Title: Deformations of the Riemann hierarchy, Miura invariants, and moduli spaces of curves

Abstract: I will talk about integrable deformations of the Riemann hierarchy that have the form of systems of conservation laws. There is a beautiful conjecture of Arsie-Lorenzoni-Moro claiming that the space of such deformations, up to Miura transformations, can be parameterized by a sequence of functions of one variable, the so-called Miura invariants. While this conjecture is open, I am interested in an explicit construction of an integrable deformation with a given sequence of Miura invariants. It occurs that the moduli spaces of curves give a powerful tool for this. I will show how to construct deformations with arbitrary constant Miura invariants and also deformations with arbitrary linear Miura invariants. The last case is particularly interesting because it includes the famous Camassa-Holm and Degasperis Procesi hierarchies, which didn't appear before in relation with the moduli spaces of curves.