Lionel Lang is currently working on his project entitled "Tropical compactifications of moduli spaces". This project is financed by the Swedish Research Council, from 2024 to 2027.

Tropical compactifications of moduli spaces

Algebraic geometry is a fundamental area of mathematics that aims at understanding the relations between shapes (geometry) and equations (algebra). Tropical geometry is a simplified version of algebraic geometry that emerged 20 years ago. There is a procedure called tropicalisation that transforms the shapes of algebraic geometry into the shapes of tropical geometry. The latter remember the core structure of their algebraic counterparts but forget about information that is often irrelevant in many problems.

Many long-standing problems in algebraic geometry have been solved using tropical geometry, making the latter a very active field of research. However, the relation between algebraic and tropical geometries still lacks theoretical foundations. Our goal is to unify these two geometries by constructing tropical compactifications of moduli spaces.

A moduli space is a set that encodes all possible shapes of the same nature. Our goal is to construct new spaces, the aforementioned compactifications, by gluing together algebraic and tropical moduli spaces, taking tropicalisation into account. These compactifications should:

  • provide the correct translation of a problem in algebraic geometry to a problem in tropical geometry
  • guarantee that a solution to the tropical problem provides a solution to the original problem.

Since problems in tropical geometry are often much easier to solve, tropical compactifications of moduli spaces have countless applications to algebraic geometry.

Sound interesting? Email Lionel Lang for more information.