## Séminaire de Géométrie et Quantification |

**Marta
Mazzocco **

(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.

.