Mean Field Equations and Inverse Wave Problems

本课题主要研究两类重要的偏微分方程——平均场方程和逆波方程。平均场方程出现在数学、数学物理和生物学中一些重要现象的研究中，如陈-西蒙斯规范场理论、广义相对论中霍金质量的刚性以及细胞移动和湍流的数学模型。本项目研究的逆波问题在光声和热声断层成像中有直接的应用，将有助于新的医学成像方法的发展，在临床应用中具有重要的潜力。医学影像方法对于疾病的早期发现、诊断和治疗至关重要。本项目为研究生提供研究训练机会。来自弱势群体和弱势背景的本科生和高中生将从拟议的研究、教育和推广计划中受益。主要研究者（PI）本着球覆盖不等式的精神，通过发展新的泛函不等式来研究平均场型方程解的对称性和唯一性。覆盖不平等性的球体最近被PI和他的合作者发现，并从此导致了几个开放问题的解决方案。PI将发展球面的奇异版本，包括不等式和非单连通区域上改进常数的不等式，并将这些不等式扩展到更高的维度。该项目在各种问题上有重要的应用，包括Nirenberg在球上规定高斯曲率的问题、Moser-Trudinger不等式、自对偶和chen - simons规范场理论、广义相对论中霍金质量的刚性、细胞迁移的Keller-Segal型自由边界模型、Onsager的涡旋理论、Toda系统和宇宙弦方程。在项目的第二部分，PI将研究反问题，即通过测量边界上的波动方程的解来确定波的来源及其在介质中的速度，这是一个长期存在的开放问题，在医学成像中具有重要应用。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。

This project focuses on the study of mean field equations and inverse wave equations, two important types of partial differential equations. Mean field equations arise in the study of several important phenomena in mathematics, mathematical physics, and biology such as Chern-Simons gauge field theories, rigidity of Hawking mass in general relativity, and mathematical models for cell mobility and turbulence. Inverse wave problems investigated in this project have immediate applications in photoacoustic and thermoacoustic tomography and will contribute to the advancement of new medical imaging methods with significant potential in clinical applications. Medical imaging methods are crucial for early detection, diagnosis, and treatment of diseases. This project provides research training opportunities for graduate students. Undergraduate students, and high school students from underrepresented groups and disadvantaged backgrounds will benefit from the proposed research, educational, and outreach plans. The principal investigator (PI) aims to study symmetry and uniqueness of solutions of mean field type equations by developing new functional inequalities in the spirit of the Sphere Covering Inequality. The Sphere Covering Inequality was recently discovered by the PI and his collaborator and has since led to solutions for several open problems. The PI will develop singular versions of the sphere covering inequality as well as inequalities with improved constants on non-simply connected regions, and extend such inequalities to higher dimensions. The project has important applications to various problems including Nirenberg's problem of prescribing Gaussian curvature on the sphere, Moser-Trudinger inequalities, self-dual and Chern-Simons gauge field theories, rigidity of Hawking mass in general relativity, Keller-Segal type free boundary models for cell mobility, Onsager's vortex theory, Toda systems, and cosmic string equations. In the second part of the project, the PI will investigate the inverse problem of determining both the sources of a wave and its speed inside a medium from the measurements of the solutions of the wave equation on the boundary, which is a long-standing open problem and has important applications in medical imaging.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.