Research in Harmonic Analysis and Partial Differential Equations

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The project concerns research in harmonic analysis, and in the analysis of partial differential equations (PDE). Harmonic analysis has played major roles in pure and applied sciences since Fourier's work on the theory of heat diffusion, continuing with the success of Schrödinger’s equation in quantum mechanics. It underlies a diverse array of tools widely used in sciences and engineering and offers the promise of further applications in the future. The research is to deal with foundational issues, which may help to underpin future applications. In PDE, the focus will be to study long-time dynamical properties, such as decay and smoothing of dispersive PDE including several fundamental equations describing diverse physical phenomena. In particular, the Dirac equation is a model for graphene, which has important applications in science and engineering. The fourth order Schrödinger equation was introduced to model the propagation of intense laser beams in a bulk medium with Kerr nonlinearity; in addition it is useful in the study of interaction of water waves. In harmonic analysis, the focus lies on questions in Euclidean spaces centered around Lebesgue norm inequalities. One subject of on-going research is the Fourier restriction phenomenon and its applications on questions in PDE and geometric measure theory. The project will involve undergraduate students in research activities through numerical projects in Illinois Geometry Lab and the mentoring of graduate students. More specifically, the research is to encompass dispersive decay and smoothing estimates and the boundedness of wave operators for dispersive PDE such as higher order Schrödinger’s equations and Dirac equations, and to study applications to the regularity properties and long-time dynamics of the nonlinear counterparts. The methods involved will include the spectral theory of self-adjoint operators and oscillatory integral estimates in Fourier analysis. Another area of research is on the fractal dimension of solution graphs of dispersive PDE, or the Talbot effect. Previously, these questions were studied in the case of periodic boundary conditions using exponential sum estimates and smoothing estimates for nonlinear equations; as part of this project, more general geometries such as the sphere and tori in higher dimensions will be investigated. In harmonic analysis the project will entail weighted restriction estimates partly relying on recent developments in decoupling theory, as well as the applications of weighted restriction estimates on questions in geometric measure theory and in dispersive PDE such as the Schrödinger’s equation with a fractal measure as potential.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.