Problems in geometric analysis

Dartmouth College, Hanover NH

Investigators

Abstract

Inverse spectral geometry is the study of the extent to which the geometry of a Riemannian manifold can be recovered from spectral data. For compact Riemannian manifolds, the natural spectral data are the eigenvalues of the Laplacian. The principal investigator along with various research collaborators will address inverse spectral problems on symmetric and locally symmetric spaces and on line bundles over Riemann surfaces. They will also study the extent to which the spectrum of a compact K\"ahler manifold determines the structure of the associated generalized Jacobian varieties and Albanese tori. For non-compact manifolds, the relevant spectral data are the scattering resonances and scattering phase. Gordon, and Webb, along with P. Perry, will continue their study of obstacles with the same scattering resonances and scattering phase. Spectral geometry, which is rooted in spectroscopy, studies the relationship between the geometry of an object such as a vibrating drum and spectral data such as the characteristic frequencies of vibration. Spectral geometry draws from many areas of mathematics such as geometric analysis, representation theory, and number theory and is an active area of interplay between mathematics and physics. This project will focus primarily on geometric aspects, frequently in settings in which Lie group representations play a central role. Symmetric spaces are the model spaces of Riemannian geometry. One focus of the research will be the question of whether symmetric spaces are spectrally distinguishable from other geometry objects. Other aspects of the research will be the relationship between the spectral data and various geometric properties such as the lengths of closed geodesics.