# Workshop Programme

## for period 31 July - 4 August 2006

### Noncommutative Geometry and Cyclic Cohomology

31 July - 4 August 2006

Timetable

 Monday 31 July 08:30-10:00 Registration 10:00-11:00 Plymen, R (Manchester) A noncommutative geometry approach to the representation theory of p-adic groups Sem 1 We will begin by recalling the periodic cyclic homology of the Hecke algebra of GL(n). The Langlands parameters occur naturally at this point [joint work with J Brodzki]. This has led us to the formulation of a conjecture, according to which there are simple geometric structures underlying the representation theory of p-adic groups. We will illustrate this conjecture with examples, including GL(n) and the exceptional group G_2. We will attempt to relate our conjecture to the Langlands-Deligne-Lusztig parameters (s,u,\rho). [joint work with Anne-Marie Aubert and Paul Baum]. 11:00-11:30 Coffee 11:30-12:30 Wodzicki, M (California) What do we know, and what we do not, about exotic traces Sem 1 12:30-13:30 Lunch at Wolfson Court 14:00-15:00 Khalkhali, M (Western Ontario) Bivariant Hopf cyclic cohomology I Sem 1 Hopf cyclic cohomology is a new cohomology theory for Hopf algebras. It was introduced by Connes and Moscovici in 1998 in the course of their study of transverse index theory on foliated manifolds. Its relation to cyclic cohomology is to some extent similar to the relation between de Rham cohomology and group and Lie algebra cohomology, though this was not the original motivation and this analogy by itself can be of no help in defining it! In the first part of this talk I will give a general introduction to Hopf cyclic cohomology, its origins, the progress made in the last few years, and our current understanding of the subject. In the second half I will present new joint work with Atabey Kaygun where a bivariant Hopf cyclic theory is developed. 15:00-15:30 Tea 15:30-16:30 Pflaum, M (John Wolfgang Goethe University) Relative pairing in cycle cohomology and divisor flows Sem 1 In the talk I elaborate on joint work with H. Moscovici and M. Lesch. We show that Melrose's divisor flow and its generalizations by Lesch and Pflaum are invariants of K-theory classes for algebras of parametric pseudodifferential operators on a closed manifold, obtained by pairing the relative K-theory modulo the symbols with the cyclic cohomological character of a relative cycle constructed out of the regularized operator trace together with its symbolic boundary. This representation gives a clear and conceptual explanation to all the essential features of the divisor flow -- its homotopy nature, additivity and integrality. It also provides a cohomological formula for the spectral flow along a smooth path of self-adjoint elliptic first order differential operators, between any two invertible such operators on a closed manifold. 17:30-18:30 Wine Reception 18:45-19:30 Dinner at Wolfson Court (Residents only)
 Tuesday 1 August 09:00-10:00 Connes, A (IHES) Dimensional regularization, chiral anomalies, and the local index formula in noncommutative geometry Sem 1 10:00-11:00 Moscovici, H (Ohio State) Noncommutative geometry on Q-spaces of Q-lattices Sem 1 This is joint work, in progress, with A. Connes on the complex geometry of the quotient space of rank 2 Q-lattices modulo commensurability. It builds on our prior work on modular Hecke algebras and their Hopf symmetry, and on the Connes-Marcolli C*-algebraic framework for Q-lattice spaces. The emerging spectral-geometric picture, modeled on the transverse geometry of a generic codimension 1 foliation, has notable arithmetic overtones. 11:00-11:30 Coffee 11:30-12:30 Baum, P (Pennsylvania State) Noncommutative algebraic geometry and the representation theory of p-adic groups Sem 1 Non-commutative geometry begins with the Gelfand theorem asserting that commutative C* algebras and locally compact Hausdorff topological spaces are the same thing. Another classical theorem states that unital commutative finitely-generated nilpotent-free algebras (over the complex numbers) is the same thing as complex affine algebraic varieties. This can be taken as the starting point for non-commutative algebraic geometry. Based on this point of view, this talk states a conjecture within the representation theory of p-adic groups. The idea of the conjecture is that a simple geometric structure underlies many delicate and inticate results in this representation theory. The above is joint work with Anne-Marie Aubert and Roger Plymen. 12:30-13:30 Lunch at Wolfson Court 14:00-15:00 Beggs, EJ (Swansea) A noncommutative sheaf theory Sem 1 Modules with flat connection over algebras with differential structure have several properties in common with sheaf theory over topological spaces. In particular they admit long exact sequences for a cohomology theory. However sheaf theory is best described by looking at its applications, one of which is the Serre spectral sequence for a topological fibration. In the case where the cohomology of the fiber is not a trivial bundle over the base, sheaf cohomology is required to make sense of the resulting cohomology theory. I will describe a noncommutative version of the Serre spectral sequence for de Rham cohomology, which uses flat connections. This will effectively specify a definition of differential fibration in noncommutative geometry. I will then consider examples, which are quantum homogenous spaces. 15:00-15:30 Tea 15:30-16:30 Sharygin, G (ITEP) Bivariant cyclic (co)homology and paitings in Hopf-type cyclic (co)homology with coefficients Sem 1 In the recent years there appeared a number of papers, generalising the Connes-Moscovici's construction of cyclic cohomology of Hopf algebras. It turned out that in this case it is possible to introduce certain coefficients module in the cyclic cohomology (see, e.g. [1]). In the present talk (poster presentation) we investigate further this construction: we show how the X-complex formalism of Cuntz and Quillen (see [2]) can be extended to embrace the coefficients, and use this result to introduce the bivariant cyclic theory with coefficients. It turns out that the composition product in bivariant theory can be used to introduce the pairing between cohomology of an algebra and a coalgebra. We show, that this construction is a generalisation of Crainic's pairing ([3]) and that under certain conditions on the coalgebra it coincides with the pairing, introduced by Khalkhali and Rangipour ([4]). This talk is partly based on the paper [5] of Igor Nikonov and G.Sh.. [1] P.M.Hajac, M.Khalkali, B.Rangipour, M.Sommerhaeuser: Hopf-cyclic homology and cohomology with coefficients; C. R. Math. Acad. Sci. Paris 338, (2004), no. 9, 667-672 (also available as preprint arXiv:math.KT/0306288 v.2) [2] J.Cuntz, D.Quillen: Cyclic homology and nonsingularity; J. Amer. Math. Soc., 8, n.2 (1995) 373-442 [3] M.Crainic: Cyclic cohomology of Hopf algebras; J. Pure Appl. Algebra, 166, (2002) 29-66 [4] M.Khalkhali, B.Rangipour: Cup Products in Hopf-Cyclic Cohomology; available as preprint at arXiv:math.QA/0411003 v1 [5] I.Nikonov, G.Sharygin: On the Hopf-type Cyclic Cohomology with Coeﬃcients, C* -algebras and Elliptic Theory Trends in Mathematics, 203–212, 2006 Birkhaeuser Verlag Basel/Switzerland 18:45-19:30 Dinner at Wolfson Court (Residents only)
 Wednesday 2 August 09:00-10:00 Karoubi, M (Universite Paris 7 - Denis Diderot) A new homology theory on rings: the stabilized Witt groups Sem 1 We define a new homology theory defined on discrete rings with involution (even when 2 is not invertible). This theory sastisfies excision, homotopy invariance, periodicity (of period 4) and other nice properties. It is closely related to Balmer's theory and to surgery groups. When A is a real or complex C*-algebra, we recover topological K-theory (up to 2-torsion). Related Links http://www.math.jussieu.fr/~karoubi/ - Web page of Max Karoubi 10:00-11:00 Kraehmer, U (Polish Academy of Sciences) On the Hochschild homology of quantum groups Sem 1 This talk reports on two papers written in collaboration with T.Hadfield. In the first we computed the Hochschild homology of the standard quantisation of SL(2) with coefficients in bimodules obtained from the algebra itself by twisting the multiplication on one side by an automorphism. It turned out that precisely for Woronowicz's modular automorphism there is a unique nontrivial Hochschild class in degree 3=dim(SL(2)), in contrast to the untwisted case where all homologies were known to vanish in degrees greater than one. In the second paper we generalised this to SL(N) by showing that the quantisations satisfy van den Bergh's analogue of Poincare duality in Hochschild (co)homology. This was extended recently by K.Brown and J.Zhang to all Noetherian Artin-Schelter Gorenstein Hopf algebras. These results clarify the purely homological relevance of the twisted coefficients whose study was originally motivated by their relation to the theory of covariant differential calculi over quantum groups. 11:00-11:30 Coffee 11:30-12:30 Yu, G (Vanderbilt) Geometrization of the Novikov conjecture for residually finite groups Sem 1 12:30-13:30 Lunch at Wolfson Court 14:00-15:00 Pal, AK (Indian Statistical Institute) Equivariant spectral triples for odd dimensional quantum spheres Sem 1 15:00-15:30 Tea 15:30-16:30 Ponge, R (Toronto) Noncommutative residue and applications in Riemannian and CR geometry Sem 1 In this talk we will present applications of the noncommutative residue in Riemannian and CR geometry. It will be divided into 3 main parts. 1. Lower dimensional volumes in Riemannian geometry. Given a Riemannian manifold (M^n,g) a natural geometric is how to define the k'th dimensional volume of $M$ for $k=1,...,n$. Classical Riemannian geometry provides us with an answer for $k=n$ only. We will explain that by extending an idea of A. Connes we can make use of the noncommutative residue for classical PsiDO's and of the framework of noncommutative geometry to define in a purely differential- geometric fashion the k'th dimensional volumes for any k. 2. Lower dimensional volumes in Riemannian geometry. CR structures naturally arise in varous contexts. We also can define lower dimensional volumes in CR geometry. This involves constructing a noncommutative residue trace for the Heisenberg calculus, which is the relevant pseudodifferential calculus at stake in the CR setting. 3. New invariants for CR and contact manifolds. We can define new global invariants of CR and contact structures in terms of noncommutative residues of various geometric projections in the Heisenberg calculus. This allows us to recover recent results of Hirachi and Boutet de Monvel and to answer a question of Fefferman. 18:45-19:30 Dinner at Wolfson Court (Residents only)
 Thursday 3 August 09:00-10:00 Nest, R (Copenhagen) Baum-Connnes isomorphism for certain quantum groups Sem 1 10:00-11:00 Tsygan, B (Northwestern) Real Riemann-Roch theorem Sem 1 I will present my recent joint work with Bressler, Kapranov and Vasserot on a higher Riemann-Roch theorem for families and its applications. 11:00-11:30 Coffee 11:30-12:30 Gorokhovsky, A (Colorado at Boulder) Formal deformations of gerbes and etale groupoids Sem 1 This is a joint work with P.Bressler, R. Nest and B. Tsygan. We give an explicit description of the differential graded Lie algebra which controls the deformation theory of gerbes and etale groupoids. 12:30-13:30 Lunch at Wolfson Court 14:00-15:00 Brodzki, J (Southampton) Property A, exactness of uniform Roe algebra, and uniform embeddings in groups Sem 1 Property A for metric spaces has been introduced by Guoliang Yu as a weaker form of the Folner condition characterizing amenable groups. This property admits a number of equivalent formulations, and it can be described in terms of certain operator algebras associated with the space. For example, a discrete group G satisfies property A if and only if its reduced C*-algebra is exact. In this talk we introduce the notion of a partial translation structure T on a metric space X, which provides an analogue of a left-right action of a group on itself. We associate a C*-algebra C*(T), which is a subalgebra of the uniform Roe algebra of X, and use it to relate the exactness of the uniform Roe algebra of X to property A. We introduce an invariant of metric spaces which provides an obstruction to the existence of a uniform embedding in a group. This talk reports on a joint work with Graham A. Niblo and Nick Wright. Related Links http://front.math.ucdavis.edu/math.OA/0603621 15:00-15:30 Tea 15:30-16:30 Kaygun, A (Western Ontario) Hopf equivariant cyclic (co)homology and Morita invariance Sem 1 We will define a Hopf equivariant cyclic (co)homology theory for Hopf module algebras and coalgebras with coefficients, and prove that our theory is Morita invariant. We will also give several spectral sequences relating our equivariant theory with Hopf cyclic (co)homology of module algebras and coalgebras. 19:30-18:00 Conference Dinner in the Dining Hall at Trinity Hall
 Friday 4 August 09:00-10:00 Rangipour, B (Ohio State) Periodic Hopf cyclic cohomology of bicrossed product Hopf algebras Sem 1 We develop intrinsic tools for computing the periodic Hopf cyclic cohomology of Hopf algebras related to transverse symmetry. Besides the Hopf algebra found by Connes and the second author in their work on the local index formula for transversely hypoelliptic operators on foliations, this family includes its `Schwarzian' quotient, on which the Rankin-Cohen universal deformation formula is based, the extended Connes-Kreimer Hopf algebra related to renormalization of divergences in QFT, as well as a series of cyclic coverings of these Hopf algebras, motivated by the treatment of transverse symmetry for nonorientable foliations. Related Links http://www.arxiv.org/abs/math.QA/0602020 - Cyclic cohomology of Hopf algebras of transverse symmetries: the codimension 1 case 10:00-11:00 Hadfield, T (Queen Mary, London) Braided homology of quantum groups Sem 1 11:00-11:30 Coffee 11:30-12:30 Brzezinski, T (Wales, Swansea) Flat connections and comodules Sem 1 This talk is motivated by a recent paper [A Kaygun and M Khalkhali, Hopf modules and noncommutative differential geometry, Lett. Math. Phys. 76 (2006), 77-91] in which Hopf modules appearing as coefficients in Hopf-cyclic cohomology are interpreted as modules with flat connections. We start by describing how all the algebraic structure involved in a universal differential calculus fits in a natural way into the notion of a coring (or a coalgebra in the category of bimodules). We recall the theorem of Roiter [A.V. Roiter, Matrix problems and representations of BOCS's. [in:] Lecture Notes in Mathematics, vol. 831, Springer-Verlag, Berlin and New York, 1980, pp. 288-324] in which a bijective correspondence is established between semi-free differential graded algebras and corings with a grouplike element. A brief introduction to the theory of comodules is given and the theorem establishing a bijective correspondence between comodules of a coring with a grouplike element and flat connections (with respect to the associated differential graded algebra) is given [T Brzezinski, Corings with a grouplike element, Banach Center Publ., 61 (2003), 21-35]. Finally we specialise to corings which are built on a tensor product of algebra and a coalgebra. Such corings are in one-to-one correspondence with so-called entwining structures, and their comodules are entwined modules. The latter include all known examples of Hopf-type modules such as Hopf modules, relative Hopf modules, Long dimodules, Doi-Koppinen and alternative Doi-Koppinen modules. In particular they include Yetter-Drinfeld and anti-Yetter-Drinfeld modules and their generalisations, hence all the modules of interest to Hopf-cyclic cohomology. In this way the interpretation of the latter as modules with flat connections is obtained as a corollary of a more general theory. (We hope to make the talk as accessible to the non-commutative geometry community as possible. In particular we hope to concentrate only on these aspects of the coring and comodule theory which should be of interest and appeal to non-commutative geometers). 12:30-13:30 Lunch at Wolfson Court 14:00-15:00 Woronowicz, SL (University of Warsaw) $GL_q (2, I\!\!R)$ and $SL_q(2,I\!\!R)$ as locally compact quantum groups Sem 1 15:00-15:30 Tea 18:45-19:30 Dinner at Wolfson Court (Residents only)