PROGRAMME

Le colloque débutera mercredi 20 octobre 2010 à 14h45 et se terminera vendredi 22 octobre 2010 à 16h15. Tous les exposés se dérouleront dans la Salle 1723 (2ème étage) du bâtiment 17 du Campus de l'UFR Sciences de l'Université de Reims (voir Rubrique "Informations Pratiques").

Mercredi 20 octobre  
14:00-14-45 Accueil des participants
14:45-15:35 Tilmann Wurzbacher On the geometry and quantization of symplectic Howe pairs
15:40-16:30 Pablo Ramacher Invariant integral operators on the Oshima compactification of a Riemannian symmetric space and their traces
  / Pause Café-Thé /  
17:00-17:50 Yannick Voglaire A new class of symplectic symmetric spaces and their deformation quantizations

Jeudi 21 octobre  
9:00-9:50 Hervé Oyono-Oyono Almost projections, propagation and large scale geometry
  / Pause Café-Thé /  
10:20-11:10 Wolfgang Bertram On conformal contractions
11:15-12:05 Paul-Émile Paradan Formal quantization of non-compact hamiltonian manifolds
  / Déjeuner /  
14:30-15:20 Benjamin Enriquez Drinfeld associators and Kashiwara-Vergne conjecture
15:25-16:15 Bachir Bekka Lattices with and without spectral gap
  / Pause Café-Thé /  
16:45-17:25 Axel de Goursac Non-formal Deformation Quantization of the Heisenberg Supergroup
20:00 Dîner au Restaurant Cosi (35 bis, rue Buirette)

Vendredi 22 octobre  
9:00-9:50 Stanislas Woronowicz Commutation relations for a pair of self-adjoint operators and quantum ax+b group
  / Pause Café-Thé /  
10:20-11:10 Nicolas Prudhon The square of the cubic Dirac operator
11:15-12:05 Martin Schlichenmaier Berezin-Toeplitz quantization of moduli spaces
  / Déjeuner /  
14:30-15:20 Mélanie Bertelson Affine connections revisited
15:25-16:15 Hideyuki Ishi Unitarizable holomorphically induced representation of the Jacobi group

RÉSUMÉS

Bachir BEKKA (Univ. Rennes) Lattices with and without spectral gap (joint work with Y. de Cornulier and with A. Lubotzky). — The existence of a spectral gap for a group action on a probability space is a strong version of ergodicity appearing in various problems. Define a lattice $\Gamma$  in a locally compact group $G$ to have a  spectral gap if the action of $G$ on the homogeneous space $G/\Gamma$ has a spectral gap. This is the case for a real Lie group $G$ ; this is also the case when $G$  is a simple algebraic group over a local field. However, when $G$ is the automorphism group of a  $k$-regular tree for $k\geq 3,$ there are lattices in $G$ without spectral gap.

Mélanie BERTELSON (Univ. Libre Bruxelles) Affine connections revisited. — In a joint work with Pierre Bieliavsky, we propose a formula for the unique torsionless affine connection preserved by the symmetries of a symmetric space. This formula, appropriately interpreted, allows to describe any affine connection on a differentiable manifold. This leads to more geometric formulations of certain objects and results in the theory of affine connections.

Wolfgang BERTRAM (Univ. Nancy-1) On conformal contractions (joint work with M. Kinyon, resp. with P. Bieliavsky). — We will describe several examples of "conformal contractions" of geometric structures: Lie groups, symmetric spaces, and associative geometries.

Benjamin ENRIQUEZ (Univ. Strasbourg) Drinfeld associators and Kashiwara-Vergne conjecture (joint work A. Alekseev and Ch. Torossian). — The Kashiwara-Vergne conjecture predicts the existence of formal series satisfying certain conditions; the existence of these formal series allows in particular to extend the Duflo isomorphism between the center of an enveloping algebra and its classical analog in case of distributions. We show that one can construct, for every Drinfeld associator, an explicit solution to the Kashiwara-Vergne conjecture. This result is based on the one hand on the interpretation of the Kashiwara-Vergne conjecture in terms of non-commutative divergence, due to Alekseev and Torossian, and on the other hand on the study of the restriction to free groups of formality isomorphisms, induced by associators, between braid groups and their infinitesimal analogs.

Axel de GOURSAC (Univ. Cath. Louvain) Non-formal Deformation Quantization of the Heisenberg Supergroup. — We present a Universal Deformation Formula for the Heisenberg supergroup and apply it on some examples.

Hideyuki ISHI (Univ Nagoya) Unitarizable holomorphically induced representation of the Jacobi group. — We realize the Jacobi group as a central extension of the Kaelher automorphism group of a certain homogeneous Kaehler manifold. Then a representation of the Jacobi group is naturally defined on the space of holomorphic sections of the quantization bundle of the Kaehler manifold. We consider the unitarizability of the representation. The resultant unitary representation appears in the branching law for the restriction of the holomorphic discrete series representation of a larger symplectic group to the Jacobi group.

Hervé OYONO-OYONO (Univ. Metz) Almost projections, propagation and large scale geometry. — Coarse spaces and discrete groups give rise to C*-algebras filtered by elements of finite propagation. In this talk, we see how almost projections with finite propagation lying in these C*-algebras allow to capture topological invariants related to large scale geometry.

Paul-Émile PARADAN (Univ. Montpellier 2) Formal quantization of non-compact hamiltonian manifolds. — Let K be a compact Lie group acting in a Hamiltonian fashion on a symplectic manifold M prequantized by a Line bundle L. When the manifold is not compact but the associated moment map is proper we will show how to define the geometric quantization of the data (K,M,L) as a generalized character of K. We will explain why this procedure satisfies the principle "Quantization commutes with Reduction". We will explore several examples from the theory of representations.

Nicolas PRUDHON (Univ. Metz) The square of the cubic Dirac operator. — In 1999, Kostant introduced a cubic Dirac operator D associated to any triple (g,h,B), where g is a complex Lie algebra provided with a bilinear symmetric non degenerate form B which is ad g-invariant, and h is a Lie subalgebra of g such that B is non degenerate on h. The square of the operator D satisfies a formula that generalizes the famous Parthasarathy formula. We will present here a new proof of this Kostant formula.

Pablo RAMACHER (Univ. Marburg) Invariant integral operators on the Oshima compactification of a Riemannian symmetric space and their traces. — We study invariant integral operators on the Oshima compactification of a Riemannian symmetric space, and characterize them within the framework of pseudodifferential operators, in particular describing the singular nature of their kernels. The study of such operators is expected to be relevant for the development of geometric scattering theory on symmetric spaces.

Martin SCHLICHENMAIER (Univ. Luxembourg) Berezin-Toeplitz quantization of moduli spaces. — As was shown by Bordemann, Meinrenken, and Schlichenmaier the Berezin-Toeplitz (BT) operator quantization and its associated star product give a unique natural quantization for a quantizable compact Kaehler manifold. In the talk an overview over BT quantization is given. The procedure is applied for the moduli space of gauge equivalence classes of SU(N) connections on a fixed Riemann surface. In the language of algebraic geometry this moduli space is the moduli space of semi-stable vector bundles over a smooth projective curve. In this context the Verlinde spaces and the Verlinde bundle over Teichmueller space show up. Recent results of J. Andersen on the asymptotic faithfulness of the representation of the mapping class group on the space of covariantly constant sections of the Verlinde bundle are presented.

Yannick VOGLAIRE (Univ. Cath. Louvain/Univ. Reims) A new class of symplectic symmetric spaces and their deformation quantizations. — We study a new class of solvable symplectic symmetric spaces stemming from elementary normal j-algebras, and show the prominent role played in their deformation quantization by linear projections from some coadjoint orbits to their tangent spaces. We discuss how these new examples question the geometrical interpretation of the above projections.

Stanislas L. WORONOWICZ (Univ. Varsovie) Commutation relations for a pair of self-adjoint operators and quantum ax+b group. — We shall discuss the precise meaning of the relation $ab=q^2ba$, where $a$ and $b$ are selfadjoint operators and deformation parameter $q$ is a number of modulus 1. There are two different ways of understanding the relation: first one related to S. Zakrzewski and the second to K. Schmudgen. We shall present the quantum $ax+b$-group, where $a$ and $b$ satisfy Schmudgen commutation relations.

Tilmann WURZBACHER (Univ. Metz/Univ. Bochum) On the geometry and quantization of symplectic Howe pairs. — We study the orbit structure and the geometric quantization of a pair of mutually commuting hamiltonian actions on a symplectic manifold. If the pair of actions fulfils a symplectic Howe condition, we show that there is a canonical correspondence between the orbit spaces of the respective moment images. Furthermore, we show that reduced spaces with respect to the action of one group are symplectomorphic to coadjoint orbits of the other group. In the Kaehler case we show that the linear representation of the pair of Lie groups on the geometric quantization of the manifold is then equipped with a representation-theoretic Howe duality.