Structure versus Randomness in Algebraic Geometry and Additive Combinatorics

代数几何和加法组合中的结构与随机性

• 批准号: 2302988
• 负责人: Guy Moshkovitz
• 金额: $ 20.14万
• 依托单位: CUNY Baruch College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-06-15 至 2026-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2302988&HistoricalAwards=false
• 关键词: Structure; versus; Randomness; Algebraic; Geometry

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.

在整个数学过程中，结构和随机性的概念普遍存在，对这两个概念之间的相互作用的研究塑造了整个研究领域。该项目的目的是获得众多经典和尖端结果的定量改进，这些结果依赖于添加剂组合学中的结构和随机性之间的相互作用，数字理论，交换代数和代数几何形状。反过来，这些改进应在计算机科学的多个领域中应用，例如编码理论，财产测试和降低。该项目进一步旨在通过使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对组合研究的访问。该项目的研究有两个主要方面。第一个侧重于多项式，旨在改善以结构-VS随机现象的结果，尤其是多项式正规化为基础的最佳界限。这里的一个主要主题是利用张量排名研究的最新进展，以及研究有关多项式的经典定理的定量版本（例如，Stillman的猜想，有限的现场无效的无史塔兹）。这项研究的第二个方面集中在一般功能上，尽管如此，在某种程度上，其行为就像多项式。这里的一个主要目的是改善或开发通用工具（例如算术规则性引理）的新变异，目的是在高阶傅立叶分析和加法组合术中朝着多项式gowers逆猜测和其他核心问题取得进展。这对NSF的法定任务和审查的范围均通过构成构成的范围和构成的作用。