The moduli spaces of stable algebraic curves with marked points form a family of topological spaces numbered by two integers: the genus and the number of marked points. The marked points can be associated with some natural complex line bundles over moduli spaces, the first Chern classes of which are called psi-classes. Witten's conjecture, proved by Kontsevich, asserts that the generating series of integrals of monomials from psi-classes over moduli spaces is a solution to the system of Korteweg-de Vries equations. We will tell you about a distant generalization of Witten's conjecture proposed by Dubrovin and Zhang. It turns out that it is possible to ins ert in to the considered integrals over moduli spaces additional cohomology classes satisfying some very simple properties, and in this case the resulting generating series again turns out to be a solution to the PDE system, but already more complicated than the KdV system. Time permitting, we will also describe how the Dubrovin-Zhang theory allows (hypothetically) to classify PDE systems that satisfy certain integrability properties.
