Sergei Natanzon | Searcharxiv

arXiv2016

BKP and projective Hurwitz numbers

We consider $d$-fold branched coverings of the projective plane $\mathbb{RP}^2$ and show that the hypergeometric tau function of the BKP hierarchy of Kac and van de Leur is the generating function for weighted sums of the related Hurwitz numbers. In particular we get the $\mathbb{RP}^2$ analogues of the $\mathbb{CP}^1$ generating functions proposed by Okounkov and by Goulden and Jackson. Other examples are Hurwitz numbers weighted by the Hall-Littlewood and by the Macdonald polynomials. We also consider integrals of tau functions which generate projective Hurwitz numbers and Hurwitz numbers related to different Euler characteristics of the base Klein surfaces.

math-ph↗

arXiv2001

Topological classification of generic real rational functions

To any real rational function with generic ramification points we assign a combinatorial object, called a garden, which consists of a weighted labeled directed planar chord diagram and of a set of weighted rooted trees each corresponding to a face of the diagram. We prove that any garden corresponds to a generic real rational function, and that equivalent functions have equivalent gardens.

math.AG↗