Chern classes of tropical manifolds

Tropical algebraic geometry is a relatively new field of mathematics concerned with understanding combinatorially defined polyhedral objects that encode interesting behaviour of algebraic varieties. Some of the most basic tropical varieties are tropical linear spaces, which have a very rich combinatorial structure governed by matroids.

Chern classes for matroids and more general tropical varieties were introduced recently. These classes have very interesting combinatorial and geometric properties – for instance, they were used in a central way by Ardila, Dehnam, and Huh in their renowned proof that the h-polynomial of the broken circuit complex of any matroid has log-concave coefficients.

The goal of this PhD project will be to push further our understanding of the geometrical and combinatorial aspects of Chern classes of tropical manifolds, and in particular, of tropical linear spaces. It will involve learning about combinatorial objects like matroids and balanced polyhedral fans, the fundamentals of tropical geometry and tropical varieties, and also some of the algebraic geometry/topology motivating the definitions of Chern classes for tropical manifolds.

This project has the advantageous feature of providing the PhD student with a broad background in various fields, such as tropical geometry, combinatorics, and algebraic geometry/topology. In addition, it can be tackled from different points of view, including computational approaches.

References:

– Diane Maclagan and Bernd Sturmfels, Introduction to tropical geometry, American Mathematical Society, 2021.

– Lucía López de Medrano, Felipe Rincón, and Kris Shaw, Chern-Schwartz-MacPherson cycles of matroids, Proceedings of the London Mathematical Society 120, no. 1, 2020.

– Lucía López de Medrano, Felipe Rincón, and Kris Shaw, Chern classes of tropical manifolds, arXiv preprint: 2309.00229, 2023.