Paul Baum, Pennsylvania State University
Expanders and K-theory for group C*-algebras

An expander or expander family is a sequence of finite graphs X1, X2, X3,…, 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. Some care must be taken with the definition of “contains”. M. Gromov outlined a method for constructing such a group. G. Arjantseva and T. Delzant completed the construction. Any group so obtained is known as a Gromov group or Gromov monster group, and these are the only known examples of a non-exact groups.

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 exact and Morita compatible intermediate crossed-product. For exact groups (i.e. all groups except the Gromov groups) there is no change in BC with coefficients.

In the corrected form of BC with coefficients a Gromov group acting on the coefficient algebra obtained from an expander is not a counter-example. Thus at the present time (May, 2014) there is no known counter-example to the corrected form of BC with coefficients. The above is joint work with E. Guentner and R. Willett. This work is based on — and inspired by — a result of R. Willett and G. Yu, and is very closely connected to results in the thesis of M. Finn-Sell. Also closely related are the ideas and results in the talk of R. Willett at this meeting.

Jean Bellissard, Georgia Institute of Technology
Computational Noncommutative Geometry: the work of Emil Prodan

The formalism of Noncommutative Geometry is applicable to problems of solid state physics involving an aperiodic distribution of atomic sites. This leads to the concept of noncommutative Brillouin zone. After reviewing this formalism, the Kubo formula for the electric conductivity is such a solid will be derived and discussed. The approximation scheme provided by E. Prodan to compute efficiently this conductivity will be justified in the case of a disordered solid. The consequences for the numerical study of the quantum Hall effect will be illustrated by a recent work of Prodan and his collaborators, showing the the noncommutative approach is much more efficient than anything known so far.

David Berenstein, University of California, Santa Barbara
From matrices to hypersurfaces and beyond

In the talk I will describe a procedure that takes a collection of 2k+1 hermitian matrices and produces a hypersurface with additional structure in a 2k+1 dimensional space. I will describe the origin of this problem in physics in the context of D-brane dynamics. I will furthermore explore this geometric problem when the matrices are random and the implications this has for the study of black holes.

Branimir Ćaćić, Texas A&M University
Twisted group algebras and Connes–Landi deformation of spectral triples

In this talk, we will survey Connes–Landi deformation of spectral triples from the perspective of Abadie and Exel's generalisation of strict deformation quantisation, which can be viewed as constructing deformed algebras as twisted group algebras with coefficients in a Fell bundle. In particular, we will discuss work towards a reconstruction theorem for toric noncommutative manifolds, and show that toric noncommutative manifolds with rational deformation parameter are almost-commutative spectral triples, at least over a certain naturally defined orbifold.

Alain Connes, Collège de France
Foiaş Lecture Series

I. Noncommutative geometry from its origins to its implications in physics

I will explain how the fact that continuous and discrete variables can coexist in the formalism of quantum mechanics at the price of non-commuting, together with the generation of time from non-commutativity, yield the new paradigm for Riemannian geometry given by spectral triples which I introduced (under another name) in the early 1980's. I will explain how this allows one to dress geometry from the quantum and how the inner fluctuations of metrics come from a natural semigroup enlarging the unitary group.

II. The music of shapes

I will explain how a simple question "where are we" forces one to rethink of geometry in an invariant and spectral manner, and why many spaces are actually perceived through their spectrum. The formalism of NCG is perfectly adapted to this perception and even standard music suggests dealing with new shapes which are quantum such as the quantum 2-spheres. I will explain the inference on physics of gravity coupled with the Standard Model and end with a still uncovered space whose music is well defined.

III. Noncommutative geometry and number theory

I will describe the interplay between noncommutative geometry and number theory. It has long been known that the noncommutative space of adèle classes of a global field provides a framework to interpret the explicit formulas of Riemann–Weil in number theory as a trace formula. In our joint work with C. Consani we showed that if one divides the adèle class space of rational numbers by the action of the maximal compact subgroup of the idèle classes one obtains, by considering the fixed points of the induced flow, the counting distribution which in turn determines, using the Hasse–Weil formula in the limit q → 1, the complete Riemann zeta function. We have recently uncovered in our joint work the "Arithmetic Site": an object of algebraic geometry of great simplicity deeply related to the noncommutative-geometric approach to the Riemann hypothesis but also to the topos theory of Grothendieck and to tropical geometry. The set of points of the arithmetic site over the maximal compact subring of the tropical semifield coincides with the noncommutative space quotient of the adèle class space of the field of rational numbers by the action of the maximal compact subgroup of the idèle class group. The square of the arithmetic site is also well-defined and the Frobenius correspondences parametrized by positive real numbers are interpreted as subvarieties of this square. I will also discuss the link between the arithmetic site and cyclic homology through its relation with the epicyclic site and our recent joint results on the cyclic homology interpretation of the Serre archimedean factors of arithmetic varieties.

Caterina Consani, Johns Hopkins University
The universal thickening of real and complex numbers: towards an archimedean analogue of Fontaine's theory.

The talk will introduce the universal thickening of real and complex numbers: these two constructions parallel, at the archimedean places, the construction of Fontaine's ring of p-adic periods BdR. (Joint work with A. Connes)

Michael R. Douglas, Stony Brook University
A survey of string theory and noncommutative geometry

Noncommutative geometry arises in various contexts in string theory: in noncommutative gauge theory, in the study of Dirichlet branes on tori and on Calabi–Yau manifolds, and in Matrix theory. We review the current status and explain some open puzzles.

Ron Douglas, Texas A&M University
Arveson's conjecture and the Grothendieck–Riemann–Roch theorem

In joint work with Xiang Tang and Guoliang Yu, techniques from several complex variables and partial differential equations are used to advance Arveson's conjecture and relate it to the Grothendieck–Riemann–Roch theorem.

Farzad Fathizadeh, University of Western Ontario
Scalar curvature, the Gauss–Bonnet theorem and the Einstein–Hilbert action for noncommutative tori

After the seminal work of A. Connes and P. Tretkoff on the Gauss-Bonnet theorem for the noncommutative 2-torus, there have been significant developments in understanding the local differential geometry of these noncommutative spaces equipped with curved metrics. In this talk, I will review a series of joint works with M. Khalkhali, in which we extend this result to general translation invariant conformal structures on noncommutative 2-tori, compute the scalar curvature, and prove the analogue of Weyl's law and Connes' trace theorem. Our final formula for the curvature matches precisely with the one computed independently by A. Connes and H. Moscovici. I will then report on our recent work on the computation of scalar curvature for noncommutative 4-tori (which involves intricacies due to violation of the Kähler condition). We show that metrics with constant curvature are extrema of the analogue of the Einstein–Hilbert action. A purely noncommutative feature in these works is the appearance of the modular automorphism from Tomita–Takesaki theory in the computations and final formulas for curvature.

Dan Freed, University of Texas, Austin
Differential K-theory and Dirac operators

I will begin with a brief general exposition of differential cohomology. Then I will specialize to differential K-theory and discuss several theorems relating pushforwards in differential K-theory to geometric invariants of families of Dirac operators.

Sherry Gong, Massachusetts Institute of Technology
The finite part of operator K-theory and traces on reduced C*-algebras for groups with rapid decay

This talk is about the part of the operator K-theory of groups arising from torsion elements in the group. We will see how idempotents arise from torsion elements in a group, and discuss the part of K-theory they generate, and in particular, how to detect such idempotents using traces. We conclude with a condition for when such elements can be detected in the case of groups of rapid decay. We further analyse traces on the reduced C*-algebras of hyperbolic groups and in doing so completely classify such traces.

Alexander Gorokhovsky, University of Colorado, Boulder
Higher analytic indices and a symbolic index pairing

An index theorem computing the pairing of an operator with an Alexander–Spanier cocycle was proven in the work of A. Connes and H. Moscovici on the Novikov conjecture. The goal of my talk is to reinterpret their theorem, extend the definition of higher indices to new situations, and to describe a theorem computing them in topological terms. This is a joint work with H. Moscovici.

Nigel Higson, Pennsylvania State University
The noncommutative geometry of tempered representations

In representation theory it is typical to concentrate on irreducible representations or at least close-to-irreducible representations. In noncommutative geometry it is typical to concentrate on representations that are projective (when viewed as modules over a group convolution algebra), or at least close-to-projective. The two perspectives meet in the theory of the discrete series representations of a reductive group, which are irreducible, but also projective over Harish-Chandra's convolution algebra of Schwartz functions. It is well known that noncommutative geometry has a lot to say about the discrete series (this includes pioneering work of Connes and Moscovici on index theory for homogeneous spaces). But in this talk I want to instead examine continuous series representations from a noncommutative-geometric viewpoint. An inspiration is Bernstein's discovery that, in the context of p-adic groups, parabolic induction carries projective modules to projective modules. A longer-term objective is to frame the Langlands classification within the context of noncommutative geometry, with possible applications to the Baum–Connes conjecture.

Michel Hilsum, CNRS, Institut de Mathématiques de Jussieu
Invariance of the Godbillon–Vey map by absolutely continuous homeomorphisms

The Godbillon–Vey map is an additive morphism from the K-theory group of the C*-algebra of a foliation to the scalar field. In this talk we shall address the invariance of this map under foliated homeomorphisms.

Masoud Khalkhali, University of Western Ontario
On the spectral action principle for Robertson–Walker metrics

The spectral action principle was introduced by Connes and Chamseddine in their study of the standard model of elementary particles. I will start by a brief review of this principle and in particular recall their calculation of the spectral action for the Dirac operator for Robertson–Walker metrics. I shall describe techniques from semi-classical analysis such as the Poisson summation formula, the Euler–Maclaurin summation formula and the Feynman–Kac formula that are employed in these calculations. I shall then report on ongoing joint work with F. Fathizadeh and A. Ghorbanpour where we extend these calculations to higher order terms.

Henri Moscovici, The Ohio State University
Hopf cyclic cohomology and characteristic classes
Talk abstract
Marcus Pflaum, University of Colorado, Boulder
Whitney functions and the real homotopy type of a semi-analytic set

In the talk we consider semi-analytic subsets of a real analytic manifold and their homology and real homotopy type. It is well-known that de Rham's Theorem does not hold true in general for singular spaces such as semi-analytic sets. We show that to remedy this one can replace the de Rham complex by the Whitney–de Rham complex to compute the singular homology of such sets. Beyond that, the Whitney–de Rham complex even determines the real homotopy type of a semi-analytic set, which extends a result by Sullivan for the de Rham complex on smooth manifolds. As an application we derive that the Hochschild homology of the differential graded algebra given by the Whitney–de Rham complex is isomorphic to the cohomology of the free loop space of the underlying space. The talk is based on joint work with B. Chriestenson.

Raphaël Ponge, Seoul National University
Noncommutative geometry and conformal geometry

In this talk I will report on a series of papers with Hang Wang, the aim of which is to study conformal geometry by using tools from noncommutative geometry, and more especially the twisted spectral triples recently introduced by Connes and Moscovici. In this talk I will present two main results. The first main result is the explicit construction of various conformal invariants as a consequence of an index formula in conformal-diffeomorphism invariant geometry. Incidentally, this provides us with a cyclic cohomology interpretation of previous results of Branson–Ørsted. The second main result is a reformulation of the Vafa–Witten inequality in conformal geometry. It is obtained as a consequence of a version of the Vafa–Witten inequality for twisted spectral triples. The latter result extends to twisted spectral triples a result of Moscovici for ordinary spectral triples. In particular, a notion of (rational) Poincaré duality for twisted spectral triples is paramount to the proof of the inequality. This notion of Poincaré duality also sheds new light on twisted spectral triples. (This is joint work with Hang Wang.)

Bahram Rangipour, University of New Brunswick
Lie–Hopf algebras and their Hopf cyclic cohomology

For any Lie–Hopf algebra we introduce a commutative DG-algebra which is quasi-isomorphic to the Hopf cyclic cohomology of the bicrossed product Hopf algebra. We investigate algebraic connection on this algebra in general and show that all geometric Lie–Hopf algebras admit such connection. We conclude that all known characteristic classes of foliations can be written explicitly as cyclic cohomology classes of the groupoid action algebra. At the end we give an example that does not quite fit in the picture and needs further developments.

Jonathan Rosenberg, University of Maryland
Levi-Civita connections for noncommutative tori

We show how to define Riemannian metrics and connections on a noncommutative torus in such a way that an analogue of Levi-Civita's theorem on the existence and uniqueness of a Riemannian connection holds. The major novelty is that we need to use two different notions of noncommutative vector field. Levi-Civita's theorem makes it possible to define Riemannian curvature using the usual formulas, without assuming a priori that the metric is a conformal deformation of a flat metric.

Dennis Sullivan, Stony Brook University
Numerical algorithms for ideal 3D fluid motion based on algebraic topology

At the continuum level, there are a skew-symmetric 3-form and a quadratic form on the space of 1-forms that are purely topological, but in the presence of a metric allow one to express fluid motion algebraically. These structures can be transferred to finite-dimensional grid approximations respecting non-linearity by making use of the idea of structures up to homotopy from algebraic topology. The goal of this work in progress is to construct consistent numerical algorithms that are typically stable for incompressible 3D fluid motion without friction.

Xiang Tang, Washington University in St. Louis
A higher index theorem for proper cocompact actions

In this talk I will desribe a cohomological formula for a higher index pairing between invariant elliptic differential operators and differentiable group cohomology classes. This index theorem generalizes the Connes–Moscovici L2–index theorem and its variants. If time permits, I will explain the extension to groupoids. This is joint work with Markus Pflaum and Hessel Posthuma.

Paula Tretkoff, Texas A&M University
An ε-knowledge introduction to some basic objects of commutative and noncommutative number theory

In this talk we give a very elementary introduction to the noncommutative 2-torus, its complex structure, classical elliptic curves and modular forms. We also recall the statement of the Riemann Hypothesis and briefly discuss the work of Bost–Connes from circa 1995 which lies at the core of a lot of subsequent noncommutative number theory. We make no attempt at completeness, technical correctness, nor at being up-to-the-minute, in this talk aimed at an audience with little or no knowledge of these objects.

Rufus Willett, University of Hawai'i
Expanders, exactness, and exotic completions

Expanders are (sequences of) highly connected, bounded degree graphs. Following work of Higson, Lafforgue and Skandalis, expanders are known to cause problems for Baum–Connes type conjectures. As expanders are generic among sequences of graphs, this suggests that one should expect a lot of problems. In this talk I’ll show that one can alter the C*-algebras used for the coarse Baum–Connes conjecture so that it becomes true for generic sequences of expanders; thus the problems are not as serious as one might think. I’ll also discuss what this has to do with exactness, leading into Paul Baum’s talk. Part of this talk is based on joint work with Paul Baum and Erik Guentner.

Zhizhang Xie, Texas A&M University
Higher rho invariants and the moduli space of positive scalar curvature metrics

Given a closed smooth manifold M which carries a positive scalar curvature metric, one can associate an abelian group P(M) to the space of positive scalar curvature metrics on this manifold. The group of all diffeomorphisms of the manifold naturally acts on P(M). The moduli group of positive scalar curvature metrics is defined to be the quotient abelian group of this action, i.e., the coinvariant of the action. In this talk, I will talk about how to use the higher rho invariant and the finite part of the K-theory of the group C*-algebra of the fundamental group of M to give a lower estimate of the rank of the moduli group. This is joint work with Guoliang Yu.