Mathematical Sciences: Harmonic Analysis, Representation Theory and Partial Differential Equations

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Professor Greenleaf will apply the theory of group representations to problems in mathematical analysis on nilpotent Lie groups and their homogeneous spaces. Particular emphasis will be placed on studying the invariant differential operators on homogeneous spaces, describing algebras of such operators in terms of Kirillov's orbit picture, and developing solvability criteria for large classes of differential operators on these spaces. Lie groups arise naturally in mathematics and physics as groups of symmetries. An interesting example that lies readily at hand consists of all rotations of a sphere, made into a group by applying rotations successively. Understanding this group is important in any situation in which spherical symmetry is present. The simplest noncommutative example of a nilpotent Lie group (the class of most immediate interest here) is the Heisenberg group of upper triangular three-by-three real matrices with ones on the diagonal. The Heisenberg group plays a fundamental role in the quantum mechanical treatment of the harmonic oscillator, as the symmetry group of a certain differential equation. The strong relationship between differential equations and the representation theory of the underlying symmetry group is at the heart of Professor Greenleaf's project.