Searching for the fundamentals of tropical geometry

Johannes Rau is a professor at the University of the Andes, Colombia. Thanks to a grant from the Knut and Alice Wallenberg Foundation, he will be a visiting professor at the Department of Mathematics, University of Gävle.

The subject studied in the project is tropical geometry. This is a simplified version of algebraic geometry, a branch of mathematics that explores the relationship between shapes and equations. While algebraic geometry originated with the ancient Greeks, as a mathematical field, tropical geometry is not even half a century old. However, it has already helped to solve some long-standing problems in algebraic geometry by translating into tropical geometry.

The tool used to perform this translation is called tropicalization; this process transforms classical geometric objects into tropical objects. Tropical objects must capture the most important characteristics of their classical counterparts and disregard any that are superfluous. This simplifies the original problem. For example, a tropical curve is never curved, as tropicalization stretches the curve so it looks more like a web of matches.

Although tropicalization is at the heart of tropical geometry, it still lacks complete theoretical foundations. The long-term goal of the project is to find a general theory for the relationship between algebraic and tropical geometry. 

The project focuses on the study of tropical compactifications of moduli spaces. A moduli space is a catalogue for geometric objects of a certain type, in which similar objects are close together. Tropical compactifications are combinations of algebraic and tropical moduli spaces, where the proximity between algebraic and tropical objects is governed by tropicalization. Therefore they serve as a habitat where algebraic and tropical objects coexist in symbiosis.