Noncommutative Geometry Seminar, Fall 2011

Schedule of Talks

September 6th: Zhizhang Xie (Vanderbilt), Index theorems on manifolds with boundary.
Abstract
September 13th: No seminar: NCG meeting in Oberwolfach.
Thursday September 22nd (joint with colloquium): Matilde Marcolli (Caltech), Quantum Statistical Mechanics, L-series and Anabelian Geometry.
Abstract
September 27th: Mark Sapir (Vanderbilt), Aspherical groups and manifolds with extreme properties.
Abstract
October 4th: No seminar: activities postponed to next week.
October 11th: Rufus Willett (Hawaii), "Uniform local amenability".
Abstract
Wednesday October 12th (joint with Topology and Group Theory seminar): Tom Farrell (SUNY Binghampton), Bundles with negatively curved fibres.
Abstract
Thursday October 13th (joint with colloquium): Tom Farrell (SUNY Binghampton), The space of negatively curved metrics.
Abstract
October 18th: No seminar: half the organizers out of town.
October 25th: No seminar: faculty meeting.
Wednesday October 26th (joint with Topology and Group Theory seminar): Rufus Willett (Hawaii), Coarse geometry of expanding graphs, and graphs with large girth.
Abstract
November 1st: Tim Ferguson (Vanderbilt), Extremal Problems in Bergman spaces.
Abstract
November 8th: Claude Schochet (Wayne State University), A report on the unitary group of a C*-algebra.
Abstract
Monday November 14th (in SC 1312): Tsuyoshi Kato (Kyoto University), Dynamical Burnside Problem.
Abstract
November 15th: Igor Mineyev (University of Illinois Urbana-Champaign), The Atiyah Conjecture.
Abstract
November 22nd: No seminar: Thanksgiving break.
November 29th: No seminar: faculty meeting.
Wednesday December 7th (joint with Topology and Group Theory seminar): Denis Osin (Vanderbilt University), Inner non-amenability and simplicity of C*-algebras of groups acting on hyperbolic spaces.
Abstract
December 13th: Tim Austin (Brown University), Continuity properties of measurable group cohomology.
Abstract

Abstracts

September 6th

Zhizhang Xie (Vanderbilt), Index theorems on manifolds with boundary.
We will review the Atiyah-Patodi-Singer index theorem and the basic properties of the eta invariant. We will then talk about some recent results in this direction inspired by the methods of noncommutative geometry.
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September 22nd

Matilde Marcolli (Caltech), Quantum Statistical Mechanics, L-series and Anabelian Geometry.
This talk is based on joint work with Gunther Cornelissen. The zeta function of a number field can be interpreted as the partition function of an associated quantum statistical mechanical system built from Artin reciprocity. While it is well known that two number fields with the same Dedekind zeta function are not necessarily isomorphic, we show using this quantum statistical mechanics point of view that isomorphism of number fields is the same as the existence of an isomorphism of character groups of the abelianized Galois groups that induces an equality of all corresponding L-series. This is in turn equivalent to the fact that number fields are isomorphic if and only if the associated quantum statistical mechanical systems are isomorphic. This can be seen as another version of Grothendieck's "anabelian" program, much like the Neukirch-Uchida theorem that characterizes isomorphism of number fields by topological isomorphism of their associated absolute Galois groups.
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September 27th

Mark Sapir (Vanderbilt), Aspherical groups and manifolds with extreme properties.
We prove that every aspherical recursively presented group embeds into a group with finite aspherical presentation complex. By results of Gromov and Davis, this implies that there exists a closed aspherical manifold of any dimension >3 (smooth in dimension >4) with universal cover of infinite asymptotic dimension, and which is a counterexample to the Baum-Connes conjecture with coefficients.
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October 11th

Rufus Willett (Vanderbilt), "Uniform local amenability".
I will introduce some new coarse geometric / group theoretic properties, which could reasonably be called 'uniform local amenability'. The main aim is to clarify the relationships between various existing properties in coarse geometry (e.g. I'll show property A implies metric sparsification implies the existence of a coarse embedding in Hilbert space, etc.), although they seem likely to be of interest in their own right. These results arose from discussions with Ján Špakula, Jacek Brodzki, Graham Niblo and Nick Wright, most or all of whom will appear on any paper that gets written.
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October 12th

Tom Farrell (SUNY Binghampton), Bundles with negatively curved fibres.
I will talk on joint work with Pedro Ontaneda. Let M be a closed smooth manifold which supports a negatively curved Riemannian metric. And let p: E to B be a smooth bundle with abstract fibre M and family of fibres E_x= p^{-1}(x), x in B. I will talk about the problem of equipping the fibres with negatively curved Riemannian metrics varying continuously with x.
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October 13th

Tom Farrell (SUNY Binghampton), The space of negatively curved metrics.
A well known problem in differential geometry is whether the space S(M) of all negatively curved Riemannian metrics on a closed smooth manifold M is always path connected. I will talk on joint work with Pedro Ontaneda motivated by this problem. We showed in particular that if the dimension of M > 9, then S(M) is either empty or has infinitely many path components.
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October 26th

Rufus Willett (Hawaii), Coarse geometry of expanding graphs, and graphs with large girth.
I will discuss expanding ('highly connected, low density') graphs, and graphs with large girth ('no small loops') from the point of view of large-scale geometry. Gromov (and Sapir) have shown that some such sequences of graphs 'embed' into (aspherical) groups, leading to several surprising results. Some of these properties are quite pathological, due to the complicated geometry of expanders, and lead to counterexamples to Baum-Connes type conjectures (among other things). The two classes of graphs in the title share many similar properties: for example, graphs with large girth are expanders 'at small scales'. However, graphs (even expanders) with large girth are in some sense relatively 'benign' from the point of view of large scale geometry, partly as they do not have a 'geometric' version of property (T). I will survey all this. Much of what I will talk about is joint work with Guoliang Yu.
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November 1st

Tim Ferguson (Vanderbilt), Extremal Problems in Bergman spaces.
I will discuss extremal problems in Bergman spaces, which are spaces of analytic functions. After discussing basic results about these problems, I will discuss regularity results relating these problems to Hardy spaces. Specifically, I will discuss a result of Ryabykh which says that if the "data" for the extremal problem lies in a certain Hardy space, the extremal function must lie in a corresponding Hardy space. I will then talk about recent extensions of this result.
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November 8th

Claude Schochet (Wayne State University), A Report on the Unitary Group of a C*-algebra.
Let A be a C*-algebra. Its unitary group, UA, contains a wealth of topological information about A. However, the homotopy type of UA is unknown even for A the 2-by-2 matrices. There are various simplifications which have been considered. The first, well-traveled road, is to pass to the homotopy groups of the unitary group of the stabilization of A, which is isomorphic (with a degree shift) to K_*(A). This approach has led to spectacular success in many arenas, as is well-known.

A different approach is to consider the rational homotopy groups of UA or, to be brave / reckless, to consider the homotopy groups of UA itself. We report on progress in the calculation of these functors for the cases A = M_n(C(X)), A a unital continuous trace C*-algebra, and more generally for A an algebra of sections of a locally trivial bundle of C*-algebras. The resultant spectral sequence obtained generalizes work of H. Federer and J. Rosenberg. Finally, we discuss work in progress and a conjecture relating differentials and Samelson products in the unitary groups.

These results are joint work and work in progress with J. Klein, G. Lupton, N.C. Phillips, S. Smith, and E. Dror-Farjoun.
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November 14th

Tsuyoshi Kato (Kyoto University), Dynamical Burnside Problem.
In this talk we introduce a new connection between geometric group theory with rational dynamics passing through tropical geometry. As an application we construct infinitely quasi recursive rational dynamics.
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November 15th

Igor Mineyev (University of Illinois Urbana-Champaign), The Atiyah Conjecture.
This talk will give an introduction to the Atiyah Conjecture (AC), originally stated by Atiyah as a question. First we will give a brief overview of results on AC, then define the deep-fall property that was used in the proof of the Hanna Neumann Conjecture. The deep-fall property is stated for left-orderable groups and it implies AC. This gives a plan to address AC in the case of left-orderable groups.
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December 7th

Denis Osin (Vanderbilt University), Inner non-amenability and simplicity of C*-algebras of groups acting on hyperbolic spaces.
I will discuss some general properties of groups acting acylyndrically and non-elementary on hyperbolic spaces. In particular, I will prove that the following conditions are equivalent for every such a group G: a) G has no non-trivial finite normal subgroups. b) G is ICC (i.e., every nontrivial conjugacy class in G is infinite) c) G is not inner amenable. d) the reduced C*-algebra of G is simple with unique trace.
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December 7th

Tim Austin (Brown University), Continuity properties of measurable group cohomology.
In a classic sequence of papers, Calvin Moore established a generalization of group cohomology to the setting of locally compact groups and Polish (that is, complete, separable and metrizable) modules over them, taking their topologies into account. His theory enjoys analogs of the standard algebraic properties of discrete group cohomology, such as long exact sequences, effaceability and a Hochschild-Serre spectral sequence, and his cohomology groups in degree two correctly classify group extensions in this topological setting.

However, various analytic properties of the theory remained unclear, such as its behaviour under direct or inverse limits, and little was known about when it coincides with other topological group cohomology theories such as that defined using continuous cochains. This talk will describe some recent progress which establishes such continuity in various cases, and also gives isomorphisms to some of those other cohomology theories, based on a simple procedure for showing that cocycles in Moore's theory can be `regularized' to exhibit more structure than assumed a priori.

I will try to make the talk mostly self-contained, but some familiarity with basic homological algebra would be helpful to the listener. It is based on joint work with Calvin Moore.
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