Mathematical Sciences: Group Representations and Partial Differential Equations

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Greenleaf will apply the theory of group representations to problems in mathematical analysis on nilpotent Lie groups and their quotient spaces. These investigations will seek a description of the algebra of invariant differential operators in terms of the coadjoint orbits in the dual of the Lie algebra of the group. Initial efforts will address a basic conjecture: The invariant differential operators on the quotient form a commutative algebra precisely when the quasi-regular representation has finite multiplicities. Success in these investigations will lead to understanding of solvability, and existence of fundamental solutions, for invariant differential operators. 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.