Noncommutative Geometry and Cyclic Cohomology
Monday 31st July 2006 to Friday 4th August 2006
08:30 to 10:00  Registration  
10:00 to 11:00 
R Plymen ([Manchester]) A noncommutative geometry approach to the representation theory of padic groups 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 padic 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 LanglandsDeligneLusztig parameters (s,u,\rho). [joint work with AnneMarie Aubert and Paul Baum]. 
INI 1  
11:00 to 11:30  Coffee  
11:30 to 12:30 
M Wodzicki ([California]) What do we know, and what we do not, about exotic traces 
INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
14:00 to 15:00 
Bivariant Hopf cyclic cohomology I 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. 
INI 1  
15:00 to 15:30  Tea  
15:30 to 16:30 
M Pflaum ([John Wolfgang Goethe University]) Relative pairing in cycle cohomology and divisor flows 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 Ktheory classes for algebras of parametric pseudodifferential operators on a closed manifold, obtained by pairing the relative Ktheory 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 selfadjoint elliptic first order differential operators, between any two invertible such operators on a closed manifold. 
INI 1  
17:30 to 18:30  Wine Reception  
18:45 to 19:30  Dinner at Wolfson Court (Residents only) 
09:00 to 10:00  Dimensional regularization, chiral anomalies, and the local index formula in noncommutative geometry  INI 1  
10:00 to 11:00 
Noncommutative geometry on Qspaces of Qlattices This is joint work, in progress, with A. Connes on the complex geometry of the quotient space of rank 2 Qlattices modulo commensurability. It builds on our prior work on modular Hecke algebras and their Hopf symmetry, and on the ConnesMarcolli C*algebraic framework for Qlattice spaces. The emerging spectralgeometric picture, modeled on the transverse geometry of a generic codimension 1 foliation, has notable arithmetic overtones. 
INI 1  
11:00 to 11:30  Coffee  
11:30 to 12:30 
Noncommutative algebraic geometry and the representation theory of padic groups Noncommutative 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 finitelygenerated nilpotentfree algebras (over the complex numbers) is the same thing as complex affine algebraic varieties. This can be taken as the starting point for noncommutative algebraic geometry. Based on this point of view, this talk states a conjecture within the representation theory of padic 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 AnneMarie Aubert and Roger Plymen. 
INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
14:00 to 15:00 
A noncommutative sheaf theory 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. 
INI 1  
15:00 to 15:30  Tea  
15:30 to 16:30 
Bivariant cyclic (co)homology and paitings in Hopftype cyclic (co)homology with coefficients In the recent years there appeared a number of papers, generalising the ConnesMoscovici'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 Xcomplex 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: Hopfcyclic homology and cohomology with coefficients; C. R. Math. Acad. Sci. Paris 338, (2004), no. 9, 667672 (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) 373442 [3] M.Crainic: Cyclic cohomology of Hopf algebras; J. Pure Appl. Algebra, 166, (2002) 2966 [4] M.Khalkhali, B.Rangipour: Cup Products in HopfCyclic Cohomology; available as preprint at arXiv:math.QA/0411003 v1 [5] I.Nikonov, G.Sharygin: On the Hopftype Cyclic Cohomology with Coeﬃcients, C* algebras and Elliptic Theory Trends in Mathematics, 203212, 2006 Birkhaeuser Verlag Basel/Switzerland 
INI 1  
18:45 to 19:30  Dinner at Wolfson Court (Residents only) 
09:00 to 10:00 
M Karoubi ([Université Paris 7  Denis Diderot]) A new homology theory on rings: the stabilized Witt groups 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 Ktheory (up to 2torsion). Related Links

INI 1  
10:00 to 11:00 
On the Hochschild homology of quantum groups 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 ArtinSchelter 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. 
INI 1  
11:00 to 11:30  Coffee  
11:30 to 12:30  Geometrization of the Novikov conjecture for residually finite groups  INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
14:00 to 15:00 
AK Pal ([Indian Statistical Institute]) Equivariant spectral triples for odd dimensional quantum spheres 
INI 1  
15:00 to 15:30  Tea  
15:30 to 16:30 
Noncommutative residue and applications in Riemannian and CR geometry
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. 
INI 1  
18:45 to 19:30  Dinner at Wolfson Court (Residents only)  INI 1 
09:00 to 10:00  BaumConnnes isomorphism for certain quantum groups  INI 1  
10:00 to 11:00 
Real RiemannRoch theorem I will present my recent joint work with Bressler, Kapranov and Vasserot on a higher RiemannRoch theorem for families and its applications. 
INI 1  
11:00 to 11:30  Coffee  
11:30 to 12:30 
Formal deformations of gerbes and etale groupoids 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. 
INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
14:00 to 15:00 
Property A, exactness of uniform Roe algebra, and uniform embeddings in groups 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 leftright 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 
INI 1  
15:00 to 15:30  Tea  
15:30 to 16:30 
A Kaygun ([Western Ontario]) Hopf equivariant cyclic (co)homology and Morita invariance 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. 
INI 1  
19:30 to 18:00  Conference Dinner in the Dining Hall at Trinity Hall 
09:00 to 10:00 
Periodic Hopf cyclic cohomology of bicrossed product Hopf algebras 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 RankinCohen universal deformation formula is based, the extended ConnesKreimer 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

INI 1  
10:00 to 11:00 
T Hadfield ([Queen Mary, London]) Braided homology of quantum groups 
INI 1  
11:00 to 11:30  Coffee  
11:30 to 12:30 
Flat connections and comodules This talk is motivated by a recent paper [A Kaygun and M Khalkhali, Hopf modules and noncommutative differential geometry, Lett. Math. Phys. 76 (2006), 7791] in which Hopf modules appearing as coefficients in Hopfcyclic 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, SpringerVerlag, Berlin and New York, 1980, pp. 288324] in which a bijective correspondence is established between semifree 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), 2135]. Finally we specialise to corings which are built on a tensor product of algebra and a coalgebra. Such corings are in onetoone correspondence with socalled entwining structures, and their comodules are entwined modules. The latter include all known examples of Hopftype modules such as Hopf modules, relative Hopf modules, Long dimodules, DoiKoppinen and alternative DoiKoppinen modules. In particular they include YetterDrinfeld and antiYetterDrinfeld modules and their generalisations, hence all the modules of interest to Hopfcyclic 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 noncommutative 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 noncommutative geometers). 
INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
14:00 to 15:00  $GL_q (2, I\!\!R)$ and $SL_q(2,I\!\!R)$ as locally compact quantum groups  INI 1  
15:00 to 15:30  Tea  
18:45 to 19:30  Dinner at Wolfson Court (Residents only)  INI 1 