Title: Moduli space of curves, tautological relations and integrable systems
Abstract: In the study of the topology of moduli space of stable curves and its tautological ring, 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, this fact brings new powerful techniques to the field. In a recent series of papers with A. Buryak, B. Dubrovin and J. Guéré, we construct an integrable system from any given cohomological field theory using various tautological classes (including the double ramification cycle) and we compare it with the more classical Dubrovin-Zhang integrable hierarchy. This comparison suggests a new, large family of conjectural tautological relations in all genera and number of marked points. I will report on our progress in proving them and on their applications.