Collaborative Research: Microlocal Concentration and Propagation in Spectral Theory

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In nature, solutions of wave-type equations appear in phenomena as far reaching as wave functions of quantum particles, the acoustics of concert halls, and scattering of gravitational waves. In these situations, there are certain fundamental frequencies and vibrational modes associated to the underlying geometry of the space. These frequencies and modes play an essential role in the understanding of the behavior of wave phenomena. They appear, for example, in the acoustical design of concert halls and the vibration of instruments. This project develops tools, based on propagation phenomena, to study the delicate relationship between the geometry of the space and behavior of the corresponding modes. This research will provide new insight into the structure of vibrational modes for general spaces. Many quantitative regularity results (as measured by L^p norms) are already known for spaces with non-positive curvature. However, these types of estimates are unavailable under less restrictive geometric assumptions. The main goals of this project are two fold: (1) to describe the concentration phenomena that result in L^p norm growth for individual modes, and (2) to use this understanding to produce estimates for modes and frequencies based solely on dynamical properties of the underlying manifold.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.