The Spectral theory and harmonic analysis of Geometric Differential operators

Spectral theory and harmonic analysis are branches of mathematics dealing with natural frequencies (eigenvalues) and modes of vibration (eigenfunctions) of systems arising in geometry, physics and engineering. As such, they have important applications in medical imaging, image compression, and optical communications. In this project I will investigate a variety of problems including quantum chaos, eigenfunction concentration, harmonic analysis on complete noncompact manifolds, and nonlinear wave propagation.