Tropical Chern classes and Chern-Weil theory

There is a natural analog of a vector bundle in the tropical setting. The Chern classes of a tropical vector bundle was introduced by L. Allermann in terms of intersection theory. In our talk we will construct the Chern classes of a tropical vector bundle in terms of differential geometry. The classical Chern-Weil theory constructs the Chern classes of a complex vector bundle in terms of differential geometry, in particular using curvature and connection of a vector bundle. We will describe the tropical analog of this construction. Also we will discuss a tropical analog of the exponential long exact sequence of a line bundle.