The seminar was organized by Robin Deeley, Erik Guentner, Rufus Willett and Allan Yashinski; please get in touch with one of us if you would like more information.
The Atiyah-Singer index theorem represents a landmark in 20th century mathematics. Since its proof in the 1960s, index theory has developed in many directions and found many applications. Understanding the index theorem is still nontrivial. However, there are a number of useful frameworks which make the Atiyah-Singer index theorem somewhat more intuitive.
I will discuss one such framework, K-homology. K-homology is the homology theory dual to the cohomology theory K-theory. However, its usefulness in index theory comes from its various realizations (e.g., Baum-Douglas cycles, Fredholm cycles, extensions, etc), explicit isomorphisms between such realizations, and explicit pairings between such cycles and K-theory cocycles. Our main goal is to view the Atiyah-Singer index theorem as a corollary of the isomorphism between geometric and analytic K-homology.
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In their papers "Mapping surgery to analysis I, II, and III", Higson and Roe construct an analytic counterpart to the classical surgery exact sequence of Browder, Novikov, Sullivan and Wall. In joint work with Magnus Goffeng, we construct a geometric version of Higson and Roe's exact sequence; by "geometric", we mean a construction based on the geometric (i.e., (M,E,\varphi)) model of K-homology due to Baum and Douglas.
I will discuss the construction of the geometric exact sequence and an application to relative eta type invariants. The former is obtained by applying a relative construction in geometric K-homology to the Baum-Connes assembly map. No knowledge of classical surgery or Higson and Roe's analytic surgery exact sequence are required for the talk.
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Smale spaces are a class of dynamical system defined by Ruelle to axiomatize properties of basic sets of an Axiom A diffeomorphism. I will introduce Smale spaces by presenting three natural examples in detail. 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 aforementioned examples. Much of this talk is based on Ian Putnam's paper "C*-algebras from Smale spaces".
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I will discuss some results of a joint project with Brady Killough and Michael Whittaker. The goal of the project is to better understand the functorial properties of the homology theory for Smale spaces introduced by Ian Putnam. This homology theory is conjecturally linked to the K-theory of the C*-algebras associated to a Smale space.
Inspired by Connes and Skandalis' notion of correspondences in KK-theory, we define correspondences in the setting of Smale spaces. The basic idea is to encode both types of functorial properties of Smale spaces (with respect to Putnam's homology theory) into a single object. The talk will be very much an introduction and will contain many examples (for the most part coming from the theory of shifts of finite type). No knowledge of Putnam's homology theory is required for the talk.
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Given a group G (e.g. the integers), the Fourier-Stieltjes algebra B(G) is a collection of bounded complex-valued functions on G that are associated to (and in some sense determine) the unitary representation theory of G. Unfortunately, B(G) is generally more-or-less impossible to completely describe. Today, I’ll introduce the algebra, say what it has to do with representation theory, and say at least the few general things about it that are possible as well as giving some examples.
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I’ll continue with an introduction to the Fourier-Steiltjes algebra B(G) of a discrete group. I’ll discuss means, which allow one to take averages of (at least some) functions on a group, and use this to discuss the difference between B(G) and the space of all bounded functions on G. I’ll then discuss what this has to do with amenability.
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I’ll give my last lecture on the Fourier-Steiltjes algebra B(G) of a discrete group. I’ll start by making what I said last time about means better. I’ll then introduce the topology on the space of irreducible representations, give some examples where things go wrong, and say what this has to do with amenability.
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We will consider the odometer dynamical system, which is a minimal Z-action on the Cantor set. In particular, the orbit equivalence relation associated to this dynamical system and tail equivalence will be compared (both at the level of relations and C*-algebras). The overall goal is to the relate the CAR algebra (which is defined "algebraically" using an inductive limit) with dynamics using the odometer system. I will endeavour to make the talk as self-contained as possible.
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George Mackey proposed a measure-theoretic correspondence between the irreducible unitary representations of a semisimple Lie group and those of its "Cartan Motion Group", a certain semidirect product group. Nigel Higson studied this correspondence at the level of the K-theory of the corresponding group C*-algebras. Here, the correspondence is perfect, and this is equivalent to the Connes-Kasparov conjecture for G (also to the Baum-Connes conjecture for G). Following Higson, we shall construct the Cartan Motion Group as a smooth deformation of G. We'll then indicate how to use this deformation to construct the proposed isomorphism of the Connes-Kasparov conjecture. For the sake of concreteness, G = SL(2,R) in this talk, though everything can be generalized to an arbitrary semisimple G.
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The talk will focus on rotation algebras (noncommutative 2-tori). The highlight will be Rieffel’s construction of nontrivial projections in rotation algebras. We’ll indicate how these projections and some knowledge of the K_0 group can be used to distinguish the irrational rotation algebras for different values of the parameter theta. This talk is intended for non-experts.
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Crystallographic groups are the groups of isometries of Euclidean space corresponding to ‘nice’ tessellations by squares, triangles, tetrahedra etc. The Baum-Connes conjecture is a conjecture about the K-theory of group C*-algebras (among other things). In principle, crystallographic groups should be one of the simplest cases of the conjecture (they were one of the first cases for which it was proved). However, it is probably fair to say that the conjecture lacks a good conceptual explanation even in this case. Inspired partly by recent work of Niblo, Wright, and Plymen, I’ll discuss some examples (with lots of pictures) and try to explain some of the issues (without assuming anyone knows much about K-theory etc.).
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I will introduce the notion of minimal dynamical systems by considering two classes of examples: odometers and irrational rotations on the circle. After discussing some ergodic theory, we will turn to the crossed product construction and the connection between the minimality of an action and the simplicity of the associated crossed product C*-algebra.
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Direct limits preserve exact sequences of (say) abelian groups, but inverse limits do not. The defect is measured by a homological gadget due to Milnor. We’ll discuss some of this in general, and more specifically in C*-algebra K-theory.
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To begin, I will introduce three examples of Smale spaces (Smale spaces are a class of hyperbolic dynamical system). For each of these examples there is a natural Z/2Z-action. These examples motivate the study of group actions on a general Smale space. I will discuss such actions both in regards to the Smale space itself and the C*-algebras associated to it. This talk is based on joint work with Karen Strung.
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In 1980, Cuntz and Krieger constructed a canonical C*-algebra from a non-degenerate square matrix with entries in { 0, 1}. These C*-algebras are now known as the Cuntz-Krieger algebras. They also showed for two non-degenerate matrices A, B with A, B, A^t, and B^t satisfying Condition (I), if the associated shift spaces X_A and X_B (i.e., consider the graph E_A with vertices \{ 1, \dots, n \} and an edge from i to j whenever A(i,j) = 1, then X_A is the collection of all bi-infinite paths in E_A) are flow equivalent, then there exists an isomorphism between the stabilized Cuntz-Krieger algebras preserving the canonical diagonal sub-algebras.
In this talk, I will discuss possible avenues to achieve the converse. In particular, I will show how to characterize diagonal preserving stable isomorphism of Cuntz-Krieger algebras via orbit equivalence of the associated one-sided shift spaces, and via equivalence of the associated groupoids. Finally, I will finish with how this result relates to achieving flow equivalence between the two-sided shift spaces.
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Given an expanding self-map f: Y to Y on a compact metric space, one can build a stationary inverse limit space X. The map f defines a dynamical system on X, which is a Smale space under suitable hypotheses. The prototypical example is the natural n-fold cover from the circle to itself given by z \mapsto z^n. The resulting inverse limit dynamical system is called a solenoid. We shall study this example in detail, and also examples where Y is a wedge of circles. Our aim will be to understand a certain C*-algebra associated to these Smale spaces by identifying a nice inductive limit structure.
The talk will be example-oriented, and no knowledge of Smale spaces is required (even for the speaker). If there is enough interest, I would like to speak again the following Thursday. The plan for that talk is to use the inductive limit structure to compute the K-theory of these C*-algebras.
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