Tropical Vector Bundles and Matroids

Explore advanced mathematical concepts in this lecture from the Geometry of Matroid Workshop, where University of Pittsburgh's Kiumars Kaveh delves into tropical vector bundles and matroids. Begin with Klyachko's classification of toric vector bundles and understand how their data can be interpreted as piecewise linear maps from toric variety fans to vector space valuations. Learn about the novel concept of piecewise linear maps from fans to Bergman fans of matroids, introducing tropical toric vector bundles (or toric matroid bundles). Examine equivariant K-theory, characteristic classes, and discover how matroids possess tautological tropical toric vector bundles over permutahedral toric variety. Investigate how corresponding equivariant K-classes and Chern classes connect to recent developments in tautological matroid classes by Berget-Eur-Spink-Tseng. Master concepts including sheaf of sections, Euler characteristic, and positivity notions such as global generation, ampleness, and nefness for tropical toric vector bundles, culminating in the exploration of vanishing higher cohomologies results from joint work with Christopher Manon.