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

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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.