Integrable Hamiltonsche Systeme und algebraische Geometrie. (Integrable Hamiltonian systems and algebraic geometry) (Q1081169)

The paper gives a brief review of the latest achievements in the theory of integrable Hamiltonian systems with both finite and infinite degrees of freedom. The author emphasises the algebraic-geometric aspects of integrability. The mathematical pendulum behaviour is discussed as an introductory example and also some classical finite-dimensional integrable cases are considered. The Lax representation for the Toda lattice is presented producing a bridge to the theory of KdV equation; the formulas of the solutions obtained in terms of Riemann's theta- functions by Novikov and others are given. The last section is devoted to the Kadomtsev-Petviashvili equation and its connections with the Schottky problem.