Séminaire de Géométrie et Quantification
(Department of Mathematical Sciences, Loughborough University)
Colliding Holes in Riemann Surfaces
Lundi 24 avril à 15h30, IHP salle 01,
Résumé: In 1997 Hitchin proved that the Riemann Hilbert correspondence between Fuchsian systems and
conjugacy classes of representations of the fundamental group of the punctured sphere is a Poisson map.
Since then, some generalisations of this result to the case of irregular singularities have been proposed by
several authors. In this talk we interpret irregular singularities as the result of collisions of boundaries in a
Riemann surface and show that the Stokes phenomenon corresponds to the presence of "bordered cusps".
We introduce the concept of decorated character variety of a Riemann surface with bordered cusps and
construct a generalised cluster algebra structure and cluster Poisson structure on it. We define the quantum
cluster algebras of geometric type and show that they provide an explicit canonical quantisation of this Poisson structure.