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, elementary particle physics, and relativity. In particular, the structures and the symmetries are interwoven, as illustrated by the fundamental work of Einstein, and later Yang-Mills, in which profoundly successful nonlinear theories for the fundamental forces in nature have been guided mathematically by the need for a large family of symmetries. In a different direction, mathematicians and physicists, often independently, have developed a vast array of powerful techniques that classify the linear realizations of the symmetries. This is the subject of "representation theory". Professor Zuckerman has done pioneering creative work in many facets of representation theory of Lie groups, and he is now applying his work to fundamental physics. Ten years ago he invented new ways to construct linear representations of Lie groups, his methods drawing from diverse, seemingly unrelated areas of mathematics. Recently, Professor Zuckerman's efforts have been directed toward their adaptation to the exciting new study of "string theory" in physics. The goal of this theory is to provide a unified description of all the forces of nature. In the current proposal he will conduct research on both linear and nonlinear manifestations of symmetry in string theory, as well as other more conventional physical theories. This includes the development of a new basis for the passage from classical to quantum physical theories, and resulting compatibility of both theories with Einstein's special and general relativity. Symmetries are realized nonlinearly at the classical level and linearly at the quantum level. This dual role of symmetry is a fundamental aspect of his work.
