RII Track-4:NSF: From Analytic Number Theory to Harmonic Analysis

University Of Mississippi, University MS

Investigators

Abstract

The fields of number theory and harmonic analysis are intricately connected. Harmonic analysis is a fundamental area of mathematics with applications across all of STEM, including number theory. Number theory examines patterns in the integers, and analytic number theory refers to a collection of methods that rely on analyzing continuous functions to understand discrete objects. Harmonic analysis provides the bridge that converts problems between the discrete and continuous settings. The Department of Mathematics at the University of Mississippi has an active analytic number theory group, but there are no harmonic analysts in the state of Mississippi. Through this fellowship, the PI will gain research expertise in harmonic analysis, especially as it applies to discrete settings. The PI will travel to the University of California, Los Angeles to collaborate with and learn from Professor Terence Tao. Tao has a world-class research group of students, postdocs, and visitors, with a strong record of collaboration and cooperative learning. The PI will bring one PhD student to Los Angeles to learn from and collaborate with the students in the UCLA analysis group. This fellowship has potential to have a lasting impact on the PI’s career, will strengthen the analytic number theory group at the University of Mississippi, and will enhance graduate and undergraduate education. This Research Infrastructure Improvement Track-4 EPSCoR Research Fellows (RII Track-4) project would provide a fellowship to an Assistant Professor and training for a graduate student at the University of Mississippi (UM). This fellowship will result in two or more collaborative research projects in the general area of decoupling and restriction theory. There is a deep connection between the circle method in analytic number theory and the method of decoupling in harmonic analysis. During the first year of this fellowship, the PI will collaborate with Tao and his research group to explore this connection by investigating discrete restriction and decoupling theorems for certain curves which lack translation-dilation invariance. In the second year, building on the previous work with discrete restriction and decoupling, the collaboration will pivot into the area of continuous restriction theory. Through this sequence of collaborations, the PI will gain a new research area and will form long-lasting collaborative partnerships with some of the strongest harmonic analysts in the country. The PI will share this new expertise with the University of Mississippi community by developing an undergraduate course and a graduate course in harmonic analysis. The undergraduate course will be useful to mathematics majors planning to go into industry, and will be of interest to science and engineering students who are studying signal processing and data transfers. The course will also create opportunities for interdisciplinary thesis projects co-advised by the PI and UM faculty in other STEM fields. The graduate course will cover harmonic analysis concepts and techniques that are relevant to current research in analytic number theory. Both courses will equip Mississippi students with valuable tools to become more effective mathematicians, scientists, and engineers. 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.