Palais will continue his work on differentiable transformation groups. The main focus of this work is to find interrelationships between these groups and other geometric concepts. In particular the concept of isoparametric submanifold will be studied. These are submanifolds of Euclidean space with simple local invariants. Although the finite dimensional case has been extensively worked on there is still much to do in the infinite dimensional setting of a Hilbert manifold. Adler and van Moerbeke will work on algebraically completely integrable systems. This work involves very difficult questions which are now seen to have close connections with problems in algebraic geometry. Shiota, an algebraic geometer, will also work on strengthening and clarifying these connections. The topics under investigation will include linearizing completely integrable systems on complex algebraic tori, expressing solutions in terms of hyperelliptic integrals, the geometry of quadrics and quartics. Following his proof of the Novikov conjecture, Shiota obtained an analogous characterization theorem for a certain class of Prym varieties. His current project is to further develop these techniques to characterize appropriate classes of principally polarized Abelian varieties.
