Toric vector bundles and buildings

The talk is a report on the ongoing work with Chris Manon on toric vector bundles. First I will discuss the well-known classification of vector bundles on toric varieties by Klyachko. Then I will explain how to regard Klyachko's data describing a vector bundle using piecewise linear maps and the so-called "buildings". This point of view promises to give a new insight on the subject and involves (higher rank) generalizations of notions of "convex polytope" and "piecewise linear function". I will try to make the talk as accessible as I can.