Teaching: On leave 2011-2012

Research Interests: Associative rings and algebras, Lie algebra, group theory

Research Profile: The specific branch of Noncommutative Algebra that I study is sometimes called Combinatorial Algebra. The term derives from its ancestor, Combinatorial Group Theory, which traditionally focuses on such topics as groups presented by generators and relators, growth in groups, the Burnside Problem, as well as computational and algorithmic problems. Similar problems are now studied for infinite-dimensional associative and Lie algebras. I am particularly interested in Burnside-type problems. 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 to both problems 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 infinitesimally-PI, generalizing Shirshov's result for PI-algebras. I have also had some success in applying this new theory to Kaplansky’s Problem which addresses the structure of group algebras whose augmentation ideal is Jacobson radical. Kaplansky’s Problem has been described as the next big hurdle in group theory after the Restricted Burnside Problem. The solution of the Restricted Burnside Problem is equivalent to saying every pro-finite group of finite exponent is locally finite. Zelmanov went on to prove even more: every periodic pro-finite group is locally finite. Together, Wilson and Zelmanov proved a related result every Engelian pro-finite is locally nilpotent. These results led to the obvious generalizations to all compact groups. I have been working on the Lie-theoretic analogues of these problems. In particular, McInnes and I have proved that every Engelian pro-finite Lie ring is locally nilpotent. It is unknown whether or not this result extends to all compact Lie rings.