Variable Coefficient Fourier Analysis

Johns Hopkins University, Baltimore MD

Investigators

Abstract

In this project the principal investigator will study several problems in geometric Fourier analysis. The settings for these problems involve objects known as geometric manifolds of dimension two or greater. Associated to a given manifold are fundamental objects called its 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 omits. Similar phenomena arise for manifolds, and the principal investigator wishes 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 project will also 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. Among the specific problems the principal investigator will study, he seeks to obtain improved estimates that measure the size and concentration of eigenfunctions. In order to do this he will develop what he calls "global harmonic analysis," which is a mixture of classical harmonic analysis, microlocal analysis, and techniques from geometry. The basic estimates that one has in mind are estimates for eigenfunctions and quasimodes, and related highly localized estimates that are sensitive to concentration. The main questions center around how the geometry and the global dynamics of the geodesic flow affect 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 and resolvent estimates coming from the metric Laplacian. The principal investigator is also interested in understanding the harmonic analysis and spectral theory of operators that arise from boundary traces.