Abstract: The talk explores an interplay between three concepts from different areas of algebra and geometry: vector bundles from geometry and topology, valuations from commutative algebra and piecewise linear functions from convex geometry. A "vector bundle" over a geometric space X (such as a manifold) is, roughly speaking, an assignment of vector spaces to each point in X. Vector bundles are a central object of study in geometry and topology. We introduce the notion of a valuation with values in piecewise linear functions and see that these are the right gadgets to classify (equivariant) vector bundles on so-called "toric varieties". Examples include classification of all (equivariant) vector bundles on a projective space. This can be regarded as a reformulation of Klyachko's famous classification of toric vector bundles. This point of view leads to far reaching extensions which I will touch on if there is time. This is joint work with Chris Manon.