Topics in Analytic Number Theory and Additive Combinatorics

Project Summary Overview: The PI proposes to continue his work surrounding additive problems with primes by developing and combining analytical and combinatorial tools. The classical approaches using the theory of L-functions and sieve theory have achieved great results in this direction, and it is expected that by incorporating tools from additive combinatorics one can go beyond what the classical methods achieve. The PI proposes to continue to investigate the theory of Gowers norms for multiplicative functions and for the von-Mangoldt function. These are ``higher-order” extensions to classical exponential sum estimates involving these functions. The study of general Gowers norms for these functions is an essential analytic input for applying additive combinatorial tools. Intellectual Merit: Analytic number theory, and its applications and interactions, are currently experiencing intensive progress, in sometimes unexpected directions. In recent years, many important classical questions have seen spectacular advances based on new techniques; conversely, methods developed in analytic number theory have led to the solution of striking problems in other fields. For instance, Green-Tao-Ziegler showed that one can count asymptotically the number of primes represented by a general class of systems of linear equations, by proving the inverse theorem for the Gowers norms. The additive combinatorial ideas in their proof, involving pseudorandomness and Gowers norms, have a strong link with topics in theoretical computer science such as extractors and expanders, and property testing. It is hoped that the need for stronger results to cater for applications in analytic number theory will drive the development of additive combinatorics, and vice versa. Broader Impacts: Via the links between additive combinatorics and theoretical computer science discussed in the intellectual merit section above, the proposed project can potentially benefit academics working in computer science and lead to applications in cryptography and financial security. The PI attended the workshop ``Pseudorandomness” held at Simons Institute for the Theory of Computing in Spring 2017, when various aspects of these links were discussed. As the PI progresses in building a research program, he requests summer research funding for one graduate student in each year of the grant. Allowing one student the opportunity to work closely with the PI over the summer would benefit both the student and the PI. The department has admitted a new graduate student, Junren Zheng, who has expressed serious interest in working with the PI. The PI has already been quite involved in mentoring graduate and undergraduate students and postdocs at his institution, the University of Kentucky. In the past year he has supervised a visiting graduate student, Mengdi Wang, and he has offered semester-long independent study opportunities to two undergraduates. He is currently mentoring a postdoc, Thomas Tran. The PI will continue to participate actively in the algebra and number theory community at the University of Kentucky and beyond. This means not only talking informally with students and young mathematicians, but also inviting young number theorists to present their work in the University of Kentucky algebra and number theory seminar.