The theory of Lie groups, named in honor of the Norwegian mathematician Sophus Lie, has been one of the major themes in twentieth century mathematics. As the mathematical vehicle for exploiting the symmetries inherent in a system, Lie theory has had a profound impact upon mathematics itself and theoretical physics, especially quantum mechanics and elementary particle physics. The research proposed by Professor Jenkins utilizes the representation theory of nilpotent Lie groups to study certain classes of differential operators. These include Schrodinger operators, which are the operators that determine the dynamical laws in quantum mechanical systems. Given a differential operator, one constructs an appropriate Lie group in which that operator is realized infinitesimally in the universal enveloping algebra. This construction leads to a convolution semigroup of probability measures having the differential operator as infinitesimal generator. In this way, Professor Jenkins uses the theory of these semigroups to study the spectrum and corresponding eigenfunction expansion of the given operator.
