Federal Grant Award Summary

The National Science Foundation (NSF), Division of Mathematical Sciences, awarded $200,000 to the University of California, Los Angeles under the Mathematical and Physical Sciences program (CFDA 47.049) for the period July 1, 2026 through June 30, 2029. The project, "Interactions of Fourier Restriction with Fractal Geometry and Number Theory," delivers mathematical research and analysis focused on understanding the refined behavior of the Fourier transform under specialized geometric conditions. The investigator applies harmonic analysis tools to address longstanding problems in number theory and fractal geometry, with particular emphasis on developing new approaches to exponential sum problems through advances in multiscale geometry of balls and tubes within Fourier restriction theory.

The award supports the development of technical mathematical understanding with broad applicability to wave propagation, oscillatory phenomena, and physics equations (including the Schrödinger equation) relevant to engineering and medical imaging applications. Beyond research deliverables, the project includes substantial human capital development through training and mentorship of graduate students and postdoctoral researchers, international collaboration with mathematical experts, and structured support activities for undergraduate transfer students and incoming graduate students. The project strengthens pathways for entry into mathematical research and facilitates the transfer of ideas across challenging areas of mathematical research.