Matroids in tropical geometry

Tropical geometry is the geometry obtained when the operations of addition and multiplication on the real numbers are replaced by the operations of minimum and addition, respectively. Tropical mathematics have been studied in many different contexts, but a deep connection to algebraic geometry has only been established in the last few decades. This development has led to numerous applications in many different areas, such as enumerative algebraic geometry, mirror symmetry, optimisation, and computational biology.

Matroids are mathematical objects that abstract many different notions of independence throughout mathematics. They are essential in tropical geometry, as they play the same role as linear subspaces in classical mathematics. The connection between tropical geometry and matroid theory is deep and strong, and has been very beneficial to both fields.

Recently, the PI and his collaborators have introduced two new notions in tropical geometry that promise to be very useful for the field: tropical ideals and tropical CSM classes. Tropical ideals serve as algebraic and combinatorial objects that keep track of the equations that define a tropical variety. Tropical CSM classes are tropical objects that carry combinatorial and topological information about any smooth tropical variety. Matroids are essential in the construction of both of these objects.

The aim of this project is to continue to develop the strong connections between matroid theory and tropical geometry, by pushing the study of these two novel tropical notions: tropical ideals and tropical CSM classes. Investigating these promising objects will push the reach of tropical geometry further, opening the door to numerous applications such as a tropical study of Hilbert schemes, a deeper exploration of realisability questions in tropical geometry, and new approaches to enumerative algebro-geometric problems.