The seminar was organized by Robin Deeley, Erik Guentner and Rufus Willett; please get in touch with one of us if you would like more information.
We discuss the construction of spectral triples in noncommutative
geometry from the point of view of the K-homology classes they represent,
with the goal of identifying what should mean the class of the Dirac operator
on a `noncommutative manifold'. We introduce a framework for discussing
this, that turns out to contain quite a number of interesting (well-known)
examples.
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Smale spaces are a class of dynamical system defined by Ruelle based on Smale's notion of an Axiom A diffeomorphism. I will introduce Smale spaces through three natural examples and also give a brief discussion of Ruelle's axioms.
From a Smale space one can construct a number of C*-algebras; each is obtained from an equivalence relation. These algebras will be discussed both in general and in the three examples. No knowledge of Smale spaces will be assumed.
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The stable C*-algebra of a Smale space is obtained the etale groupoid associated to the stable equivalence relation. When the original system is mixing it is simple, separable, nuclear, and stably finite. I will outline the construction of an explicit inductive limit decomposition of the stable C*-algebra when the stable sets are totally disconnected. From this, one can often compute the K-theory of the stable algebra. The talk will be example based. In particular, no knowledge of Smale spaces is required. This talk is based on joint work with Allan Yashinski.
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When considered up to a natural notion of equivalence introduced by Murray and von Neumann, projections in C*-algebras can be thought of as generalized ‘dimensions’: a basic example is the n by n matrices, where equivalence classes of projections correspond to the set {0,1,…,n}, with the correspondence given by taking the rank of a projection.
I’ll briefly discuss some broad conjectures relating projections in uniform Roe algebras — C*-algebras associated to discrete metric spaces — to the underlying geometry, and then discuss a fairly detailed classification of projections for the (discretized) line and plane. Some of this is based on joint work with Aaron Tikuisis.
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A band operator on l^2(N) is a bounded operator that, when represented as a matrix in the natural way, only occupies finitely many diagonals. One can make sense of this for more general metric spaces than the natural numbers. It is then natural, and useful, to ask how much of the geometry of the underlying space is ‘remembered' by the band operators (or, the analogous and harder question for norm limits of such operators).
I’ll discuss some answers to this question, and some key analytic facts about band operators that make the answers work (and are of interest in their own right). Part of this is based on old joint work with Jan Spakula, part of it on recent joint work of Jan Spakula and Aaron Tikuisis that has nothing to do with me, and part of it is inspired by a question of Stuart White.
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The crossed product structure is a nice (alternative) way to view groupoid C*-algebras. We construct the reduced crossed product C*-algebra and show that it is isomorphic to the reduced transformation groupoid C*-algebra.
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We will define higher-order Dehn functions of groups, and we will discuss how polynomial bounds on them imply the Banach strong Novikov conjecture.
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I’ll discuss two ‘niceness' notions for dynamical systems, one based on asymptotic dimension (due to Erik Guentner, Guoliang Yu, and myself), and another on amenability (due to David Kerr), and their connections to C*-algebra theory. I’ll discuss these in terms of ‘towers’, which are an old notion in dynamics that gives a convenient way to formalize the combinatorics involved; their use in this context is due to David Kerr, and a lot of the way I’ll present the material in this talk comes out of discussions with David.
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