9402994 Helgason Helgason will continue his work on existence questions for invariant differential operators and on operational problems for integral operators on symmetric spaces. Vogan will continue his work on representations of special topological groups. At present the most urgent problem is to find all the ways in which a given group can be realized as a group of unitary operators on a Hilbert space. Vogan will concentrate on realizations when the group is the unitary group of an indefinite Hermitian quadratic form. 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, the representation theory of Lie groups has had a profound impact upon mathematics itself, particularly in analysis and number theory, and upon theoretical physics, especially quantum mechanics and elementary particle physics. ***
