Mathematical Sciences: Automorphic Representations of Semisimple Lie Groups

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The principal investigator plans to work on problems in mathematical analysis concerning the representation theory of (semisimple) Lie groups. A suitable example of the latter is the group of rotations of a sphere. Groups like this are important because they occur in many areas of mathematics (e.g. geometry, differential equations, algebraic number theory, mathematical physics) as groups of symmetries. Representation theory allows one to take advantage of symmetries in solving problems. More specifically, this project will study representations that come from spaces of automorphic forms (invariant analytic functions) on the quotient of the group modulo a discrete subgroup. The aim is to obtain explicit formulas for the multiplicities of known irreducible representations in these automorphic representations, based on algebraic properties of the discrete group and information about how it sits in the ambient Lie group.