Variable Coefficient Fourier Analysis

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The PI will study several problems in Geometric Fourier Analysis. The settings for these problems involve geometric manifolds of dimension two or more. Associated to a given manifold are fundamental objects called eigenfunctions. These are the fundamental modes of vibration of the manifold, and they are the higher dimensional analogs of the familiar trigonometric functions for the circle. Designers of musical instruments are well aware that the shape of, say, a drum or the soundboard of stringed instrument affects the basic tones that it emits. Similar phenomena arise for manifolds, and the project aims to study precisely how their shapes, such as how they are curved, affect the resulting eigenfunctions. Just as in music, one particularly expects different shapes and geometries to become more apparent in the behavior of the fundamental modes of vibration as the frequency becomes larger and larger. These eigenfunctions are solutions of a differential equation that is similar to the wave equation, and the PI also wishes to study similar problems involving it. The general theme is to study how solutions of wave equations are affected by their physical backgrounds, such as whether or not black holes are present or whether the background becomes very close to a vacuum near infinity. The PI will be supervising graduate students in domains related to the proposed research. Such activities will be supported by the award.Among the specific problems to be studied, the project aims to obtain improved estimates that measure the size and concentration of eigenfunctions. In order to do this, the PI will develop what could be called "global harmonic analysis", which is a mixture of classical harmonic analysis, microlocal analysis and techniques from geometry. The basic estimates to have in mind are Lp-estimates for eigenfunctions and quasimodes, and related highly localized L2 estimates that are sensitive to concentration. The main questions center around how the geometry and the global dynamics of the geodesic flow affects the estimates and the kinds of functions that saturate them. The latter issue is closely related to the much studied (but still not well understood) questions of concentration, oscillation and size properties of modes and quasimodes in spectral asymptotics. These questions are also naturally linked to the long-time properties of the solution operator for the wave equation, Schrodinger equation and resolvent estimates coming from the metric Laplacian. The PI is particularly interested in high frequency solutions and improving existing results under geometric assumptions.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.