#### 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!

Done