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 K-theory (up to 2-torsion). Related Links http://www.math.jussieu.fr/~karoubi/ - Web page of Max Karoubi INI 1 10:00 to 11:00 U Kraehmer (Polish Academy of Sciences)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 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. INI 1 11:00 to 11:30 Coffee 11:30 to 12:30 G Yu ([Vanderbilt])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 R Ponge ([Toronto])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 B Rangipour ([Ohio State])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 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 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 T Brzezinski ([Wales, Swansea])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), 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). INI 1 12:30 to 13:30 Lunch at Wolfson Court 14:00 to 15:00 SL Woronowicz ([University of Warsaw])$GL_q (2, I\!\!R)$ 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