Mean Field Equations and Inverse Wave Problems

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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.