Mathematical Sciences: Analysis on Nilpotent Lie groups

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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. The simplest noncommutative example of a nilpotent Lie group 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. Nilpotent Lie groups are sufficiently well understood that one can do analysis on them in almost the same level of detail as Fourier analysis in the classical, Euclidean setting. The project of Professor Jenkins has two main objectives. The first and more important goal is to classify Gelfand pairs (roughly, a Lie group together with a compact subgroup of automorphisms satisfying a certain commutativity condition) in the nilpotent case. The second is to find out more about the space of spherical functions relative to such a pair.