• Non-commutative Geometry and its Applications
• Workshop: Non-commutative geometry's interactions with mathematics

• Schedule
• Participants
• Recorded Talks

The non-commutative torus is commonly described as a cocycle quantization of the group (C*-) algebra of the abelian group ℤ². In the first part of the talk I will explain how, using the WBZ transform of solid state physics, finitely generated projective modules over the NC-torus can be interpreted as deformations of vector bundles on elliptic curves by the action of a 2-cocycle, provided that the deformation parameter of the NC-torus and the modular parameter of the elliptic curve satisfy a non-trivial relation. I will then discuss the relation between (formal) deformations of vector bundles on the torus and cochain twists based on the Lie algebra of the 3-dimensional Heisenberg group. Based on a joint work with G. Fiore and D. Franco.

We introduce a concept of noncommutative minimal surfaces in the Weyl algebra, and show that one may prove a noncommutative analogue of Weierstrass’ representation theorem. This result enables us to provide a multitude of explicit examples, appearing as analogues of classical minimal surfaces. Furthermore, we develop a theory of curvature on associated projective modules appearing naturally as tangent bundles in this context, and show that the curvature may be explicitly computed in several examples. Our results may potentially be interesting for physics since noncommutative analogues of harmonic maps appear as solutions to equations of motion in certain contexts.

Projectively invariant elliptic operators appear in the mathematical description of magnetic fields, in particular in some models of the fractional quantum Hall effect. From the geometric point of view, these operators give very interesting invariants analogous to those studied in L²-index theory for covering spaces, or more generally higher index theory.

In this talk, we will describe the construction of eta and rho invariants for Dirac operators, on the universal covering of a closed manifold, that are invariant under the projective action associated to a 2-cocycle of the fundamental group. We will see how to prove an Atiyah-Patodi-Singer index theorem in this setting, and discuss some of the geometric properties of the rho invariant.

(joint work with Charlotte Wahl)

This is a joint with Francesca Arici and Giovanni Landi.

We construct an analogue of the Gysin sequence for circle bundles, now for q-deformed lens spaces in the sense of Vaksman-Soibelman. Our proof that the sequence is exact relies heavily on the non commutative APS index theory of Carey, Phillips, Rennie and subsequent coworkers.

In recent work Debord and Skandalis realized pseudodifferential operators (on an arbitrary Lie groupoid G) as integrals of certain smooth kernels on the adiabatic groupoid of G. We propose an alternative definition of pseudodifferential calculi (including nonstandard calculi like the Heisenberg calculus) arising from tangent groupoids. The aim is to simplify the construction of such calculi, by reducing the problem to the construction of the appropriate tangent groupoid.

We show how to develop a Morse theory along the leaves of a measured foliation. In particular, we prove Morse inequalities for foliations and show how to compute the L²-cohomology of a foliation from the singularities of a leafwise Morse function.

I will present my joint work with M. Khalkhali in which we study curved geometry of the noncommutative four torus, where its flat geometry is perturbed by a Weyl conformal factor. We consider the Laplacian associated with the curved geometry and use Connes' pseudodifferential calculus to find explicit formulas for the terms in the small time asymptotic expansion of its heat kernel which correspond to the volume and the scalar curvature. The analog of the Einstein-Hilbert action is then considered and it is shown that flat metrics are the critical points for this action. We also define a noncommutative residue and prove the analog of Connes' trace theorem, which provides a convenient tool for computing the Dixmier trace of pseudodifferential operators of order -4 on the noncommutative four torus.

We construct the join of noncommutative Galois objects (quantum torsors) over a Hopf algebra H. To ensure that the join algebra enjoys the natural (diagonal) coaction of H, we braid the tensor product of the Galois objects. Then we show that this coaction is principal. Our examples are built from the noncommutative torus with the natural free action of the classical torus, and arbitrary anti-Drinfeld doubles of finite-dimensional Hopf algebras. The former yields a noncommutative deformation of a non-trivial torus bundle, and the latter a finite quantum covering. (Based on joint work with L. Dabrowski, T. Hadfield and E. Wagner.)

The Kasparov absorption (or stabilization) theorem states that any countably generated Hilbert C*-module is isomorphic to a direct summand in a standard module. In this talk, I will generalize this result by incorporating a densely defined derivation on the base C*-algebra. The extra compatibility assumptions needed are minimal: It will only be required that there exists a sequence of generators with inner products in the domain of the derivation. As an application, I will show how to construct densely defined connections (or correspondences) on Hilbert C*-modules. These connections can in turn be used to define selfadjoint and regular "lifts" of unbounded operators which act on an auxiliary Hilbert C*-module.

We discuss the asymptotic behavior of the eigenvalues of the Laplacian on a Riemannian compact foliated manifold when the metric is blown up in the directions normal to the leaves (in the adiabatic limit).

This problem can be considered as an asymptotic spectral problem on the leaf space of the foliation. One can use the ideas of noncommutative geometry of foliations to introduce some notions and make some conjectures related to the adiabatic limits. We review known results and discuss some open problems in this direction.

Familiar examples of bivariant homology theories include KK-theory and local cyclic homology. There is another one called noncommutative stable homotopy that is a universal example is a certain sense. They are defined on the category of noncommutative pointed compact spaces (or C*-algebras). I am going to talk about a method to generalize these constructions beyond noncommutative compact spaces that is extremely desirable from the viewpoint of homotopy theory. Some computations and applications will also be discussed.

(Joint work with A. Rennie)

By considering the general properties of approximate identities in Lipschitz algebras, we are able to present a unified approach to characterising completeness of spectral metric spaces, existence of connections on modules, and the lifting of Kasparov products to the unbounded category. In particular we show that given any two composable Kasparov classes, we can find unbounded representatives whose product can be constructed to yield an unbounded representative of the Kasparov product.

Using a homological invariant together with an obstruction class in a certain Ext²-group, we may classify objects in triangulated categories that have projective resolutions of length two. This invariant gives strong classification results for actions of the circle group on C*-algebras, C*-algebras over finite unique path spaces, and graph C*-algebras with finitely many ideals.

Extension of the M. Kontsevich formality theorem to gerbes necessitates extension of the notion of Deligne 2-groupoid to L^∞-algebras. For a differential graded Lie algebra g which is zero below the degree -1, the nerve of the Deligne 2-groupoid is homotopy equivalent to the Hinich simplicial set of g-valued differential forms. This leads to a formulation of the formality theorem in the presence of a grebe on the manifold in question. We will explain the notions involved in this result and sketch the application to the formality theorem.

The global symbol calculus for pseudodifferential operators on tori can be generalised to noncommutative tori. In this global approach, the quantisation map is invertible and traces are discrete sums. On the noncommutative torus, Fathizadeh and Wong had characterised the Wodzicki residue as the unique trace which vanishes on trace-class operators. In contrast, we build and characterise the canonical trace on classical pseudodifferential operators on a noncommutative torus, which extends the ordinary trace on trace-class operators. It can be written as a canonical discrete sum on the underlying toroidal symbols. On the grounds of this uniqueness result, we prove that in the commutative setup, this canonical trace on the noncommutative torus reduces to Kontsevich and Vishik's canonical trace which is thereby identified with a discrete sum. By means of the canonical trace, we derive defect formulae for regularised traces on noncommutative toris. The conformal invariance of the zeta-function at zero of the Laplacian on the noncommutative torus is then a straightforward consequence. This is based on joint work with Cyril Lvy and Carolina Neira Jimnez.

In this talk we will consider smooth actions of Lie groupoids on manifolds. Using cyclic cohomological methods, we will establish local higher index formulas for certain pseudodifferential-type operators.

The talk is based on joint work with H. Posthuma and X. Tang. We consider a proper cocompact action of a Lie groupoid and define a higher index pairing between invariant elliptic differential operators and smooth groupoid cohomology classes. We prove a cohomological index formula for this pairing by applying the van Est map and algebraic index theory. The meaning of the index pairing and our index formula is also discussed by considering examples.

A Cheeger space is a smoothly stratified pseudomanifold which is in general non-Witt but that admits an additional structure along the strata that allows for the definition of ideal boundary conditions. An interesting example is given by the reductive Borel-Serre compactification of a Hilbert modular surface. In this talk I will explain recent results, in collaboration with Albin, Leichtnam and Mazzeo and, in part, with Banagl, concerning the topology and the analysis of Cheeger spaces. I will concentrate on the geometric consequences of our analysis; in particular I will explain how it is possible to define on a Cheeger space a homology L-class and thus higher signatures à la Novikov. One of our main result is the stratified homotopy invariance of the higher signatures for Cheeger spaces with fundamental group satisfying the rational injectivity of the assembly map in K-theory; put it differently, I will show how the usual Strong Novikov Conjecture implies the stratified homotopy invariance of these higher signatures.

I will present a recent definition of correspondence for locally compact groupoids with Haar systems due to R. D. Holkar and based on previous work by M. Buneci and P. Stachura, which makes the groupoid C*-algebra construction a functor from the category of groupoids to the category of C*-algebras and implements many classical cases of induced representations. The case of Lie groupoids will be emphasized.

If A is a C*-algebra which is Poincaré dual to its opposite algebra, we show how Poincaré duality can be proved for the crossed product of A by an automorphism. Time permitting, I will discuss generalisations. This is joint work with Dave Robertson and Aidan Sims.

We give an equivariant index theorem in the framework of Heisenberg calculus for foliations, using zeta functions and excision in cyclic cohomology. In particular, this approach leads to a new proof of the transverse index theorem of Connes and Moscovici, in which we obtain directly a characteristic class formula for the index.

I will explain the construction of the recently introduced algebra of projective pseudodifferential operators related to a bundle of simple algebras. I will then give a proof that the Wodzicki residue of pseudodifferential projections vanishes in case the dimension of the manifold is odd. This is related to the vanishing of the eta residue density. Based on a joint work with Joerg Seiler.

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. Following the work of Weinberger and Yu, 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 talk is based on joint work with Guoliang Yu.

This talk will be a survey of an ongoing project on the relationship between noncommutative geometry and quantum group theory, in collaboration with Christian Voigt. We will begin by describing a construction of a Dirac-type class for the full flag variety of quantum SU(3) as a bounded Fredholm module. This is weaker than a spectral triple, but sufficient to obtain, for instance, the Baum-Connes Conjecture for the discrete quantum dual of SU_q(3). The construction will lead us to discuss principal series representations for the quantized complex semisimple Lie groups SL_q(n,C). We construct intertwining operators for these representations which are analogous to their classical counterparts.