Sergei Lando: Integrable hierarchies of PDEs in mathematical physics and combinatorics of the symmetric groups

The KP hierarchy is a completely integrable system of quadratic, partial differential equations that generalizes the Korteweg-deVries hierarchy. Although its origins belong in mathematical physics, it possesses solutions that are of combinatorial nature. Moreover, equations of this hierarchy can be treated as recurrence relations for certain enumerative problems. We will explain how one can obtain solutions to KP from combinatorics of symmetric groups, and (after Goulden, Jackson, Bender, Gao, Richmond) how these results can be applied to enumerate maps on surfaces.