This is a joint seminar with the universities Blaise Pascal in Clermont-Ferrand and Paul Verlaine in Metz. This seminar, which takes about once a month on Fridays at 2pm , brings together geometers, algebraists and analysts, at the University of Potsdam, Building 2 Room 0.15 .
Sessions:
7 June 2013 14:00 : Frédéric Patras (CNRS-Université de Nice) About logarithmic derivatives
Abstract: The Dynkin idempotent plays a fundamental role in the theory of Lie algebras where it encodes logarithmic derivatives as well as in many applications of the theory of noncommutative representation of symmetric algebras, bases of Lie algebras, Campbell-Hausdorff formulae, quantum field theory... I shall give a brief presentation of the classical theory of Dnykin operators and of recent results (joint work with M. Bauer, R. Chetrite, K. Ebrahimi-Fard and F. Menous).
5 June 2013 9:00 : Dominique Manchon (University of Clermont-Ferrand) Combinatorics of the Magnus expansion.
Abstract: Consider the first-order homogeneous differential equation X'=AX with initial value X(0)=1, for functions with values in a unital, not necessarily commutative algebra. The logarithm of the solution is expressed as a formal series known as the Magnus expansion, after W. Magnus (1954). We will describe the various algebraic structures underlying this expansion, which involves Rota-Baxter, dendriform, pre-Lie and tridendriform algebras. This is a survey of several papers with Kurusch Ebrahimi-Fard.
5 June 2013 10:30 : Susama Argawala (University of Hamburg) Combinatorial Dyson Schwinger equations and applications
Abstract: The Dyson Schwinger equations define important relations for perturbative field theories. In this talk, I review the combinatorial avatar of these relations in the context of rooted trees. I develop their applications to field theory calculations, multiple polylogarithms, and informations theory.
10 May 2013 : René Schulz (Göttingen) Global Fourier integral operators via tempered oscillatory integrals with inhomogeneous phase functions
Abstract: The theory of global Fourier integral operators is a field of active research, with many open questions. In our approach, we study certain families of oscillatory integrals, parametrised by phase functions and amplitude functions globally defined on the Euclidean space, which give rise to tempered distributions, avoiding the standard homogeneity requirement on the phase function. The singularities of these distributions are described both from the point of view of the lack of smoothness as well as with respect to the decay at infinity. In particular, the latter will depend on a version of the set of stationary points of the phase function, including elements lying at the boundary of the radial compactification of the Euclidean space. We then consider classes of global Fourier integral operators on the Euclidean space, defined in terms of kernels of the form of such oscillatory integrals. As an example we consider the solution operator of the Klein Gordon equation. This talk is based on joint work with Sandro Coriasco from the University of Torino, Italy.
1 March 2013 : Olivier Gabriel (Göttingen) Lie group actions, spectral triples and generalised crossed products
Abstract: The aim of this talk is to generalise the constructions of spectral triples on noncommutative tori and Quantum Heisenberg Manifolds (QHM) to broader settings. After a few reminders about noncommutative tori and spectral triples, we prove that an ergodic action of a compact Lie group G on a unital C*-algebra A yields a natural spectral triple structure on A. In the second part, we investigate "permanence properties" for the previous sort of spectral triples. We first introduce the notion of Generalised Crossed Product (GCP) and illustrate it by the case of QHM. A GCP contains a sub-C*-algebra called its "basis". A spectral triple on the basis can induce a spectral triple on the GCP, under some assumptions which we make explicit. This talk is based on work in progress in collaboration with M. Grensing. If time permits, we will relate these new results to our previous work in this direction.
1 February 2013 : Sara Azzali (Paris VII) R/Z K-theory and the relative eta invariant
Abstract: With a flat vector bundle on a compact manifold M one can associate a class in the K-theory of M with R/Z coefficients. A theorem by Atiyah, Patodi and Singer proves that the pairing of this class with a K-homology class [D] is equal, in R/Z, to the relative eta invariant of D. As suggested in the original paper by Atiyah-Patodi-Singer, the R/Z K-theory group also has a von Neumann-algebraic description. It is in this setting that we propose a canonical construction for the K-theory class associated with a flat vector bundle, which generalizes the APS's one. We compute its pairing with [D] as a Kasparov product, recovering the result of APS, along with the equality to a type II spectral flow, originally proved by Douglas-Hurder-Kaminker et al..
7 December 2012 : Charlotte Wahl (Hannover) Rho-invariants and the classification of differential structures on closed manifolds
Abstract: In her talk, Sara Azzali explained the use of rho-invariants associated to the spin Dirac operator for the classification of metrics with positive scalar curvature. Analogously, rho-invariants associated to the signature operator can be used to distinguish differential structures on closed manifolds. However, since the signature operator is not invertible in general, their study tends to be more difficult. I will discuss three types of rho-invariants - the L2-rho-invariants of Cheeger and Gromov, Lott's higher rho-invariants and the rho-invariants associated to 2-cocycles studied by Sara Azzali and myself and I will explain what is known about their properties for the signature operator.
16 November 2012 : Sara Azzali (Paris VII) Eta invariants and positive scalar curvature
Abstract: The Atiyah-Patodi-Singer index theorem and eta invariants of Dirac operators can be used to distinguish an infinite number of "different" metrics with positive scalar curvature on a spin manifold. We shall first explain the basic ideas of these classical results (due to Gromov-Lawson, Botvinnik-Gilkey, Leichtnam-Piazza and Piazza-Schick). We shall further prove a new result for manifolds having fundamental group of product type, by means of an index theorem which holds for operators on the covering that are invariant under a projective action of the fundamental group. The eta invariant we use to distinguish metrics with positive scalar curvature is then associated to a 2-cocycle on the fundamental group (joint work with Charlotte Wahl).
12 October 2012: Elmar Schrohe (University of Hannover) A Families Index Theorem for Boundary Value Problems
Abstract: Boutet de Monvel's calculus provides a pseudodifferential framework which encompasses the classical differential boundary value problems. In an extension of the concept of Lopatinski and Shapiro, it associates to each operator two symbols: a pseudodifferential principal symbol, which is a bundle homomorphism, and an operator-valued boundary symbol. Ellipticity requires the invertibility of both. If the underlying manifold is compact, elliptic elements define Fredholm operators. Boutet de Monvel showed how then the index can be computed in topological terms. The crucial observation is that elliptic operators can be mapped to compactly supported $K$-theory classes on the cotangent bundle over the interior of the manifold. The Atiyah-Singer topological index map, applied to this class, then furnishes the index of the operator. In joint work with Melo, Nest, and Schick, it was shown how C$^*$-algebra K-theory, can be used to give a proof of Boutet de Monvel's index theorem for boundary value problems. Similar techniques could now be applied to yield an index theorem for families of Boutet de Monvel operators. The key ingredient of our approach is a precise description of the K-theory of the kernel and of the image of the boundary symbol.
22 June 2012: Christian Becker (University of Potsdam) Cheeger-Chern-Simons theory and geometric string structures
Abstract : The Chern-Weil construction on a principal G bundle with connection associates to an invariant polynomial on the Lie algebra a closed di erential form on the base. There are two well-known re.nements of this construction: the Chern-Simons form on the total space and the Cheeger-Simons character on the base. We combine these constructions to obtain a further re.nement: the Cheeger-Chern-Simons construction. A String structure in the topological sense on a principal Spin(n) bundle is a degree 3- cohomology class on the total space that yields the preferred generator of H3(Spin(n);ZZ) on each fiber. By a geometric String structure, we understand a re.nement of a String structure in the topological sense to a di erential cohomology class. The Cheeger-Chern- Simons shows a way to de.ne such geometric String structures. In the talk, I will explain the notion of di erential characters (with sections along a smooth map). I will review the classical Chern-Weil, Chern-Simons and Cheeger-Simons constructions and show how they .t together in the Cheeger-Chern-Simons construction. This construction then establishes a notion of geometric String structure." In this talk I shall review the formulation for Hopf principal bundles with quantum group symmetries, and describe how it is possible to introduce Hodge dualities on quantum SU(2) and S2 spheres, their corresponding Laplacians, and to couple them to gauge connections.
9. March 2012: Paulo Carrillo Rousse (University of Toulouse) Twisted K-theory for foliations and wrong way functoriality
Abstract: I will report on a joint work with Bai-Ling Wang (ANU Canberra) on index theory for twisted foliations. Twisted K-theory was first defined by Donovan and Karoubi in the early seventies but it has attracted the attention of mathematicians only in the last two decades mainly because of the interest physicists have focused in this theory. Originally defined for spaces, twisted K-theory can be defined for "noncommutative spaces" such as the space of leaves of a regular foliation (Laurent-Gengoux, Tu and Xu). In this talk I will explain how its possible to do index theory for foliated spaces with a twisting (could be given for exampel by some Dixmier-Douady class), in particular I will sketch a very geometric proof of a Connes-Skandalis longitudinal index theorem in this context. This last result will allow us to discuss the first step into a complete wrong way functoriality theory for twisted foliations. If we have time I will give the example of the construction of an assembly Baum-Connes map for twisted foliations.
10 February 2012: Camille Laurent (University of Metz) Global "action-angles" variables on Poisson manifolds
Abstract: The local "action-angle" theorem says that there is only one integrable system whose leaves are compact. There are nevertheless cohomological obstructions to the global triviality of these systems. This is well-known for symplectic manifolds. For Poisson manifolds, it is more appropriate to start from a so-called "noncommutative" integrable system. (Note that the terminology "noncommutative" is somewhat misleading since it has nothing to do whatsoever with quantisation; rather, it is related to "overdetermined".) The local case was studied a few years ago by Eva Miranda, Pol Vanhaecke and myself. Together with Rachelk Cseiro, Rui Fernandes, Daniel Sepe and Pol Vanhaecke we now investigate global constructions.
2. December 2011: Alessandro Zampini (University of Munich): Laplacians on quantum Hopf fibrations
Abstract: In this talk I shall review the formulation for Hopf principal bundles with quantum group symmetries, and describe how it is possible to introduce Hodge dualities on quantum $SU(2)$ and $S^2$ spheres, their corresponding Laplacians, and to couple them to gauge connections.
4. November 2011: Nicolas Prudhon (University of Metz) : On the Dirac-Kostant operator
Abstract: In 1999, B. Kostant introduced a Dirac operator associated with any spectral triple (g,h,B), where (g, B) is a complex quadratic Lie algebra, h a Lie subalgebra of g on which B is non degenerate. Konstant showed that the square of D satisfies a formula generalising that of Parthasarathy. We give a proof of this formula, which is less technical than that of Kostant.
