Title: Tropical geometry of b-Hurwitz numbers
Abstract: The Goulden-Jackson b-conjecture is a remarkable open problem in algebraic combinatorics. It predicts an enumerative meaning for the coefficients of the expansion of a certain expression of Jack symmetric functions. Major progress was made in recent work of Chapuy and Dołęga, which led to the introduction of b-Hurwitz numbers. These invariants are generalisations of classical Hurwitz numbers obtained by including non-orientable surfaces. They are polynomials in a parameter b, which measures the "non-orientability" of the coverings involved. In this talk, we develop a tropical theory of b-Hurwitz numbers and the recently introduced monotone b-Hurwitz numbers. As part of this development, we resolve an open question posed by Chapuy–Dołęga, as well as an open question posed by Bonzom–Chapuy–Dołęga. This talk is based on joint work in progress with Raphaël Fesler, Maksim Karev and Hannah Markwig.