Kazarian and Lando Seminars "Combinatorics of Vassiliev invariants". Speaker: Raphaël Fessler (HSE)

Let’s take a permutation of n elements with k independent cycles. This permutation can be represented by a product of m transpositions. Once we have this representation we can build a surface, of genus g=1+(m-n-k)/2, with k boundary components. Furthermore, there is an embedded graph with n vertices and m edges, such that all the vertices are on the boundary and their positons are given by the cyclic structure, and the edges correspond the transpositions. The surface is obtained by gluing ribbons along the edges. This is the oriented Hurwitz theory.

We can extend this construction to the case of twisted ribbons. It will give us a possibly non orientable surface. We can at the same time construct the orientation cover of this surface, with also an embedded graph. We thus can build an analogue of the oriented Hurwitz theory, which is called twisted Hurwitz theory. In this talk we will explain the geometric construction above, then recall what is the oriented Hurwitz theory (Hurwitz numbers, cut-and-join, Schur functions) and explain the analogue for the twisted Hurwitz theory.

The talk is based on a joint work with Y.Burman.