Mathematics

The first part of this course will introduce the moduli spaces of stable curves and the main results about their geometric structure and their Deligne-Mumford compactification. Studying the topology of such spaces leads directly to the question of describing their cohomology and intersection theory. We will introduce some examples of cohomlogy classes on the moduli spaces and eventually the notion of tautological ring as a natural subring of cohomology whose structure is sufficiently well behaved. Finally we will introduce the tool of cohomological field theories as natural families of cohomology classes compatible with the boundary structure of moduli spaces. Intersection theory of such classes are connected to the theory of integrable systems of PDEs, as first revealed by Witten's conjecture and then by the work of Dubrovin and Zhang. A more recent approach by Buryak and myself gives more direct evidence to why this phenomenon occurs.