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; 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.

该项目涉及在傅立叶分析和几何测量理论的界面上进行的研究。傅立叶分析研究功能与其傅立叶变换之间的关系。函数的傅立叶变换以粗略的术语表示通过频率叠加的函数。几何测量理论研究了转换下的集合和措施的几何特性。在这方面，分形集或具有高度不规则的几何形状的集合特别感兴趣。最近，傅立叶分析与几何测量理论之间的联系已导致两个领域的实质进步。该项目探讨了这两个字段之间的相互作用，以及可能应用到其他字段（例如动态和数理论）。该项目还支持研究生和早期职业数学家的研讨会：这些事件将促进指定的研究领域内的数学专业知识，将为参与者的专业培训做出贡献，并将促进新的研究合作。计划研究的一个组成部分涉及对函数的质量进行研究，在分形集中，球体上支持傅立叶变换。 另一个组件研究了通过弯曲歧管参数参数的某些线性或非线性图下的分形集合的尺寸。最终的组件涉及Kakeya的猜想，该猜想询问是否包含各个方向的单元线段必须有多大。这三个组成部分虽然很明显，但高度相互关联，并且预计每个领域的进步都将为所有这些领域的正在进行的工作提供信息。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的评估评估标准来通过评估来支持的。