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. Critical to the use of Lie theory are the linear realizations of the symmetries, called representations. In recent years, the algebraic approach to representation theory of semi-simple Lie groups has grown in power and importance, and it dominates in the attempt to classify all the irreducible representations. Professor Shelton is an expert in this methodology. His work involves an array of interactions with other areas of mathematics, including homological algebra, spectral sequences, and sheaves of differential operators. He has begun a systematic study of the difficult problem of computing extensions between universal highest weight modules in various categories. He has solved this problem for certain special classes of groups, most notably those associated to Hermitian symmetric pairs. He proposes to extend his work to the general category of semi-simple Lie groups.
