This research will investigate the structure of semisimple Hopf algebras, of Hopf algebra orders in group algebras, and of semisimple Hopf algebras which are semi-direct products. The research will focus on the structure of finite-dimensional semisimple Hopf algebras and the classification of such Hopf algebras in low dimension. In particular, a resolution of the Kaplansky conjecture will be sought. This research is in the general area of noncommutative ring theory. The research is concerned with the structure of Hopf algebras. An improved understanding of cosemisimple Hopf algebras could be of fundamental importance not just in Hopf algebra theory but in other fields as well. This an extremely difficult area but new techniques which have recently been introduced indicate that substantial progress can be expected.
