Volumes and intersection theory on moduli spaces of Abelian differentials

Computing volumes of moduli spaces has significance in many fields. For instance, Witten's conjecture regarding intersection numbers on the Deligne–Mumford moduli space of stable Riemann surfaces has a fascinating connection to the Weil–Petersson volume, which motivated Mirzakhani to give a proof via Teichmueller theory, hyperbolic geometry, and symplectic geometry. The initial two other proofs of Witten's conjecture by Kontsevich and by Okounkov–Pandharipande also used various ideas in combinatorial ribbon graphs, Gromov–Witten theory, and Hurwitz theory. In this talk I will introduce an analogue of Witten's intersection numbers on moduli spaces of Abelian differentials to compute the Masur–Veech volumes induced by the flat metric associated with Abelian differentials. This is joint work with Martin Moeller, Adrien Sauvaget, and Don Zagier (arXiv:1901.01785).