Singularities in Harmonic Analysis and the Calculus of Variations

调和分析和变分法是数学分析的两个最古老的分支。谐波分析是研究函数的尺度不变性质，如小提琴弦的振动；它是现代图像和信号处理的基础。变分法是对能量最小化的研究；例如，挂在电线上的肥皂膜形成一个使表面张力最小的形状。这个项目的目的是双重的：首先，使用谐波分析来研究能量最小化器在扰动下的行为（就像微风中的肥皂膜）。其次，使用变分法的技术来研究谐波分析中的一个中心问题，即扩散（即热量在房间内传播）如何检测几何形状。这种协同作用不仅会带来数学上的突破，还会提高我们使用这些数学思想来预测物理现象的能力。本项目为研究生提供研究训练机会。研究者感兴趣的是最小化点的奇点（最小化点不光滑的地方，就像肥皂泡相互接触时形成的角落）。特别是，研究者希望开发工具来研究这些在扰动（无论是能量还是初始条件）下持续存在的奇点。为了开发这些工具，需要谐波分析和几何测量理论的新思想。研究者还对椭圆微分方程的解（它是扩散模型）如何检测欧几里德空间中域的几何形状的问题感兴趣。近年来，有重大突破表明均匀扩散可以检测到一些几何特征。研究人员认为，从变分学的想法可以扩展这一突破，以了解非均匀扩散可以检测到更精细的几何细节。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。

Harmonic analysis and the calculus of variations are two of the oldest branches of mathematical analysis. Harmonic analysis is the study of the scale invariant properties of functions such as the oscillation of violin strings; it underpins much of modern image and signal processing. The calculus of variations is the study of energy minimizers; for example, a soap film hanging off a wire forms a shape which minimizes the surface tension. The purpose of this project is twofold: first to use harmonic analysis to study the behavior of energy minimizers under perturbations (like a soap film in a mild wind). Second, to use techniques from the calculus of variations to study a central problem in harmonic analysis, namely how does diffusion (i.e. heat spreading through a room) detect geometry. This synergy should not only lead to mathematical breakthroughs but also refine our ability to use these mathematical ideas to predict physical phenomena. The project provides research training opportunities for graduate students.The investigator is interested in singularities of minimizers (places where the minimizer fails to be smooth, like the corners soap bubbles form when they touch each other). In particular, the investigator wants to develop tools to study these singularities which are persistent under perturbations (either of the energy or the initial conditions). To develop these tools, new ideas from harmonic analysis and geometric measure theory are necessary. The investigator is also interested in the problem of how solutions to elliptic differential equations (which model diffusion) detect the geometry of domains in Euclidean space. Recently, there have been major breakthroughs showing that homogeneous diffusion can detect some geometric features. The investigator believes that ideas from the calculus of variations can extend this breakthrough to understand what finer geometric details inhomogeneous diffusion can detect.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.