A day of tropical geometry

This talk will be a refresher on algebraic geometry, tropical geometry, and everything in between. I will review several approaches to tropical geometry, and delve into other areas of combinatorial algebraic geometry such as the theory of matroids.

Gromov-Witten theory probes the geometry of manifolds by studying the space of algebraic curves in that manifold. In the last decade, it has emerged that matroids, which are combinatorial abstractions of a certain class of manifolds, possess many of the structures that one typically attaches to manifolds. This is true even when there is no direct connection to geometry. I will discuss a new direction in this interplay between matroids and geometry, by defining and examining the Gromov-Witten theory of matroids. The talk is based on joint work with Jeremy Usatine (Brown).

An early insight in tropical geometry was that matroids (a combinatorical abstraction of linear independence) play the role of linear spaces. In this talk I will give an overview of the recent work on tropical schemes, which builds on this to develop a theory of tropical commutative algebra that uses matroids, which leads to associated algebraic geometry. Recent highlights include descriptions of the cycles given by tropical schemes, and progress on the strong Nullstellensatz, both joint with Felipe Rincon.