Dhruv Ranganathan: The tropical geometry of Hurwitz numbers

Hurwitz theory, or the study of covers of the Riemann sphere, intertwines representation theory, algebraic geometry, and combinatorics. I will give an introduction to tropical geometry, which lies at the intersection of geometry and combinatorics, by considering the manner in which it can answer questions in Hurwitz theory. I will focus on the Hurwitz counting problem, which tropical geometry re-envisions as a counting problem for certain piecewise linear functions on finite graphs, and try to explain the different ways in which these ideas can arise. I will use these ideas to explain a beautiful piecewise polynomiality property of Hurwitz numbers. Time permitting, I will discuss generalizations and analogues in other settings. I will not assume any sophisticated algebraic geometry. Students are particularly welcome to attend!