CAREER: Eigenfunctions, Weyl Laws, and Random Waves

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A vast array of physical phenomena, ranging from the propagation of waves to the location of quantum particles, is dictated by the behavior of some special functions that solve what is known as the Helmholtz equation. The overarching goal of this research program is to understand how properties of these functions depend upon the underlying geometry of the space being studied. The project includes two main educational efforts. The first is the creation of working groups for incoming graduate students; its goal is to prepare the students to develop a strong set of skills to apply for grants and to prepare them for teaching. The second is the development of online materials for a course on semiclassical analysis that will be available for students to use whenever the in-person version of the course is not offered. Understanding the behavior of Laplace eigenfunctions is of fundamental importance in mathematical physics and has been studied since the 1700s. In particular, the challenging study of concentration properties of high energy eigenfunctions has been the subject of extensive work, and the tools developed for studying Laplace eigenfunctions have had a profound impact on nearly every area of spectral theory and geometric analysis. This project concerns a framework developed by the investigator to extract information on the structure of eigenfunctions from their concentration and propagation behavior in phase space via the use of semiclassical analysis. The current research aims to adapt and develop these methods to reach a deeper understanding of eigenfunction behavior by studying their pointwise growth, L^p-norms, associated two-point Weyl Laws, and applications to random waves.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.