Mathematical Sciences: Applications of Lie Sphere Geometry to the Study of Taut and Dupin Submanifolds

托马斯·塞西尔将继续他对紧致超曲面的长期研究。关注的焦点将是寻找这些超曲面、Dupin超曲面和等参超曲面之间的相互关系。这项研究将在最近与合作者使用李球几何技术的基础上进行。紧超曲面是指在其上每个欧几里德距离函数都具有Morse不等式所要求的最小临界点的超曲面。这种超曲面是紧的，因为它们具有最小的绝对总曲率。紧张性也与Dupin超曲面的概念有关。要研究的主要问题之一是嵌入到球面中的Dupin超曲面是否必须是紧的。

Thomas Cecil will continue his long term investigations of taut hypersurfaces. The focus of attention will be on finding interrelationships between these, Dupin hypersurfaces and isoparametric hypersurfaces. This research will follow on from recent work, with collaborators, using the techniques of Lie sphere geometry. A taut hypersurface is one on which every Euclidean distance function has the minimum number of critical points required by the Morse inequalities. Such hypersurfaces are tight in the sense that they have minimum absolute total curvature. Tautness is also related to the notion of Dupin hypersurfaces. One of the principal questions to be studied is whether a Dupin hypersurface embedded in the sphere must be taut.