Interactions of Fourier Restriction with Fractal Geometry and Number Theory

The investigator conducts mathematical research in the subareas of number theory and fractal geometry using tools from harmonic analysis. A central tool in harmonic analysis is the Fourier transform, which is used in engineering, medical imaging (MRI), and physics (to study wave propagation and oscillatory phenomena). This project seeks to understand refined behavior of the Fourier transform under special geometric conditions which are present in important applications, such as the wave and Schrödinger equations in physics. The broad applicability of the Fourier transform means that technical understanding on the harmonic analysis side leads to progress on longstanding problems in other areas. The detailed Fourier transform properties that the investigator studies will strengthen connections and enable the transfer of ideas between neighboring but challenging areas of mathematical research. The project offers training opportunities for graduate students and postdoctoral researchers and supports collaboration with experts from around the world. Drawing on personal experience, the Principal Investigator (PI) organizes activities supporting undergraduate transfer students and incoming graduate students, helping strengthen pathways into mathematical research. This project investigates how recent advances in the multiscale geometry of balls and tubes can be incorporated into Fourier restriction theory. A main motivation is to develop new routes for attacking longstanding exponential sum problems in number theory which fall within the scope of restriction theory but remain beyond the reach of classical estimates. Functions with frequencies lying on curved surfaces decompose into pieces which are localized to thin tubes pointing in different directions, demonstrating Kakeya-type behavior. Recent breakthroughs on the Kakeya-type questions lying at the heart of the Fourier restriction conjecture provide new tools to establish sharper oscillatory estimates and refined decoupling inequalities. Building on these developments, the PI addresses several central problems, including special cases of the Mizohata-Takeuchi conjecture in the plane and moment estimates for cubic Weyl sums in one and two dimensions. The general Mizohata-Takeuchi conjecture is closely connected to the Kakeya conjecture (open in dimensions greater than 3). Leveraging tools from high-low decoupling, introduced by the PI with Larry Guth and Hong Wang, combined with new geometric insights about weights and wave packets, the project will initiate a systematic study of cases in which Mizohata-Takeuchi estimates can be resolved. For cubic Weyl sums, the project investigates whether the main obstacles are geometric or arithmetic in nature. 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. NSF Award ID: 2556201 | Program: 01002627DB NSF RESEARCH & RELATED ACTIVIT | Principal Investigator: Dominique Maldague | Institution: University of California-Los Angeles, LOS ANGELES, CA | Award Amount: $200,000 View on NSF Award Search: https://www.nsf.gov/awardsearch/show-award/?AWD_ID=2556201 View on Research.gov: https://www.research.gov/awardapi-service/v1/awards/2556201.html