Structure versus Randomness in Algebraic Geometry and Additive Combinatorics

结构和随机性的概念在整个数学中无处不在，对这两个概念之间相互作用的研究塑造了整个研究领域。这个项目的目标是获得对无数经典和前沿结果的量化改进，这些结果依赖于加法组合学、数论、交换代数和代数几何中结构和随机性之间的相互作用。反过来，这些改进应该在计算机科学的多个领域中得到应用，例如编码理论、属性测试和去随机化。该项目还旨在通过让学生参与研究项目，并为学生和公众创造教育机会，扩大获得组合学研究的机会。本项目的研究主要有两个方面。第一个重点是多项式，旨在改善基于结构与随机性现象的结果的最佳界限，特别是多项式的正则化。这里的一个主要主题是利用张量秩研究的最新进展，以及关于多项式的经典定理的量化版本的研究(例如，Stillman猜想，有限域Nullstellensatz)。这项研究的第二个方面集中在一般函数上，但在某种程度上，这些函数的行为类似于多项式。这里的一个中心目标是改进或开发通用工具的新变体，如算术正则性引理，目标是在高阶傅立叶分析和加性组合学中的多项式Gowers逆猜想和其他中心问题方面取得进展。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.