Matroid polytopes and tropical geometry

The proposed project lies in the area of discrete mathematics, more precisely in geometric combinatorics with a view towards tropical geometry. The cornerstones are polyhedral geometry and matroid theory. The focus is on tropical linear spaces, matroid (flag) polytopes and their combinatorial structure.

These objects appear in various guises in algebraic geometry and optimization, and further have found applications in biology, economics and physics.Recent results successfully combine matroid theory with commutative algebra and polyhedral geometry.

The purpose of this project is to extend this approach and to exploit the rich geometric structure of (valuated) matroids in order to make substantial progress on current questions and conjectures in polytope theory, matroid theory, tropical geometry and their applications.Our goal is to sharpen those fundamental tools and techniques that polyhedral and tropical geometry provide and which will contribute to the future development of the area.In particular, we are aiming to prove longstanding conjectures on matroid invariants, a decomposition theorem for valuated matroids, and structural results for secondary polytopes, and moduli spaces of tropical linear flags.

To obtain our mathematical goals we will design new supplementary software components.