Tautological relations and double ramification cycles with spin parity

Abstract

In this thesis, we address two problems concerning the tautological rings of the moduli space of curves. The first problem concerns expressing the Chern classes of the Hodge bundle within the tautological ring in terms of decorated strata classes of a particular form. The second problem we address is a proof of a conjecture by Costantini-Sauvaget-Schmitt which concerns two different expressions of the so-called spin double ramification cycles, the spin version of Pixton's formula and the spin version of star graph expression. After proving the conjecture of Costantini-Sauvaget-Schmitt, we explain in the appendix how these expressions can be used to compute all spin classes of strata of k-differentials.