the verlinde formula is a celebrated explicit computation of the dimension of the space of sections of certain positive line bundles over the moduli space of semistable vector bundles over an algebraic curve. i will describe a recent generalization of this formula in which the moduli of vector bundles is replaced by the moduli of semistable higgs bundles, a moduli space of great interest in geometric representation theory. a key part of the proof is a new ``virtual non-abelian localization formula" in k-theory, which has broader applications in enumerative geometry. the localization formula is an application of the nascent theory of theta-stratifications, and it serves as a new source of applications of derived algebraic geometry to more classical questions.