On the Tautological Ring of Moduli Spaces of Riemann Surfaces.

This thesis "On the Tautological Ring of Moduli Spaces of Riemann Surfaces." gives an overview of known results about the relations on the tautological ring of Moduli spaces. First, we introduce the Moduli Space of Stable Curves as an Orbifold of a etale groupoid, a generalization of complex manifold and orbit space of a group. Then we define the Tautological Ring on it as a subring of the Cohomology Ring. Finally, we present the work of Pandharipande, Pixton and Dvonkine in \cite{3-spin}, they discovered a set of relations on the Tautological Ring that is, up to date, the largest known. This set is obtained by Cohomological Field Theories, a tool to compute and glue cohomology classes in a Tautological way, and a group action on CohFTs. Using Teleman’s characterization of Cohomological Field Theories in a specific case, they manage to deduce an explicit formula for a suitable modification of Witten’s 3-spin CohFT. This turns out to vanish non trivially, providing a set of relations.