Speakers:
Adrien Boulanger (Marseille, I2M)
Thérèse Falliero (Avignon, LMA)
Clotilde Fermanian (Angers, Larema)
Pedro Freitas (Lisbonne, IST)
Nadine Große (Freiburg, Math. Inst.)
Francis Nier (Paris, LAGA)
Anna Roig Sanchis (Nice, LJAD)
Talks (preliminary version):
A.Boulanger. Autour de la question du chaos quantique.
Dans un premier temps, et pour un auditoire large, on discutera de questions liées au chaos quantique et de leurs enjeux. On verra notamment comment celles-ci sont reliées à la dynamique du flot géodésique. On discutera ensuite, si le temps nous le permet, de l'application du chat quantique, un modèle plus compliqué conceptuellement mais plus aisé à manipuler.
T.Falliero. Eisenstein series and hyperbolic Riemann surfaces
The spectrum of the Laplace-Beltrami operator for a compact Riemann surface is discrete, but this is no longer the case as soon as you remove one point from it. In the case of cusps, Eisenstien series, that are classical objects in analytic number theory, appear to describe the continuous part of the spectrum. With the help of hyperbolic Eisenstein series and families of degenerating Riemannian surfaces, we will attempt to give a lighting on this appearance.
C.Fermanian. Semiclassical methods on filtered manifolds
In this presentation, we will discuss recent results about the construction of a symbolic pseudo-differential calculus on equirigular filtered manifolds, and their extension towards a semi-classical regime. The fundamental example of such manifolds are contact manifolds. This calculus uses the Harmonic analysis of nilpotent Lie groups which are associated with the tangent space of these manifolds, once endowed with the structure of a filtrated algebra (the Heisenberg group in the case of contact manifolds). These results are issued from collaborations with Veronique Fischer (University of Bath), Steven Flynn (University College London) and Lino Benedetto (ENS and université d’Angers).
P.Freitas. Pólya's conjecture on manifolds
Pólya's conjecture states that eigenvalues of the Dirichlet Laplacian on a bounded Euclidean domain are above the corresponding first term in Weyl's law. Pólya formulated the conjecture in 1954, and proved it in 1961 in the particular case of tiling domains, that is, domains D such that "an infinity of domains congruent to D (we admit also congruence by symmetry) cover the whole plane without gaps and without overlapping." In the intervening decades some progress has been made, but the conjecture remains open in general. We will give an overview of some of the main results that have been obtained, and then discuss what may be expected once we leave the Euclidean setting. More precisely, by deriving both qualitative and (sharp) quantitative results, we will provide a basic description of what the essential ingredients for the conjecture to hold are.
N.Große. On the Lp-spectrum of the Dirac operator
We study the p-dependence of the spectrum of the Dirac operator. In particular, we give sufficient conditions for the Lp-spectrum to be independent on p on noncompact manifolds. As an application we use this result to compute the L2-spectrum of classes of manifolds by calculating the L1-spectrum. We also talk about different notions of essential Lp-spectra. This is joint work with Nelia Charalambous.
F.Nier. New results about the low lying spectrum of Witten and Bismut Laplacians
After recalling rapidly what are Witten and Bismut Laplacians and their natural parameter dependent version (related to the physical parameters temperature and friction), I will present our result obtained firstly with D. Le Peutrec and C. Viterbo (Witten Laplacian) and secondly with X. Sang and F. White (Bismut Laplacian) in the "low temperature and high friction" regimes.
A.Roig Sanchis Spectrum of the Laplacian on random hyperbolic 3-manifolds
Given a hyperbolic manifold, the spectrum of its Laplacian contains a lot of information about its geometric structure. A natural question is to study its behaviour as the complexity of the manifold increases. A way to do so, is by using random constructions. In this talk, I will discuss a work in progress joint with Will Hide, Bram Petri and Joe Thomas concerning the behaviour of the spectral gap -the first non-zero eigenvalue- of a model of random hyperbolic 3-manifolds.