Combinatorial integrability -III

The minicourse will be devoted to a description of combinatorial solutions to integrable hierarchies of Kadomtsev--Petviashvili type that arise naturally in enumeration of various topological and algebro-geometric objects. A preliminary layout includes - Permutations and their decompositions into products of transpositions (simple Hurwitz numbers, Okounkov's theorem, Hurwitz formula, Bousquet-M\'elou--Schaeffer formula, cut-and-join equation) Symmetric group representations (diagonilizability of the cut-and-join operator, the group algebra of the symmetric group, Schur polynomial, Jucys--Murphy elements) The semiinfinite Grassmannian and the Kadomtsev--Petviashvili hierarchy (Pl\"ucker embeddings, semiinfinite planes in the space of Laurent series, Orlov--Shcherbin family of solutions) Ramified coverings of the 2-sphere (coverings and ramified coverings, Hurwitz numbers and ramified coverings, Caley formula, genus expansion)