CAREER: Oscillatory Integrals and the Geometry of Projections

本计画主要研究傅立叶分析与几何量测理论之介面。傅里叶分析研究函数和它的傅里叶变换之间的关系。函数的傅里叶变换，粗略地说，通过频率的叠加来表示函数。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形状的集，在这方面特别令人感兴趣。最近，傅立叶分析和几何测度理论之间的联系导致了这两个领域的实质性进展。这个项目探讨了这两个领域之间的相互作用，沿着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的研讨会：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（傅立叶分析）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涉及研究一个函数的质量，在球体上支持傅立叶变换，在分形集上。 另一部分研究了在某些线性或非线性映射下，由弯曲流形参数化的分形集的维数。最后一个部分涉及Kakeya猜想，它询问如果一个集合在每个方向上都包含一个单位线段，那么它必须有多大。这三个组成部分，虽然不同，是高度相互关联的，并在每个领域的进展预计将通知正在进行的工作在所有这些领域。这个奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的知识价值和更广泛的影响审查标准。

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.