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Abstract: I will speak about Toeplitz-Pimsner algebras of product systems of Hilbert bimodules over group-embeddable monoids. Given a product system $\E$ over a submonoid of a group $G$, with coefficient algebra $A$, we show that, under technical assumptions, the fixed-point algebra $\Toepr(\E)^G$ of the gauge action of $G$ is nuclear if and only if $A$ is nuclear. Assuming in addition that $G$ is amenable, we conclude that the Toeplitz algebra is nuclear if and only if $A$ is nuclear.

I will also discuss exactness results and applications to product system over abelian monoids, $ax+b$-monoids of integral domains and Baumslag--Solitar monoids $BS^+(m,n)$ that admit an amenable embedding, which we provide when $m$ and $n$ are relatively prime. This is recent joint work with Elias Katsoulis and Camila Sehnem.

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.