Weighted Hurwitz numbers & Topological Recursion
Learning Seminar
Organisers: Dan Betea, Danilo Lewański.
Contacts: danilo.lewanski@studio.unibo.it
Where: Max Planck Institut für Mathematik, Vivatsgasse 7 5311 Bonn, Germany.
When: Started on May the 9th. Usually Thursdays 13:00 - 15:00.
GOAL OF THE SEMINAR:
Hurwitz theory provides an excellent set of enumerative geometric problems related to topological recursion theory.
The reason is two-fold: these problems are numerous, and they are related — this makes possible to learn from the already solved problems to better approach the unsolved ones.
Several of them have been proved to satisfy topological recursion, for others the statement remains conjectural and in some cases has been tested computationally.
In 2016 Alexandrov, Chapuy, Eynard, and Harnad (ACEH) propose a general formula (ACEH Formula) for the topological recursion spectral curve valid for a certain class of Hurwitz problems, and investigate it throughout three papers. The main goal of the learning seminar is to understand this trilogy of papers.
The artworks "Vershik - Kerov - Logan - Shepp sunset", and "Cardioid inside an Aztec diamond", by Dan Betea. The former is exhibited at the Institut Henri Poincaré, Paris.
1. Hurwitz problems and topological recursion: state of art and ACEH formula.
Danilo Lewański. May the 9th, 2019.
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.
2. Dirac electrons’ sea, Hurwitz theory, and quantum curves.
Danilo Lewański. May the 16th, 2019.
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…
3. Short review of integrable hierarchies.
Séverin Charbonnier. May the 23th, 2019.
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.
4. Toda, Plancherel, and Hurwitz.
Dan Betea. June the 13th, 2019.
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. Using the same techniques and following Okounkov's "Infinite wedge and random partitions" https://arxiv.org/abs/math/9907127, we compute the correlation functions for the Plancherel measure and show they satisfy the same Toda hierarchy. We further discuss the probabilistic interest in the Plancherel measure (and if time permits, its variants and generalizations), connecting longest increasing subsequences of random permutations to largest eigenvalues of GUE random matrices.