Putting the “volume” back in “volume polynomials”

Recent developments in tropical geometry and matroid theory have led to the study of “volume polynomials” associated to tropical varieties, the coefficients of which record all possible degrees of top powers of tropical divisors. In this talk, I’ll discuss a volume-theoretic interpretation of volume polynomials of tropical fans; namely, they measure volumes of polyhedral complexes obtained by truncating the tropical fan with normal hyperplanes. I’ll also discuss how this volume-theoretic interpretation inspires a general framework for studying an analogue of the Alexandrov-Fenchel inequalities for degrees of divisors on tropical fans. Parts of this work are joint with Anastasia Nathanson, Lauren Nowak, and Patrick O’Melveny.