Gemeinsam mit den Universitäten von Clermont-Ferrand und Metz in Frankreich veranstalten wir einmal im Monat ein Seminar,
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Das Seminar wird per Videokonferenz einmal im Monat Freitags um 14. c.t. im Haus 2 Raum 0.15 stattﬁnden. Das thematische Spektrum breitet sich von der Algebra und nichtkommutative Geometrie bis zur mathematischen Physik aus.
René Schulz (Göttingen)
Global Fourier integral operators via tempered oscillatory integrals with inhomogeneous phase functions
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.
Max Planck Institut (Einstein Institut für Gravitationsphysik) Golm / Raum 1.63
Olivier Gabriel (Göttingen)
Lie group actions, spectral triples and generalised crossed products 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.
Charlotte Wahl (Hannover)
Rho-invariants and the classification of differential structures on closed manifolds
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.
Freitag, den 16. November
Sara Azzali (Paris VII)
Eta invariants and positive scalar curvature
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).
Freitag den 9. März 2012
Paulo CARRILLO ROUSE (Toulouse)
Twisted K-theory for foliations and wrong way functoriality
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 on 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 one can carry out index theory for foliated spaces with a twisting (e.g., given 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.
Freitag den 10. Februar 2012
Camille Laurent (Universität 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.
Die letzen Beiträge bisher:
Freitag den 2. Dezember 2011
Alessandro Zampini (Universität München)
Laplacians on quantum Hopf ﬁbrations
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.
Freitag den 4. November 2011
Nicolas Prudhon (Universität Metz)
Sur l’opérateur de Dirac-Kostant / On the Dirac-Kostant operator (pdf Datei zur Verfügung)
En 1999, B. Kostant introduit un opérateur de Dirac D associe tout triplet (g,h,B), où - (g,B) est une algèbre de Lie quadratique complexe - h est une sous-algèbre de Lie de g sur laquelle B est non dégénérée. Kostant montre alors que le carré de D vériﬁe une formule qui généralise la formule de Parthasarathy. Nous donnons ici une démonstration de cette formule, moins calculatoire que celle de Kostant.
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.