Abstract: In 1979, Novikov conjectured the following remarkable relation between algebraic-geometry and the theory of integrable models: the Jacobians of smooth algebraic curves are precisely those indecomposable principally polarized abelian varieties (ppav) whose theta-functions provide solutions to the Kadomtsev-Petviashvili (KP) equation. Novikov's conjecture was proved by T. Shiota in 1986.

The goal of this talk is to present a circle of ideas and methods which are instrumental for applications of algebraic-geometry to soliton theory, and conversely, pave the way for the solutions of some 120 years-old problems in algebraic geometry using the theory of integrable systems. Among the latter is the characterization of Jacobian varieties as ppav whose Kummer variety admits a trisecant line, and of the Prym varieties as ppav whose Kummer variety admits a pair of symmetric quadrisecants.

At the heart of this recent progress is the notion of abelian pole systems generalizing the pole systems such as Calogero-Moser system arising in the theory of elliptic solutions to the basic soliton hierarchies. We also present recent results in this direction on the characterization of Jacobians of curves with involution, which were motivated by the theory of two-dimensional integrable hierarchies with symmetries.