CAREER: Oscillatory Integrals and Applications

本研究项目旨在了解振荡积分。这些对象位于现代调和分析的核心，在数学、物理和邻近领域中无处不在。傅里叶变换可能是最著名的振荡积分算子，它不仅在数学和物理中发挥着重要作用，而且在图像处理和数据分析等应用中也发挥着重要作用。这样的积分也可以用来理解方程的整数解，并表示微分方程组的解，如波动方程和薛定谔方程。这一领域的核心是对振荡积分的大小和衰减特性有一个很好的了解。该项目寻求开发该领域的新工具，用于调查更具挑战性的问题。这些工具不仅来自分析的主题，还来自其他数学领域，如实代数几何、微分几何、有限域理论、几何测度论和模型理论。在教育方面，该项目包括组织初级研究人员的研究研讨会，以及与伯克利数学圈的合作，为小学、初中和高中的学生提供了解现代数学的新机会。该项目涉及重要的振荡积分算子的有界性。要考虑的主题包括抛物面的傅里叶限制，Parsell-Vinogradov流形的傅立叶限制，以及Bochner-Riesz乘子。还将研究振荡积分的应用，并展望在数论和几何测度论(Falconer距离猜想，Kakeya集)等领域的新应用。数学工具箱涉及分析(尺度归纳法和解耦)、代数几何(多项式方法、实代数几何和o-极小几何)、微分几何、组合学(多线性Kakeya、有限域工具和和积理论)和几何测度论(径向投影)。该项目的教育部分包括为初级研究人员举办的振荡积分研讨会，以及对伯克利数学圈的组织和发展做出的贡献。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.