CAREER: Oscillatory Integrals and the Geometry of Projections

职业：振荡积分和投影几何

• 批准号: 2238818
• 负责人: Hong Wang
• 金额: $ 55.48万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-15 至 2028-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2238818&HistoricalAwards=false
• 关键词: CAREER; Oscillatory; Integrals; Geometry; Projections

This project involves research at the interface of Fourier analysis and geometric measure theory. Fourier analysis studies the relation between a function and its Fourier transform. The Fourier transform of a function, in rough terms, represents the function via a superposition of frequencies. Geometric measure theory studies the geometric properties of sets and measures under transformations. Fractal sets, or sets with highly irregular geometry, are of particular interest in this regard. Recently, the connection between Fourier analysis and geometric measure theory has led to substantial progress in both fields. This project explores the interaction between these two fields, along with possible applications to other fields such as dynamics and number theory. The project also supports workshops for graduate students and early-career mathematicians: these events will promote mathematical expertise within the indicated research areas, will contribute to the professional training of participants, and will foster new research collaborations.The project combines work in restriction theory (within Fourier analysis) and the theory of projections (within geometric measure theory). One component of the planned research involves the study of the mass of a function, with Fourier transform supported on the sphere, on a fractal set. Another component investigates the dimensions of fractal sets under certain linear or nonlinear maps parametrized by curved manifolds. A final component concerns the Kakeya conjecture, which asks how large must a set be if it contains a unit line segment in every direction. These three components, while distinct, are highly interrelated, and progress in each area is anticipated to inform ongoing work in all of these areas.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.

本项目涉及傅立叶分析与几何测量理论的交叉研究。傅里叶分析研究一个函数和它的傅里叶变换之间的关系。一个函数的傅里叶变换，粗略地说，就是通过频率的叠加来表示这个函数。几何测度论研究集合和测度在变换下的几何性质。分形集，或具有高度不规则几何的集，在这方面特别有趣。近年来，傅里叶分析与几何测量理论之间的联系使这两个领域都取得了实质性的进展。该项目探索了这两个领域之间的相互作用，以及可能应用于其他领域，如动力学和数论。该项目还支持研究生和早期职业数学家研讨会：这些活动将促进指定研究领域的数学专业知识，将有助于参与者的专业培训，并将促进新的研究合作。该项目结合了限制理论（傅里叶分析）和投影理论（几何测量理论）的工作。计划研究的一个组成部分是研究一个函数的质量，傅里叶变换支持在球上，分形集合上。另一部分研究了由弯曲流形参数化的线性或非线性映射下分形集的维数。最后一个组成部分与Kakeya猜想有关，该猜想问的是，如果一个集合在每个方向上都包含一个单位线段，它必须有多大。这三个组成部分虽然不同，但高度相互关联，预计每个领域的进展将为所有这些领域正在进行的工作提供信息。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。