Structure versus Randomness in Algebraic Geometry and Additive Combinatorics

结构和随机性的概念在数学中无处不在，对这两个概念之间相互作用的研究塑造了整个研究领域。该项目的目标是获得大量经典和前沿结果的定量改进，这些结果依赖于加性组合学、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，这些改进应该应用于计算机科学的多个领域，如编码理论、属性测试和非随机化。该项目进一步旨在通过让学生参与研究项目以及为学生和公众创造教育机会来扩大组合学研究的机会。本课题的研究主要有两个方面。第一个集中在多项式上，旨在改进基于结构vs随机现象的结果的最佳边界，特别是多项式的正则化。这里的一个主要主题是利用最近在张量秩研究方面的进展，以及对多项式经典定理的定量版本的研究（例如，Stillman猜想，有限域Nullstellensatz）。本研究的第二个方面集中在一般函数上，尽管如此，在某种程度上表现得像多项式。这里的中心目标是改进或开发通用工具的新变体，如算术正则引理，目标是在多项式高尔斯反猜想和高阶傅立叶分析和加性组合学中的其他中心问题上取得进展。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.