Symplectic forms in the theory of solitons

We develop a Hamiltonian theory for 2D soliton equations. In particular, we identify the spaces of doubly periodic operators on which a full hierarchy of commuting flows can be introduced, and show that these flows are Hamilt onian with respect to a universal symplectic form omega = Res(infinity) [Ps i(0)(*)delta L boolean AND delta Psi(0)]dk. We also construct other higher order symplectic forms and compare our formalism with the case of 1D solito ns. Restricted to spaces of finite-gap solitons, the universal symplectic f orm agrees with the symplectic forms which have recently appeared in non-li near WKB theory, topological field theory, and Seiberg-Witten theories. We take the opportunity to survey some developments in these areas where sympl ectic forms have played a major role.