Mathematical Sciences: Research in Global Analysis

帕莱将继续他在可微变换群方面的工作。这项工作的主要焦点是找出这些群与其他几何概念之间的相互关系。特别是，我们将研究等参子流形的概念。它们是具有简单局部不变量的欧几里德空间的子流形。虽然有限维情形已经得到了广泛的研究，但在Hilbert流形的无限维情形下仍有许多工作要做。Adler和van Moerbeke将研究代数完全可积系统。这项工作涉及非常困难的问题，这些问题现在被认为与代数几何中的问题有密切的联系。代数几何学家Shiota也将致力于加强和澄清这些联系。所研究的主题将包括将复代数环上的完全可积系统线性化，用超椭圆积分来表示解，二次和四次曲线的几何。在证明了Novikov猜想之后，Shiota得到了一类Prym变种的类似刻画定理。他目前的项目是进一步开发这些技术来表征适当类别的主要极化的阿贝尔变种。

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.