This mathematics research project is in the area of Fourier analysis: this is a branch of mathematics that plays an important role in the development of mathematical and physical theories. One aspect of the research pursued in this project provides the mathematical foundation for the study of light and sound waves. Fourier analysis continues to play a significant role in deepening one's understanding of the equations that model this behavior. In particular, these investigations yield further insight as to how the presence of a hard boundary surface influences the development of waves. For example, if one listens to the symphony in an auditorium, the sounds heard are affected by the manner in which the acoustic waves reflect off the walls. In this sense, it can be important to understand how the shape of the hall influences its acoustics. While this is, of course, a classical problem, there is more to be understood in terms of how these interactions influence dispersive properties. Moreover, this line of work is important in the analysis of closely related nonlinear equations arising from fiber optics, relativity, and water waves, where there is much to be done in understanding and limiting the various types of instabilities that can occur. A second aspect of this research project seeks to understand the link between geometry and the behavior of vibrational modes. This is closely related to so-called Chladni plates, where one vibrates a metal plate with sand on it and studies the patterns formed by the accumulation of sand, corresponding to the lines on which the plate does not move. Here it is interesting to consider how the shape of the plate influences the patterns that evolve and to estimate their length. These investigations are closely related to themes in quantum chaos and semiclassical analysis.
This mathematical research project in harmonic analysis seeks to understand dispersive properties of light, sound, and quantum waves in various settings such as in curved backgrounds, in nonhomogeneous media, and in the presence of boundary conditions. These dispersive properties are, in turn, influenced by the behavior of paths of least action, so the approaches here rely partially on methods in microlocal analysis, where one seeks to understand such wave propagation in phase space. This wave behavior can be modeled by solutions to partial differential equations, possibly satisfying certain boundary conditions. Of particular interest is to understand how dispersive properties affect basic regularity estimates for solutions to these equations. These regularity estimates actually stem from Fourier restriction theory, and in particular from the classical theorems of Stein, Tomas, and Strichartz. While the non-Euclidean character of these problems means that the classical Fourier transform is not directly applicable, harmonic analysis is nonetheless fundamental to the proposed research. Indeed, the common methods employed by the principal investigator for understanding wave propagation such as Fourier integral operators, the Hadamard parametrix, and wave-packet methods, rely on Fourier analysis to a strong degree. Moreover, the oscillatory integrals that arise in applying these methods are very close to those encountered in harmonic analysis, hence the research here deepens our understanding of the classical theory.
