The Polytope of All Matroids - A Framework for Matroid Theory Conjectures

Explore a 47-minute lecture where Luis Ferroni from the Institute for Advanced Study delves into the fascinating world of matroid polytopes during IPAM's Computational Interactions between Algebra, Combinatorics, and Discrete Geometry Workshop. Discover a novel polytope construction that encodes all matroids of fixed size and rank, where each matroid appears as a lattice point and special matroids correspond to polytope vertices, introducing a new concept of extremality. Learn how linear maps in the polytope's ambient space relate to valuative invariants on matroids, and understand why extremal matroids should be the first candidates examined when investigating conjectures about matroid invariant positivity. Examine specific examples including the counterexamples to the Ehrhart positivity conjecture and the Merino-Welsh conjecture on Tutte polynomials, both corresponding to polytope vertices. Gain insights into a significant breakthrough in matroid theory through the discovery of a valuative invariant that serves as a representability test, demonstrating the existence of valuative invariants that remain non-negative for realizable matroids but not in general cases.