Flat Connections, Braid Groups and Quantum Groups

In this talk, I will describe how quantum groups serve as a useful means of expressing the monodromy of certain integrable, first order PDE’s. A fundamental, and paradigmatic result in this context is the Kohno–Drinfeld theorem. Roughly speaking, it asserts that the representations of Artin’s braid groups on n strands given by the universal R-matrix of a quantum group describe the monodromy of the Knizhnik–Zamolodchikov (KZ) equations, a flat connection on the configuration space on n points in C.