Mathematical Sciences: Spectral Geometry & Representation Theory on Higher Step Riemannian Nilmanifolds

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9409209 GORNET For (M,g) a closed Riemannian manifold, denote by spec(M,g) the collection of eigenvalues with multiplicities of the associated Laplace-Beltrami operator acting on smooth functions on M. Gornet's research focuses on the question "What geometric information is contained in the spectrum of a manifold?" In particular, she has developed a new construction for producing pairs of isospectral Riemannian nilmanifolds. This construction has produced new pairs of isospectral manifolds exhibiting many properties not previously found. Such examples are the only way to discover properties not determined by the spectrum. Geometric properties Gornet is currently studying include the length spectrum (the collection of lengths of closed geodesics) and the marked length spectrum (which also includes information on the free homotopy classes of the closed geodesics.) Techniques used in Gornet's research come from Riemannian geometry, representation theory, and Lie groups. Gornet proposes to use the Research Planning Grant to explore new techniques in representation theory and in the study of closed geodesics to supplement her current background in spectral and Riemannian geometry. ***