Combinatorial Algebraic Geometry: Tropical Vector Bundles and their Chern Classes

We introduce the notion of a tropical vector bundle (of rank n) over a tropical cycle X, which is equivalently interpretable as a polyhedral complex living "above" X or GL_n-torsor. Primarily relying on the more-accessible combinatorial definition, we go on to look at morphisms, rational sections, and pull-backs of these bundles. After some examples, we will discuss how one can take local intersection products and construct Chern classes -- concluding the talk with a classification of all vector bundles on a (tropical) elliptic curve up to isomorphism, which will (surprisingly!) coincide with Atiyah's 1957 result in classical algebraic geometry.