4/23/07 ======================================================================== Fifth Spring Institute on Noncommutative Geometry and Operator Algebras May 7-16, 2007, Vanderbilt University ======================================================================== Abstracts ========= Claire Debord, Universite Blaise Pascal, Clermont-Ferrand Title : On index theory for stratified pseudomanifolds Abstract : We will see how one can define a good notion of tangent space for a general stratified pseudomanifold X. The tangent space will no longer be a vector bundle but a groupoid whose C*-algebra is Poincare dual to the algebra C(X) of continuous functions on X. We will show how, in the case of a conical pseudomanifold, this notion of tangent space enables us to get a topological index theorem. Precisely, we will see a variant proof of Atiyah-Singer index theorem using groupoids and we will show that all the ingredients (analytical index, Thom isomorphism and topological index) as well as the proof of the index theorem generalize easily to conical pseudomanifolds as soon as one uses our notion of tangent space. --------------------------------------------------------------------------- Alexander Gorokhovsky, University of Colorado, Boulder Formal deformations of gerbes on smooth manifolds Abstract: We begin by reminding the definitions of gerbes and their formal deformations and then proceed to give an explicit construction of a differential graded Lie algebra controlling these deformations. This is a joint work with P. Bressler, R. Nest and B. Tsygan. --------------------------------------------------------------------------- Atabey Kaygun, University of Western Ontario Masoud Khalkhali, University of Western Ontario From cyclic cohomology to Hopf cyclic cohomology Abstract: In this series of 8 lectures we are planning to give an introduction to cyclic cohomology and Hopf cyclic cohomology. The lectures will not assume any prior knowledge of the subject and we shall start at a fairly basic level with cyclic cohomology. The emphasis will be on examples, main ideas and general patterns as well as applications. References include: 1. Alain Connes, Noncommutative differential geometry, Publ. Math. IHES no. 62 (1985), 41-144. (ftp://ftp.alainconnes.org/noncommutative_differential_geometry.pdf) 2. Alain Connes, Noncommutative geometry, Academic Press (1994). (ftp://ftp.alainconnes.org/book94bigpdf.pdf) 3. Alain Connes and Henri Moscovici, Cyclic cohomology and the transverse index theorem, Comm. Math. Phys. 198 (1998), no. 1, 199--246. (http://arxiv.org/abs/math.DG/9806109) The first 5 lectures will be delivered by Masoud Khalkhali and the last 3 lectures by Atabey Kaygun. Topics covered will include: - Examples of cyclic cocycles in geometry, analysis, and physics; Connes-Chern character, - Computational tools: Connes' spectral sequence, cyclic modules, smooth algebras, relations of cyclic cohomology with de Rham cohomology, group and Lie algebra cohomology, - Hopf algebras and quantum groups in noncommutative geometry, Connes-Moscovici's Hopf cyclic cohomology, - The evolution of Hopf cyclic cohomology from Connes-Moscovici to the HKRS formalism and the bivariant Hopf cyclic cohomology, - Morita invariance, excision, cup products and characteristic classes in Hopf cyclic cohomology. --------------------------------------------------------------------------- Ulrich Kraehmer, IMPAN, Warsaw Title: On piecewise trivial Hopf-Galois extensions Abstract: Faithfully flat Hopf-Galois extensions provide noncommutative geometry analogues of principal fibre bundles. The base and total space of the bundle become replaced by associative algebras and the structure group by a Hopf algebra. Using fibre products and more generally flabby sheaves of algebras as substitutes of principal fibre bundles that are glued by trivial pieces, we define a notion of local triviality of comodule algebras and prove that these are faithfully Hopf-Galois. (based on joint work with P.Hajac, R.Matthes and B.Zielinski) --------------------------------------------------------------------------- Giovanni Landi, University of Trieste; Andrzej Sitarz, Jagiellonian University Mini-course: An Introduction to equivariant spectral triples Outline: 1. Motivations (Classical spin geometry, Dirac operator) and definition (all sign tables, reality structure) 2. Fredholm modules <-> Spectral triples Properties of spectral triples: regularity, dimension spectrum, Poincare' duality .... 3. Spin manifolds are spectral triples - idea of the proof Sketch (idea) of the reconstruction theorem ( commutative spectral triples are spin manifolds) 4. Some examples of NC spectral triples (finite, Moyal, noncommutative tori, isospectral noncommutative manifolds) 5. Hopf algebras as symmetries. Equivariance for spectral triples 6. The spectral triple of SU_q(2); detailed construction 7. Local index formula, abstract algebra of PDO --------------------------------------------------------------------------- Matilde Marcolli, Max-Planck Institute, Bonn Noncommutative geometry and the standard model with neutrino mixing Abstract: This talk is based on the joint work with Ali Chamseddine and Alain Connes (hep-th/0610241), where we present an effective unified theory based on noncommutative geometry for the standard model with neutrino mixing, minimally coupled to gravity. The unification is based on the symplectic unitary group in Hilbert space and on the spectral action. It yields all the detailed structure of the standard model with several predictions at unification scale. Besides the familiar predictions for the gauge couplings as for GUT theories, it predicts the Higgs scattering parameter and the sum of the squares of Yukawa couplings. From these relations one can extract predictions at low energy, giving in particular a Higgs mass around 170 GeV and a top mass compatible with present experimental value. The geometric picture that emerges is that space-time is the product of an ordinary spin manifold (for which the theory would deliver Einstein gravity) by a finite noncommutative geometry F. The discrete space F is of KO-dimension 6 modulo 8 and of metric dimension 0, and accounts for all the intricacies of the standard model with its spontaneous symmetry breaking Higgs sector. --------------------------------------------------------------------------- Sergey Neshveyev, University of Oslo On the Drinfeld associator in the real parameter case. Abstract: I will review results from Drinfeld's theory of quasi-Hopf algebras needed for the next talk by Lars Tuset. Subtle points in the passage from formal to real parameter will be emphasized. --------------------------------------------------------------------------- Michael Puschnigg, Universite de la Mediterranee, Marseille Cyclic cohomology and the Chern-Connes character. Abstract: Cyclic cohomology was invented as a target for generalized Chern characters on various forms of K-theory. We will report on cyclic cohomology theories for topological algebras and will discuss the construction and basic properties of the Chern-Connes character on (bivariant) operator K-theory. --------------------------------------------------------------------------- Bahram Rangipour, Ohio State University Hopf cyclic cohomology of Connes-Moscovici Hopf algebras Abstract: We continue our investigation on Hopf cyclic cohomology of bicrossed product Hopf algebras by introducing a new computational tool. We apply it in, both absolute and relative, cohomology of the CM Hopf algebras to obtain a basis for these cohomologies. As another application we show that all non-periodic cyclic cohomology groups of CM Hopf Hopf algebras are finite dimensional and present for the first time a precise basis of the classes in codimension 1 case. --------------------------------------------------------------------------- Georgy Sharygin, ITEP, Moscow Chern classes of principal bundles and twisting cochains. Abstract: In this talk we shall give few formulas, expressing the Chern classes of a principale bundle in terms of the group cocycle, which determines it. This approach is based on the notion of "twisting cochain" of a bundle. In particular, we show that there exist formulas, expressing all Chern classes of a bundle in terms of its cocycle. --------------------------------------------------------------------------- Lars Tuset, Oslo Quantum groups are non-commutative manifolds in Connes' sense. Abstract: For the q-deformation G_q, 0