Isoparametric hypersurfaces in spheres

The problem of classifying isoparametric hypersurfaces in spheres is still not com-
pletely solved. Recent contributions to the case with six dierent principal curvatures
contain fundamental gaps. In my talk I present a new method for the classication of
isoparametric hypersurfaces of spheres with g = 6 dierent principal curvatures all of
multiplicity m = 1. Furthermore, I explain a new strategy for the proof of the remaining
open case with g = 6, namely (g, m) = (6, 2).