In this first talk of the seminar, we give a resumè of the known results about topological recursion for Hurwitz problems, we introduce ACEH formula and specialise it to previously known cases, we state for which class of problems ACEH formula is proved, and for which larger class could also hold.
In the last talk we have seen how Hurwitz numbers have a representation theoretic expression. This formulation can be used to express them further as operators shuffling around Maya diagrams defined on Dirac electrons’ sea. This have deep consequences at the level of integrable hierarchies (but we will see it in detail only in a future talk). On the other hand, these operators turn out handy for computational purposes. In particular, there is a way to derive quantum curves…
This talk is dedicated to a short review of the KP hierarchy and the 2d Toda lattice hierarchy, of their formulation in terms of Hirota equation, as well as of the geometric interpretation of the solutions as a Sato Grassmannian.
Following Okounkov's "Toda equations and Hurwitz numbers" https://arxiv.org/pdf/math/0004128.pdf, we show that the generating series for certain ramified coverings of the Riemann sphere (arbitrary ramification type over 0 and infinity, simple ramifications elsewhere) is a tau function for the Toda lattice hierarchy. We use and recall basic fermionic Fock space techniques in the process.