Abstracts for NCGOA 2013

Transcription

1 Abstracts for NCGOA 2013 Mini-courses Lecture 1: What is K-theory and what is it good for? Paul Baum (Penn State University) Abstract: Lecture 1 - This talk will consist of four points: 1. The basic definition of K-theory 2. A brief history of K-theory 3. Algebraic versus topological K-theory 4. The unity of K-theory The talk is intended for non-specialists. assumed. Only a general mathematical background will be Lecture 2: Expanders, exact crossed-products, and K-theory for group C algebras Paul Baum (Penn State University) Abstract: Lecture 2 - An expander is a sequence of finite graphs X 1, X 2, X 3,... which is efficiently connected. A discrete group G which contains an expander in its Cayley graph is a counter-example to the Baum-Connes (BC) conjecture with coefficients. M. Gromov outlined a method for constructing such a group G. Arjantseva and T. Delzant completed the construction. The group so obtained is known as the Gromov group and is the only known example of a non-exact group. The left side of BC with coefficients sees any group as if the group were exact. This talk will indicate how to make a change in the right side of BC with coefficients so that the right side also sees any group as if the group were exact. This corrected form of BC with coefficients uses the unique minimal intermediate exact crossed-product. For exact groups (i.e. all groups except the Gromov group) there is no change in BC with coefficients. In the corrected form of BC with coefficients the Gromov group acting on the coefficient algebra obtained from an expander is not a counter-example. Thus at the present time (April, 2013) there is no known counter-example to the corrected form of BC with coefficients. The above is joint work with E. Kirchberg and R. Willett. This work is based on and inspired by a result of R. Willett and G. Yu. 1

2 Positive Energy Representations and K-Theory Nigel Higson (Penn State University) Abstract: Let G be a compact Lie group. Its loop group LG is the infinite-dimensional space of smooth maps from the circle into G (the group operation is pointwise multiplication). The loop group has a series of representations - the positive energy representations - whose theory very closely parallels the representation theory of the compact group G. I ll try to explain some of these connections, especially as they relate to K-theory, following the approach of Freed, Hopkins and Teleman. Variants of K-theory and connections with noncommutative geometry and physics Jonathan Rosenberg (University of Maryland) Abstract: The topology literature is full of different variants of topological K-theory: K, KO, KSp, KSC, KR, and twisted versions of all of these. We will explain what these are, how they arise from noncommutative geometry, and how they have recently arisen in physics from the classification of D-branes in string theory. Cracks in topological rigidity Shmuel Weinberger (University of Chicago) Abstract: Topological rigidity is an analogue of the Baum-Connes conjecture, and is motivated by Mostow rigidity and Margulis superrigidity. I will discuss some of the places where this heuristic reasoning fails, and the rigidity shows some cracks that are rather different than the places where strong forms of the Baum-Connes conjecture fails. One-hour talks A differential complex for groups acting on CAT(0)-cube complexes Jacek Brodzki (University of Southampton) Abstract: This talk will introduce a very natural and interesting differential complex associated with a CAT(0)-cube complex. The construction builds on and extends ideas first introduced by Pytlik and Szwarc for the free group and extended by Julg and Valette in the case of groups acting on trees. We will discuss applications of this construction for the study of K-homology of groups acting on such complexes. This talk is based on joint work with Erik Guentner and Nigel Higson. 2

3 Positive scalar curvature and a new index theory for noncompact manifolds Stanley Chang (Wellesley College) Abstract: In this talk, I will describe joint work with Weinberger and Yu. We develop a new index theory for noncompact manifolds endowed with an admissible exhaustion by compact sets. This index theory allows us to provide examples of noncompact manifolds with interesting positive scalar curvature phenomena. K-theory of group rings and operator ideals Guillermo Cortiñas (Universidad de Buenos Aires) Abstract: The Baum-Connes conjecture predicts that the topological K-theory K (C r (G)) of the reduced C -algebra of a locally compact group G equals the G-equivariant topological K- homology of the classifying space for proper actions, KG (ȨG). If G is discrete, K G (ȨG) can be interpreted in terms of algebraic K-homology in at least two ways. One way is as the G-equivariant algebraic K-homology H G (ȨG, K(Ķ)) of ȨG with coefficients in the algebraic K-theory of the ideal Ķ of compact operators; the algebraic K-theory isomorphism conjecture predicts that H G (ȨG, K(Ķ)) = K (Ķ[G]), (1) the algebraic K-theory of the group algebra with Ķ-coefficients. Another way is as H G (ȨG, KH(Ļ 1 )), the G-equivariant homotopy algebraic K-homology with coefficients in the ideal of trace-class operators; the KH-isomorphism conjecture predicts that H G (ȨG, KH(Ļ 1 )) = KH (Ļ 1 [G]), (2) the homotopy algebraic K-theory of the group algebra with trace-class coefficients. In the talk we shall discuss ongoing joint work with Gisela Tartaglia on establishing (1) for a-t -menable groups, and on relating (2) to the algebraic K-theory Novikov conjecture. Invariants of continuous fields C -algebras Marius Dadarlat (Purdue University) Abstract: I will report on joint work with Ulrich Penning. We show that the automorphism group Aut(A) of a strongly self-absorbing C -algebra A is contractible and that the automorphism group Aut(A K) of the stabilization of A has the homotopy type of a CW-complex whose homotopy groups we compute. We extend the classification of locally trivial bundles of C -algebras with compact operators K as fibers by Dixmier and Douady to the case where the fibers are isomorphic to a stabilized strongly self-absorbing C -algebra A K. We prove that the classifying space for these bundles BAut(A K) has the structure of an infinite loop space with respect to the tensor product. Thus, bundles with fibers A K are classified by a generalized continuous cohomology theory which is computable via the Atiyah-Hirzebruch spectral 3

4 sequence. This allows us to introduce rational characteristic classes for such bundles. A necessary and sufficient K-theoretical condition for local triviality is given for continuous fields with fibers A K over spaces of finite covering dimension. K-theory for crossed products by group actions on totally disconnected spaces and of semi-group algebras Siegfried Echterhoff (University of Münster) Abstract: In this lecture we give a report on joint work with Joachim Cuntz and Xin Li on the computation of the K-theory for crossed products by certain actions of groups on totally disconnected spaces. We apply the results to the computation of the K-theory for certain semigroup C -algebras. In particular, we obtain explicit computations for the ax+b-semigroups R R, where R is the ring of integers in a number field. Bounded noncommutative geometry and boundary actions of hyperbolic groups Heath Emerson (University of Victoria) Abstract: We use the canonical Lipschitz geometry available on the boundary of a hyperbolic group to construct finitely summable Fredholm modules over the crossed product of the boundary action. The procedure yields a full set of representatives of the K-homology of the crossed product, and all of them are p-summable for p in an appropriate range related to the geometry of the boundary. By restricting these boundary Fredholm modules to the reduced group C -algebra of the group, we obtain classes in the K-homology of the (reduced) group C -algebra. This implies various results about the (reduced) representation ring, for example that every class in the representation ring of a classical hyperbolic group of dimension n is represented by a n + summable Fredholm module generalizing an observation of Connes. Classification of simple C -algebras of generalized tracial rank one Guihua Gong (Puerto Rico University) Abstract: In this talk, we will present a classification results for a class of unital simple C - algebras. As suggested by their Elliott invariants, this class of C*-algebras may include all unital simple separable nuclear stable finite Z-stable algebras (here Z is the Jiang Su algebra). 4

5 Formality for deformations of gerbes Alexander Gorokhovsky (University of Colorado at Boulder) Abstract: After reviewing the general theory of deformations, I will discuss the classification of deformations of gerbes on smooth manifolds, and then describe the L -algebra controlling deformations of gerbe and explain the analogue of the Kontsevich formality theorem for gerbes. This is a joint work with P. Bressler, R. Nest and B. Tsygan Fredholm module associated to the boundary of a rank one symmetric space Abstract: Pierre Julg (Université d Orléans) Hamiltonian deformation of groups acting on trees Tsuyoshi Kato (Kyoto University) Abstract: One can construct continuous deformations of groups by embedding a discrete group acting on a tree into the automorphism group over the infinite projective space. We verify a phenomena that any deformations by a class of Hamiltonian diffeomorphisms must be uniformly infinite in our sense. In particular there are no finite deformation in the class. As an application of moduli theory over the infinite dimensional spaces, we obtain a new inequality between cobordism invariant and iteration norms. Semigroup C -algebras Xin Li (University of Münster) Abstract: The semigroup C -algebra of a cancellative semigroup is the C -algebra generated by the regular representation of the semigroup. The first example of such a C -algebra is the classical Toeplitz algebra. Recently, there has been a revival of the subject of semigroup C -algebras due to interesting new classes of examples, which can be divided into two types: Semigroups which naturally appear in group theory, and semigroups attached to rings from number theory or algebraic geometry. The goal of the talk is to give an overview of these recent developments. 5

6 Persistent approximation property for C -algebras with propagation Hervé Oyono-Oyono (University of Metz) Abstract: The study of elliptic differential operators from the point of view of index theory and its generalisations to higher order indices gives rise to C -algebras where propagation makes sense and encodes the underlying large scale geometry. Prominent examples for such C -algebras are Roe algebras, group C -algebras and crossed product C -algebras. Unfortunately, K-theory for operator algebras does not keep track of these propagation properties. Together with G. Yu, we have developed a quantitative version of K-theory that takes into account propagation phenomena. In this lecture we explain that in many cases, these quantitative K-theory groups approximate in a particular relevant way the K-theory. We also discuss connection with the Baum-Connes and the Novikov conjecture. On the geometry and stratification theory of orbit and inertia spaces of proper Lie groupies Markus Pflaum (University of Colorado at Boulder) Abstract: In the talk, the geometry and stratification theory of the orbit space and the inertia space of a proper Lie groupoid will be examined. We construct an explicit Whitney stratification of the orbit space and, under the assumption that the groupoid is the action groupoid of a compact Lie group action, a Whitney stratification of the inertia space. As a consequence, it is proved that both the orbit and the inertia space are triangulable differentiable stratified spaces. In addition, de Rham type theorems are shown, and some applications are given. The results have crucial applications for studying the noncommutative geometry and geometric analysis of proper Lie groupoids and their quotient spaces. The talk is on joint work with H.Posthuma and with X.Tang, as well as with C.Farsi, and Ch.Seaton. Positive scalar curvature, surgery classification of manifolds and Dirac type operators I and II - Part II Paolo Piazza (Sapienza Università di Roma) Abstract: In these talks reporting on joint work with Thomas Schick, we will discuss exact sequences helping in the classification of positive scalar curvature metrics (the Stolz sequence) and in the classification of manifolds of a given homotopy type (Wall s surgery sequence). We will show how higher index theory of Atiyah-Patodi-Singer type directly allows to map Stolz sequence for metrics of positive scalar curvature to a sequence in K-theory due to Higson and Roe, which includes the Baum-Connes assembly map. The construction heavily relies on ideas from large scale index theory (coarse index theory). We will also explain the analogous result for Wall s surgery sequence, thus giving an alternative treatment to a theory due to Higson and Roe (developed in their papers Mapping surgery to analysis I-III ). Again, our treatment is based directly on the index theory of the signature operator. 6

7 Dynamical Systems on Spectral Metric Spaces Kamran Reihani (Northern Arizona University) Abstract: Spectral triple is the fundamental object of the metric aspects of Connes noncommutative geometry. A spectral metric space is a spectral triple (A, H, D) with additional properties guaranteeing that the Connes metric on the state space of A induces the weak*-topology. It is, in fact, the noncommutative analog of a complete metric space. Assuming that (A, H, D) defines a spectral metric space and G is a group of automorphisms of A, we will address the problem of whether there is a natural spectral triple for the crossed product algebra C (G, A) that can characterize the metric properties of the dynamical system (G, A). We will introduce a solution to this problem when a single automorphism of A generates G as an equicontinuous family of quasiisometries. We will also address the converse problem, namely, when a spectral metric space for the crossed product gives rise to one for A. This talk is based on a joint work with Jean Bellissard and Matilde Marcolli. Analytic structure invariants John Roe (Penn State University) Abstract: This talk will be an introduction to the secondary analytic invariants which quantify the reasons for the vanishing or invariance properties of primary (higher) indices. Positive scalar curvature, surgery classification of manifolds and Dirac type operators I and II - Part I Thomas Schick (Georg-August-Universität Göttingen) Abstract: In these talks reporting on joint work with Paolo Piazza, we will discuss exact sequences helping in the classification of positive scalar curvature metrics (the Stolz sequence) and in the classification of manifolds of a given homotopy type (Wall s surgery sequence). We will show how higher index theory of Atiyah-Patodi-Singer type directly allows to map Stolz sequence for metrics of positive scalar curvature to a sequence in K-theory due to Higson and Roe, which includes the Baum-Connes assembly map. The construction heavily relies on ideas from large scale index theory (coarse index theory). We will also explain the analogous result for Wall s surgery sequence, thus giving an alternative treatment to a theory due to Higson and Roe (developed in their papers Mapping surgery to analysis I-III ). Again, our treatment is based directly on the index theory of the signature operator. 7

8 Uniform Local Amenability Jan Spakula (University of Münster) Abstract: Yu s property A is a coarse-geometric variant of amenability and has numerous characterizations, for instance it is equivalent to C -exactness for discrete groups. In this talk, I will discuss one a closely related property: Uniform Local Amenability. This new property is particularly well suited for showing that spaces do not have it; it readily applies to expanders and families of graphs with large girth. This is a joint work with J. Brodzki, G. Niblo, R. Willett and N. Wright. T -duality for Langlands dual groups Erik van Erp (University of Hawaii) Abstract: I ll discuss joint work with Calder Daenzer. We show that Langlands duality for complex reductive Lie groups can be implemented by T -dualization, for groups whose simple factors are of Dynkin type A, D, or E. Geometric quantization via coarse geometry Mathai Varghese (University of Adelaide) Abstract: Using coarse geometry, we propose a very general notion of quantization of a Hamiltonian G-manifold (G a Lie group acting properly). The corresponding, quantization commutes with reduction conjecture, will be formulated and the proof in several important special cases sketched. Applications to the representation theory of semisimple Lie groups will be outlined, which includes a new coarse Dirac induction map and conjecture. This is joint work (in progress) with Peter Hochs. Localized indices for elliptic operators on properly cocompact G-manifold Hang Wang (Tsinghua University) Abstract: The focus of my talk is index formulas of elliptic operators on a manifold X, where a discrete group G acts properly and cocompactly. The L 2 -index of an elliptic operator P is to compose K 0 (C (G)) R, induced by the von Neumann trace N G R, with the higher index of P. In the talk, we shall introduce a type of localized indices which generalize the notion of the L 2 -index. The associated index formulas give a finer description of Kawasaki s orbifold index formula for the quotient orbifold X/G. This is a joint project with Bai-Ling Wang. 8

9 A Dirac-type class in the K-homology of a quantum flag variety Robert Yuncken (Université Blaise Pascal) Abstract: Compact semisimple Lie groups and their flag varieties admit well-known deformations to noncommutative spaces. Seeing as the classical objects are some of the nicest examples of smooth manifolds, one would like to be able to describe their quantized cousins as noncommutative manifolds. But despite more than a decade of work, examples of geometric spectral triples (à la Connes) have only been found in relatively few cases. We will start off by discussing this problem, and then define a bounded spectral triple (K-homology class) of Dirac-type for the quantized flag variety of SU(3). This class has application to the Baum-Connes Conjecture for discrete quantum groups. (Joint work with Christian Voigt, Glasgow). A Lichnerowicz vanishing theorem for foliations Weiping Zhang (Nankai University) Abstract: We will explain a generalization of the Lichnerowicz-Hitchin vanishing theorem to the case of foliations. As a consequence, we show that there is no foliation of positive leafwise scalar curvature on any torus. Our proof, which is inspired by the analytic localization techniques developed by Bismut and Lebeau, also applies to give a new proof of the Connes vanishing theroem without using any noncommutative geometry. 9