A promenade through the history of the classification of isoparametric hypersurfaces

-연사 : Prof. Quo-Shin Chi (Washington University in St. Louis, USA)
-일시 : 5월 27일(금) 17시
-장소 : 이화포스코관 151호
-abstract :

Isoparametric surfaces in the Euclidean 3-space, defined by two PDEs, arose in the study of geometric optics in 1918; the notion of an isoparametric hypersurface can thus be defined on any Riemannian manifold.
The classification of isoparametric hypersurfaces in the Euclidean n-space started in 1937 by T. Levi-Civita, to be followed by the beautiful investigations of E. Cartan into the hyperbolic and the spherical cases.
The spherical case turned out to be remarkably deep. As the ensuing study from 1940 to this date has witnessed, the spherical case is at the crossroad of several important areas of geometry and topology, such as the representation theory of Lie groups, symmetric spaces, algebraic topology, homotopy theory, etc., let alone differential geometry itself.
I will bring another field of mathematics, namely, commutative algebra and algebraic geometry, to the crossroad that turns out to play a decisive role for the classification problem, which has almost been completed barring three remaining cases. The talk will be non-technical, through which I intend to introduce glimpses of how important areas of mathematics interplay with the beautiful isoparametric geometry.