The tropical geometry of Hurwitz numbers

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