Title: An invitation to Tropical Geometry and non-Archimedean Combinatorics

Abstract: Tropical Geometry has been the subject of great amount of recent activity over the last decade. Loosely speaking, it can be described as a piecewise linear version of algebraic geometry. It is based on tropical algebra, where the sum of two numbers is their maximum and the product is their sum. This turns polynomials into piecewise-linear functions, and their zero sets into polyhedral complexes. These tropical varieties retain a surprising amount of geometric information about their classical counterparts.

Non-Archimedean analytic geometry, as developed by Berkovich, is a variation of classical complex analytic geometry for non-Archimedean fields such as p-adic numbers. Solutions to a system of polynomial equations over these fields form a totally disconnected space in their natural topology. The process of analytification adds just enough points to make them locally connected and Hausdorff. The resulting spaces are technically difficult to study but, notably, their heart is combinatorial: they can be examined through the lens of tropical and polyhedral geometry.

In these talks, I will give a gentle introduction to these two subjects and will illustrate this powerful philosophy through concrete examples, including elliptic curves, the tropical Grassmannian of planes of Speyer-Sturmfels, and a compactification of the well-known space of phylogenetic trees of Billera-Holmes-Vogtmann and the tropical compactification of the moduli space of rational n-marked curves. I will also state some open questions in these subjects.

These lectures will serve as an overview of a topics course that will run in Spring 2016.