The tropical geometry of monotone Hurwitz numbers

Abstract: Hurwitz numbers are important enumerative invariants in algebraic geometry. They count branched maps between Riemann surfaces. Equivalently, they enumerate factorizations in the symmetric group. Hurwitz numbers were introduced in the 1890s by Adolf Hurwitz and became central objects of enumerative algebraic geometry in the 1990s through close connections with the so-called Gromov-Witten theory. This interplay between Hurwitz and Gromov-Witten theory is an active field of research and led to, among other things, the celebrated ELSV formula. In the last decade, many variants of Hurwitz numbers have been introduced and studied. In particular, the question of connections between these variants of Hurwitz numbers and Gromov-Witten theory is of great interest. So-called monotone Hurwitz numbers , which originate from the theory of random matrices, are among the most studied variants of Hurwitz numbers. This talk is a progress report of our larger program in which we study the connections between monotone Hurwitz numbers and Gromov-Witten theory by combinatorial methods of tropical geometry, and whose long-term goal is a proof of the still open conjecture of an ELSV - type formula for double monotone Hurwitz numbers. The talk is based in part on joint work with Reinier Kramer and Danilo Lewanski.

algebraic geometrycombinatorics

Audience: researchers in the topic

Online Nottingham algebraic geometry seminar

Series comments: Online geometry seminar, typically held on Thursday. This seminar takes place online via Microsoft Teams on the Nottingham University "Algebraic Geometry" team.

For recordings of past talks, copies of the speaker's slides, or to be added to the Team, please visit the seminar homepage at: kasprzyk.work/seminars/ag.html