Mathematical Sciences: Partial Differential Equations and Several Complex Variables

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The principal investigator will study several questions arising from the analogues in several complex variables of the problem: can you hear the shape of a drum? One question is whether there is an analogue of the Riemann-Roch Theorem for CR manifolds. Another concerns the small-time off-diagonal behavior of the heat kernel on CR manifolds. Investigations of normal forms for rigid hypersurfaces will continue. And she will search for an algebra of pseudodifferential operators which contains a parametrix for the Kohn Laplacian on CR manifolds of finite type. Can you hear the shape of a drum? Was it circular or square? This question was first addressed in a mathematical context by V. Kac and continues to inspire intense study. Ramifications to signal processing theory are manifold. The principal investigator will continue her study of the complex analogues to this question, a field of study in which she already is the foremost authority.