Structure versus Randomness in Algebraic Geometry and Additive Combinatorics

Notions of structure and of randomness are pervasive throughout mathematics, and the study of the interplay between these two notions has shaped entire areas of research. The goal of this project is to obtain quantitative improvements to a myriad of both classical and cutting-edge results that rely on the interplay between structure and randomness in additive combinatorics, number theory, commutative algebra, and algebraic geometry. In turn, these improvements should have applications in multiple areas of computer science such as coding theory, property testing, and derandomization. This project further aims to expand access to combinatorics research, both by involving students in research projects, and by creating educational opportunities for students and the general public. The research in this project has two main aspects. The first is focused on polynomials, aiming to improve the best bounds known for results that build on the structure-vs-randomness phenomenon, and in particular, on regularization of polynomials. A main theme here is leveraging recent progress in the study of tensor ranks, and in the study of quantitative versions of classical theorems about polynomials (e.g., Stillman's conjecture, finite-field Nullstellensatz). The second aspect of this research is focused on general functions that, nonetheless, behave like polynomials to some extent. A central aim here is to improve or develop new variants of general-purpose tools such as the arithmetic regularity lemma, with the goal of making progress towards the Polynomial Gowers Inverse conjecture and other central questions in higher-order Fourier analysis and additive combinatorics. 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.