Secants of Abelian Varieties, Theta functions, and Soliton Equations

This paper is a survey on relations between secant identities and soliton equations and applications of soliton equations to problems of algebraic geometry, i.e., the Riemann-Schottky problem and its analogues. A short introduction into the analytic theory of theta functions is also given.Comment: 69 pages, accepted by Russian Mathematical Survey