Variable Coefficient Fourier Analysis

Johns Hopkins University, Baltimore MD

Investigators

Abstract

The principal investigator 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 omits. Similar phenomena arise for manifolds, and we wish 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 in the project, the principal investigator desires to obtain improved estimates for the nodal sets (zero sets) of eigenfunctions. There is a conjecture of Yau asserting that the size of this codimension-one set should be comparable to its frequency. Although it has been fully settled in the real analytic setting, less is known for smooth manifolds. The principal investigator has already shown that there are connections between this problem and Lebesgue space estimates for eigenfunctions that can detect certain types of concentration. Several problems in the project involve developing this active field. The principal investigator would also like to obtain improved bounds for so-called period integrals of eigenfunctions under the assumption of negative curvature. This assumption is known to be necessary, and the problem measures the random cancellation of eigenfunctions along geodesics. There are connections with this problem and analytic number theory, but to date the results that the principal investigator has recently obtained using harmonic analysis are the best known. He would like to combine them with number theory techniques to try to obtain improved bounds. There are connections between these problems and other problems that the project will study involving estimates for wave equations ("Strichartz estimates") and microlocal analysis, especially propagation of singularities.