100, rue des maths 38610 Gières / GPS : 45.193055, 5.772076 / Directeur : Thierry Gallay

Journée L'espace de Teichmüller quantique

Vendredi, 16 Janvier, 2009 (Toute la journée)
Description : 

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 ».


Type: 
Journées financées par des ANR
logo uga logo cnrs