Classification of isoparametric hypersurfaces

等参超曲面是主曲率不变的球面中的超曲面。已知的例子要么来自对称空间的各向同性表示，要么来自Clifford代数的表示。由对称空间产生的等参超曲面是对称空间的各向同性表示的主要轨道。与此相反，Clifford代数中的大多数例子都不允许有可迁等距群。等参超曲面的分类是子流形几何中一个重要的公开问题。该项目的一个目标是对具有四个不同主曲率的等参超曲面进行分类。与等参子流形紧密相连的是与射影空间具有相同有理上同调的流形：此类流形的微分同胚分类是一个公开问题，特别是类似于射影平面的流形的分类是公开的。我们计划实现这样的分类。

Isoparametric hypersurfaces are hypersurfaces in spheres with constant principal curvatures. The known examples either come from isotropy representations of symmetric spaces or from representations of Clifford algebras. The isoparametric hypersurfaces arising from symmetric spaces are the principal orbits of the isotropy representation of the symmetric space. In contrast to this, most of the examples arising from Clifford algebras do not admit a transitive isometry group. The classification of isoparametric hypersurfaces is one of the major open problems in submanifold geometry. One aim of the project is the classification of isoparametric hypersurfaces with four distinct principal curvatures. Closely connected to isoparametric submanifolds are manifolds which have the same rational cohomology as a projective space: The diffeomorphism classification of such manifolds is an open problem; in particular, the classification of manifolds which are like projective planes is open. We plan to achieve such a classification.