, T. McNicholl, A. Melnikov, and T. Thewmorakot.

Abstract: One of the few cases where the Chow groups of quadrics over an arbitrary field are known is that of excellent quadrics, including the special case of Pfister quadrics. The computation of the Chow groups of Pfister quadrics also gives partial information about the Chow groups of Pfister neighbours, the forms stably birational to Pfister forms. By understanding how the Chow groups behave under field extensions, we can obtain important information about the quadric over the ground field, and by extending to an algebraic closure, we obtain interesting numerical invariants of quadratic forms such as the J-invariant. It appears the J-invariant can be useful in determining whether a form is a virtual Pfister neighbour (a form which over some extension becomes a Pfister neighbour), and we are currently interested in just how much information can be obtained from it.

Abstract: Given an almost periodic C*-algebraic dynamical system (A,\sigma) we define a new C*-algebra, the crystal of A, whose states parametrize the ground states of A, and which, in several classes of examples, has the same K-theory as A. This addresses a question of Connes about whether a quantum system can be cooled down until it becomes quasi-classical without losing its main equilibrium and topological features. Our work generalizes the two main classes of motivating examples: equilibrium states of Toeplitz-Pimsner algebras of correspondences (from work with Neshveyev) and of Toeplitz algebras of ax+b monoids, (from work with Raeburn and with Cuntz and Deninger) for which the K-theory was computed by Cuntz, Echterhoff, and Li. This is joint work with Sergey Neshveyev and Makoto Yamashita at Oslo.

Abstract: Given an almost periodic C*-algebraic dynamical system (A,\sigma) we define a new C*-algebra, the crystal of A, whose states parametrize the ground states of A, and which, in several classes of examples, has the same K-theory as A. This addresses a question of Connes about whether a quantum system can be cooled down until it becomes quasi-classical without losing its main equilibrium and topological features. Our work generalizes the two main classes of motivating examples: equilibrium states of Toeplitz-Pimsner algebras of correspondences (from work with Neshveyev) and of Toeplitz algebras of ax+b monoids, (from work with Raeburn and with Cuntz and Deninger) for which the K-theory was computed by Cuntz, Echterhoff, and Li. This is joint work with Sergey Neshveyev and Makoto Yamashita at Oslo.

Abstract: Given an almost periodic C*-algebraic dynamical system (A,\sigma) we define a new C*-algebra, the crystal of A, whose states parametrize the ground states of A, and which, in several classes of examples, has the same K-theory as A. This addresses a question of Connes about whether a quantum system can be cooled down until it becomes quasi-classical without losing its main equilibrium and topological features. Our work generalizes the two main classes of motivating examples: equilibrium states of Toeplitz-Pimsner algebras of correspondences (from work with Neshveyev) and of Toeplitz algebras of ax+b monoids, (from work with Raeburn and with Cuntz and Deninger) for which the K-theory was computed by Cuntz, Echterhoff, and Li. This is joint work with Sergey Neshveyev and Makoto Yamashita at Oslo.

Abstract: "(Strong) Morita equivalence of C*-algebras was introduced by M. Rieffel as a generalization of Morita equivalence of rings, and is an extremely useful tool in studying K-theory, ideals and representations of C*-algebras. It was generalized by Combes ('84) to 'Morita equivalence of C*-algebras with a group action' (i.e. equivariant Morita equivalence). One of Combes results is that, if C*-dynamical systems are Morita equivalent, then their (full/reduced) crossed-product C*-algebras are Morita equivalent.

In this talk, I will introduce another generalization to 'Morita equivalence of C*-correspondences.' I show that if two C*-correspondences are Morita equivalent, then the Toeplitz algebras are Morita equivalent. This will start with some background on C*-correspondences and Toeplitz algebras, a proof-sketch of the main result, and some examples."

Abstract: "(Strong) Morita equivalence of C*-algebras was introduced by M. Rieffel as a generalization of Morita equivalence of rings, and is an extremely useful tool in studying K-theory, ideals and representations of C*-algebras. It was generalized by Combes ('84) to 'Morita equivalence of C*-algebras with a group action' (i.e. equivariant Morita equivalence). One of Combes results is that, if C*-dynamical systems are Morita equivalent, then their (full/reduced) crossed-product C*-algebras are Morita equivalent.

In this talk, I will introduce another generalization to 'Morita equivalence of C*-correspondences.' I show that if two C*-correspondences are Morita equivalent, then the Toeplitz algebras are Morita equivalent. This will start with some background on C*-correspondences and Toeplitz algebras, a proof-sketch of the main result, and some examples."

Abstract: I will start with a short review of dimension groups (no background needed) and show how, in the case of a minimal action of the group of integers on a Cantor set, how they arise in a natural way (helped, in part, by the associated crossed product C*-algebra). This further leads to a complete model for such systems. In the second talk, I will discuss how these ideas can be extended, in a surprising way, to minimal actions of the group Z^2. This is joint work with Thierry Giordano (Ottawa) and Christian Skau (Oslo).

Abstract: I will start with a short review of dimension groups (no background needed) and show how, in the case of a minimal action of the group of integers on a Cantor set, how they arise in a natural way (helped, in part, by the associated crossed product C*-algebra). This further leads to a complete model for such systems. In the second talk, I will discuss how these ideas can be extended, in a surprising way, to minimal actions of the group Z^2. This is joint work with Thierry Giordano (Ottawa) and Christian Skau (Oslo).

Abstract: I will start with a short review of dimension groups (no background needed) and show how, in the case of a minimal action of the group of integers on a Cantor set, how they arise in a natural way (helped, in part, by the associated crossed product C*-algebra). This further leads to a complete model for such systems. In the second talk, I will discuss how these ideas can be extended, in a surprising way, to minimal actions of the group Z^2. This is joint work with Thierry Giordano (Ottawa) and Christian Skau (Oslo).