Stokes phenomenon, dynamical r-matrices and Poisson geometry

Abstract

In this thesis, we study the Poisson geometry of moduli spaces of flat and meromorphic connections over Riemann surfaces, the latter involves the Stokes phenomenon. The aim is to understand some new achievements in this direction from the perspective of mathematical physics. The main results of this thesis are: (1) a construction of a gauge transformation between the standard classical r-matrix and the Alekseev-Meinrenken dynamical r-matrix, using the Stokes data of a certain irregular Riemann-Hilbert problem; (2) an extension to the quantum analogue of the Stokes phenomenon, and its relation with the Yang-Baxter equation; (3) a new finite dimensional description of the Atiyah-Bott symplectic form on the moduli spaces of flat connections over surfaces, using generalised dynamical r-matrices induced by gauge fixing procedures.