DMS-9501131 Gelfand Investigation of mathematical aspects arising in the study of integrable equations is proposed. Emphasis will be placed on the following problems: (a) The nonlinearization of Fourier transforms and their use for solving initial and boundary value problems for certain physically significant equations (b) Algebraic and geometrical aspects related to integrability, including the algebra of Chern and Chern-Simons classes, Crofton K-densities, and integrable surfaces; (c) Classification of the finite bi-hamiltonian problems and their connection with geometry . In recent years important developments in the study of certain nonlinear equations have occurred. In particular it has been found that there exists a general method for solving certain equations. It turns out that this method can be considered as the application of nonlinear Fourier transforms. These nonlinear transforms will be used to study several physically important equations. It turns out that integrable equations have a rich underline algebraic and geometric structure, which also will be investigated.
