Mathematics

The famous Witten's conjecture, proposed in 1991, claims that the generating series of intersection numbers on the moduli spaces of stable algebraic curves is a solution of the Korteweg - de Vries (KdV) hierarchy. This result has now a lot of generalizations, including the theory developed by B. Dubrovin and Y. Zhang, that explains that any partition function from a large class of partition functions, controlling topological invariants of the moduli spaces of curves, is a tau-function of a hierarchy of KdV type. A new direction was opened in a recent work of R. Pandharipande, J. Solomon and R. Tessler, who considered the intersection numbers on the moduli spaces of Riemann surfaces with boundary. They conjectured that the generating series of these intersection numbers is a solution of a certain hierarchy, closely related to the KdV hierarchy. We proved this conjecture in my joint work with R. Tessler. I will talk about this result and its generalizations from my joint works with  A. Alexandrov, A. Basalaev, E. Clader and R. Tessler.