Manchester Geometry Seminar 2013/2014

Matrix models are integrals over (usually Hermitian) $N\times N$ matrices weighted by some potential function. In spite of their simplicity they manifest both conformal symmetries and integrability properties, and some of these models are related to geometry (notably the Kontsevich matrix integral is the generating function for intersection indices on Riemann surfaces with marked points). In my talk I will concentrate on the method of "topological recursion" originated in papers by L. Ch., B. Eynard, and N. Orantin, which allows constructing a consistent $1/N$-expansion for integrals over Hermitian $N\times N$ matrices with arbitrary potentials in terms of algebro-geometrical quantities related to the spectral curve of the corresponding model. This method was proved to be universal as it has been successfully applied to solving a variety of models of theoretical physics and mathematics. I will also briefly describe the results of a recent paper by L. Ch., B. Eynard and S. Ribault in which this method was generalized to the case of quantum Liouville theory.