Combinatorial Algebra

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First posed early in the twentieth century, the famous Burnside Problem for groups asked: Is every finitely generated periodic group finite? The Kurosh-Levitzki Problem is an associative algebra analogue: Is every finitely generated nil algebra finite-dimensional? Counterexamples were first constructed by Golod and Shafarevich in the 1960's. It was natural, therefore, to reformulate these problems with additional hypotheses in order to obtain a positive solution. Zelmanov won the prestigious Fields Medal in 1994 for his proof that every finitely generated residually finite group is finite, thereby solving the so-called Restricted Burnside Problem for groups. I have extended Zelmanov's seminal work in group and Lie theory to a single unified theory that has direct applications to other branches of algebra. In particular, I have proved that a finitely generated nil algebra is finite-dimensional if it is infinitesimially PI. This is a significant generalisation Zelmanov's results. I have also had some success in applying my new theory to Kaplansky's Problem which addresses the structure of group algebras whose augmentational ideal is Jacobson radical. Zelmanov has referred to Kaplansky's Problem as the next big hurdle in group theory after the Restricted Burnside Problem.