Mathematical Sciences: Research in Global Analysis

Palais将继续他关于可微变换群的研究。这项工作的主要重点是找到这些组和其他几何概念之间的相互关系。特别要研究等参子流形的概念。这些是欧几里得空间的子流形具有简单的局部不变量。尽管有限维的情况已经得到了广泛的研究，但在希尔伯特流形的无限维情况下仍有很多工作要做。Adler和van Moerbeke将研究代数上的完全可积系统。这项工作涉及到非常困难的问题，这些问题现在被认为与代数几何中的问题有密切的联系。作为一名代数几何学家，盐田也将致力于加强和澄清这些联系。研究的主题将包括复代数环面上的完全可积系统的线性化，用超椭圆积分表示解，二次和四分次几何。在他对诺维科夫猜想的证明之后，盐田得到了一类Prym变种的类似表征定理。他目前的项目是进一步发展这些技术，以表征主要极化阿贝尔变体的适当类别。

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.