Non-abelian theta functions and the theta map Abstract: Abelian theta functions are quasi-periodic holomorphic
functions, classically associated to an algebraic curve C . In modern terms, they are sections of certain line bundles on a complex torus, the Jacobian, which parametrizes C*-bundles on C. This point of view gives rise to a natural generalization, replacing C* by any reductive group. The corresponding sections, called non-abelian theta functions, play an important role in quantum field theory. In the talk I will survey what is known about their algebro-geometric properties in the case of classical groups.