CAREER: Building bridges between number theory and harmonic analysis

Carnegie Mellon University, Pittsburgh PA

Investigators

Abstract

Two areas of mathematics often viewed as separate are number theory and analysis. Number theory deals with properties of whole numbers, such as quickly factoring large numbers into primes, and underpins computer security and the behavior of black holes. Harmonic analysis can be viewed as mathematically making a sculpture out of unformed clay; this field takes a complicated function and breaks it up into simple pieces – its harmonics – and is central to medical imaging and quantum states. Imagining and building a new landscape where areas sometimes perceived as disconnected can harmonize is at the forefront of this project: Instead of exploiting known connections, new ones will be created centered around three distinct areas. Firstly, quantitative behavior of discrete operators with curvature will be undertaken; while relevant to many applications, the novelty here will be developing the infrastructure to use both analysis and number theory from the perspective of both fields. The second part will be a continuation of the discovery of hidden number theoretic properties in the framework underlying harmonic analysis, furthering this mathematical grid system in both areas. Finally, determining the behavior of a random object of algebraic interest, such as a polynomial, is a central problem in mathematics. Integrating Fourier analysis into this playing field, will enable a better prediction of this randomness in a wide variety of ways. Undergraduates will be heavily involved in research, especially in the second part, via a summer and semester research and training program. The project consists of three main programs, all on the intersection between harmonic analysis and number theory. Firstly, operator bounds for discrete operators involving integration over curved submanifolds will be obtained. This is a delicate topic in continuous analysis and the discrete setting introduces new challenges and obstacles. A development of number theoretic tools in tandem with the analysis will dramatically push forward understanding of these objects. Additionally, these bounds are connected to lattice point counts and information on this front will be quantified via refined analysis of exponential sums, including discrete restriction problems for a variety of surfaces that are translation dilation invariant. Secondly, a continued uncovering of a hidden number theoretic structure and techniques in the realm of dyadic analysis will be pursued, leading to a myriad of structure theorems. Thirdly, the program in Fourier analytic techniques in arithmetic statistics will use novel insights from analysis to improve a variety of counts of algebraic interest, such as fields and polynomials. Prior investigations here have already been successful as recent results include a major step in resolving a conjecture of van der Waerden on Galois groups of random polynomials and counting number fields of bounded discriminant. Investigations along these lines will be continued. 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.