Alladi Ramakrishnan Hall

The tropical geometry of Hurwitz numbers

Dhruv Ranganathan

Cambridge

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!