Given a group, one can associate to it a group ring, which is formed from complex linear combinations of group elements. The idempotent conjecture is that if a group has no elements of finite order, then the only elements in the group ring which are equal to their own square should be zero or one. This is a completely algebraic statement, which can be re-phrased by saying that there are only trivial solutions to a certain finite set of equations. However, suitably completing the algebraic group ring can turn it into a geometric object, and the idempotent conjecture can be reformulated in geometric terms. If the group is commutative, the idempotent conjecture is known to be true, and to be equivalent to the connectedness of the n-dimensional torus. I propose to develop analytical tools related to the operator norm completion of the group ring to get a better understanding of a more general class of groups, which includes, for example, the special linear group SL(n,Z).
Groups arise in many scientific fields because they encode symmetries of physical, biological or other systems, therefore it is important to study them. It is impossible to say anything useful about all groups, so we divide them into classes and study each class separately. I am interested in classes of groups that arise as symmetries of geometric objects. Understanding something about the geometry of these objects often can be translated into algebraic information about the group. I plan to develop tools from algebra and analysis to study these geometric objects, with a particular emphasis on understanding geometric properties which only become apparent on a large scale.
