Moduli space of curves, tautological relations and integrable systems

Consider the moduli space of stable curves, whose points parametrize all possible smooth or nodal genus g compact Riemann surfaces with n marked points whose group of automorphisms is finite. The topology of this by now classical object is still mostly unknown. It's cohomology (or Chow) ring structure, for instance, is even today quite out of reach. A natural subring that contains most of the geometrically interesting classes, called the tautological ring, is more, yet not completely, under control. For instance, in the presentation of the tautological ring in terms of generators and relations, we are not quite sure we found all possible relations and various conjectures, all interconnected, have been proposed (by Faber, Pandharipande, Pixton and others). Inside this fashinating playground, a surprising feature is the appearence of integrable systems of PDEs (typically in terms of generating functions of intersection numbers of various types of cohomology classes). Beside being a remarkable bridge towards mathematical physics, the realization of this fact brought new powerful techniques to the field. After an overview of the subject, I will report on our recent results on this topic, in a series of papers with A. Buryak, B. Dubrovin and J. Guéré.