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Unité Mixte de Recherche CNRS 5582 Université Grenoble I

UFR de Mathématiques

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Journée "L’espace de Teichmüller quantique"

Une journée ``L’espace de Teichmüller quantique’’ aura lieu le vendredi 16 janvier 2009 dans la salle 04 de l’Institut.

Programme de la journée :

11h -12h Rinat KASHAEV
Section Mathématiques, Univ. de Genève		 Quantum Teichmüller space I

Résumé : Let \Sigma be an oriented surface of non-positive Euler characteristic with one puncture. Let {\mathcal T}_\Sigma be the Teichmüller space of hyperbolic structures on \Sigma. By using Penner’s coordinates for the decorated Teichmüller space, we obtain a parameterisation of the simplectic space {\mathcal T}_\Sigma\times H^1(\Sigma,{\mathbb R}) where the surface mapping class group is realised by rational transformations, and the simplectic structure is given in the canonical form. The combinatorial data needed for this parameterisation is called a decorated ideal triangulation given by an ideal triangulation of the surface, a distinguished corner in each ideal triangle, and a total order of the set of ideal triangles. This parametrisation naturally generalises to the case of arbitrary finite number of punctures.

14h -15h Rinat KASHAEV
Section Mathématiques, Univ. de Genève		 Quantum Teichmüller space II

Résumé : The surface mapping class group can naturally be extended into a groupoid of decorated ideal triangulations, and the latter admits a particular presentation, which permits us to define an algebraic structure called semisymmetric T-matrix in the way that any realisation of such structure permits us to construct a certain representation of the groupoid of decorated ideal triangulations. In this way, the quantisation problem of the Teichmüller space is formulated as the existence problem for certain semisymmetric T-matrix.

15h15 -16h15 Rinat KASHAEV
Section Mathématiques, Univ. de Genève		 Quantum Teichmüller space III

Résumé : By using our parameterisation of the space {\mathcal T}_\Sigma\times H^1(\Sigma,{\mathbb R}) and its symplectic structure, we construct a particular semisymmetric T-matrix realising thereby the quantisation program of the Teichmüller space. As a result we obtain a unitary projective representation of the surface mapping class group in a Hilbert space.

L’atelier est partiellement financé par le projet ANR « Repsurf ».