Questions at the Interface of Analysis and Number Theory

Carnegie Mellon University, Pittsburgh PA

Investigators

Abstract

Harmonic analysis and number theory are fundamental fields of mathematics that are used to describe and interpret many real-world phenomena. Harmonic analysis involves breaking up a mathematical object such as a function into pieces that are easier to understand. The beauty of this area is that the pieces are oftentimes simple, yet represent the whole with accuracy. Number theory involves deceptively simple statements about the integers, easy to test, yet often difficult to prove. Though seemingly disparate, analysis and number theory share many interactions. For instance, one can use intricate analysis of complex functions to answer fundamental questions about prime numbers. This project explores a variety of problems at the interface of these two areas. In particular, the PI will consider discrete variants of operators in analysis, which enjoy applications in fields such as medical imaging and cosmology. To analyze these operators, continuous techniques often fail, and one has to develop number theoretic techniques adapted to the underlying geometry of the analytic problem. The PI seeks to provide new bounds, new techniques, sharper analysis and broader connections. The PI also plans to bring Fourier analysis, a fundamental decomposition of the time-frequency domain, such as that used to understand waves, into the emerging field of arithmetic statistics. Here she seeks to provide sharp counts of a wide variety of objects of arithmetic interest, such as elliptic curves used in cryptography. As a broader impact, the PI will spark new mathematical conversations between analysts and number theorists and also improve the educational and scientific climate for underrepresented groups. This project addresses several fundamental questions at the interface of analysis and number theory. Firstly, the PI pursues bounds for discrete variants of continuous operators in harmonic analysis that involve integration over a curved subvariety. These bounds provide quantitative distributional facts about the underlying Diophantine equations that define these varieties, which makes them different from their continuous counterparts. In particular, since continuous techniques usually do not carry over in this setting, the PI will develop refined number theoretic techniques to bound several operators, including multilinear spherical variants, variants defined over the primes, and higher codimensional analogues. In particular, the higher codimensional study should open new avenues of problems as very little is known in this setting. Solving these problems has connections to discrete geometry, lattice point counts of surfaces, and Falconer's distance conjecture. In another series of problems, the PI will pursue "sparse bounds" for both continuous and discrete operators. Sparse bounds are a refinement of Lebesgue space bounds that allow one to deduce weighted estimates. Finally, the PI plans to pursue a far reaching program in arithmetic statistics. This is an area greatly developed on the algebraic side recently. The PI plans to inject Fourier analytic techniques to obtain precise lattice point counts that are adaptable to take advantage of the power of the algebraic techniques and push those bounds even further. In particular, the PI hopes to obtain counts on certain objects such as elliptic curves, with an eye to not only developing techniques, but also fostering interactions between number theorists and analysts in new ways. 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.