Additive and Combinatorial Problems in Groups

群中的加法和组合问题

• 批准号: 2246921
• 负责人: Thai Hoang Le
• 金额: $ 21.9万
• 依托单位: University of Mississippi
• 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: 2023-06-01 至 2026-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2246921&HistoricalAwards=false
• 关键词: Additive; Combinatorial; Problems; Groups

This project is jointly funded by the Combinatorics program, the Established Program to Stimulate Competitive Research (EPSCoR), and the Algebra and Number Theory program. In this project, the PI aims to study certain problems in groups. A group is a mathematical structure in which we can add and subtract two elements, just like in the integers. Groups are fundamental objects in mathematics and ubiquitous in real life; for example, the numbers on a 12-hour clock form a group of 12 elements. The addressed problems are of combinatorial nature; for example, if we take any subset of a group, then as long as the subset is ''large'' in some sense (without any assumption on its structure), we predict that some pattern will emerge. Graduate students will be trained as part of this proposal. The PI will also be involved in K-12 outreach.The groups studied in this project are: (1) The integers and their polynomial ring analogs, for which techniques from number theory apply, (2) finite groups, including vector spaces over finite fields, for which techniques from algebra apply, and (3) more general amenable groups, for which techniques from Fourier analysis and ergodic theory apply. Problems studied in this proposal include the following: If a subset A of a group G is large in some sense (e.g. A has positive density, or comes from a finite partition of G), then can one find structures in sumsets such as A-A and A+A-A? This question is intimately related to various notions of recurrence in ergodic theory and topological dynamics. If A is a sparse set, e.g. a subset of the primes, then this question will lead to patterns hitherto unexplored in the primes. The PI will also study other additive problems such as additive bases, essential components, and exponential sums using various techniques.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.

该项目由组合学计划，刺激竞争研究的既定计划（EPSCoR）以及代数和数论计划共同资助。在这个项目中，PI的目的是在小组中研究某些问题。群是一种数学结构，我们可以在其中添加和减去两个元素，就像整数一样。群是数学中的基本对象，在真实的生活中无处不在;例如，12小时时钟上的数字形成了12个元素的群。所解决的问题是组合性质的;例如，如果我们取一个组的任何子集，那么只要该子集在某种意义上是“大的”（没有对其结构的任何假设），我们就预测会出现某种模式。研究生将接受培训，作为本提案的一部分。PI也将参与K-12的推广。在这个项目中研究的群体是：（1）整数及其多项式环的类似物，从数论的技术应用，（2）有限群，包括有限域上的向量空间，从代数的技术应用，和（3）更一般的顺从群，从傅立叶分析和遍历理论的技术应用。在这个建议中研究的问题包括：如果一个群G的子集A在某种意义上是大的（例如A有正密度，或来自G的有限划分），那么可以找到结构的和集，如A-A和A+A-A？这个问题与遍历理论和拓扑动力学中的各种递归概念密切相关。如果A是一个稀疏集，例如素数的子集，那么这个问题将导致迄今为止尚未探索的素数模式。PI还将使用各种技术研究其他加性问题，如加性基础、基本成分和指数和。该奖项反映了NSF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。