CAREER: Oscillatory Integrals and Applications

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

This research project aims to understand oscillatory integrals. These objects lie at the heart of modern harmonic analysis and are ubiquitous in mathematics, physics, and neighboring areas. The Fourier transform, perhaps the best-known oscillatory integral operator, plays a fundamental role not only in mathematics and physics but also in applications such as image processing and data analysis. Such integrals can also be used to understand integer solutions of equations, and to express solutions to differential equations such as the wave and Schrödinger equations. Central to this area is a good understanding of the size and decay properties of oscillatory integrals. The project seeks to develop new tools in the field that will be used to investigate more challenging questions. These tools will come not only from the subject of analysis, but also from other mathematical areas such as real algebraic geometry, differential geometry, theory of finite fields, geometric measure theory, and model theory. On the educational side, the project involves the organization of research workshops for junior researchers, as well as work with the Berkeley Math Circle, providing new opportunities for elementary, middle, and high school students to get to know about modern mathematics. This project concerns boundedness properties of important oscillatory integral operators. Topics to be considered include Fourier restriction for the paraboloid, Fourier restriction for Parsell-Vinogradov manifolds, and the Bochner-Riesz multiplier. Applications of oscillatory integrals will also be investigated; new applications in areas such as number theory and geometric measure theory (Falconer's distance conjecture, Kakeya sets) are anticipated. The mathematical toolbox involves analysis (induction on scales and decoupling), algebraic geometry (the polynomial method, real algebraic geometry, and o-minimal geometry), differential geometry, combinatorics (multilinear Kakeya, finite field tools, and sum-product theory), and geometric measure theory (radial projection). The educational components of the project include workshops on oscillatory integrals for junior researchers and contributions to the organization and development of the Berkeley Math Circle. 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.