The University of Michigan Combinatorics Seminar
Tropical geometry is a recipe for finding combinatorial analogues of constructions from algebraic geometry. Applying this recipe to linear spaces and Grassmannians, we discover tropical linear spaces and the tropical Grassmannian. Tropical linear spaces behave strikingly like standard linear spaces: there are analogues of orthogonal complement, intersection and span. Their combinatorics are governed by decompositions of matroids into smaller matroids. Studying these decompositions leads us to discover a matroid invariant which is best described through the geometric notions of Grassmannians and K-theory. Computing this invariant leads us back to the combinatorics of symmetric functions and Erhart polynomials, and suggests new questions to ask about Erhart polynomials and matroids.
This talk will not assume any knowledge of algebraic geometry, or of matroids. Some of the earlier work I will discuss was done jointly with Bernd Sturmfels.