Additive and Combinatorial Problems in Groups

该项目由组合学计划、已建立的激励竞争研究计划(EPSCoR)以及代数和数论计划共同资助。在这个项目中，PI旨在分组研究某些问题。群是一种数学结构，我们可以在其中增加和减少两个元素，就像在整数中一样。组是数学中的基本对象，在现实生活中无处不在；例如，12小时时钟上的数字构成了12个元素的组。所讨论的问题是组合性质的；例如，如果我们取一个群的任何子集，那么只要该子集在某种意义上是“大的”(不对其结构作任何假设)，我们就预测会出现某种模式。作为这项提议的一部分，研究生将接受培训。这个项目研究的群是：(1)整数和它们的多项式环模拟，应用数论的技巧；(2)有限群，包括有限域上的向量空间，应用代数的技巧；以及(3)更一般的从属群，应用傅立叶分析和遍历理论的技巧。这个方案所研究的问题包括：如果群G的子集A在某种意义上是大的(例如，A具有正密度，或者来自G的有限划分)，那么人们能否在和集中找到结构，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常返性概念密切相关。如果A是稀疏集，例如素数的子集，那么这个问题将导致素数中迄今未探索的模式。PI还将使用各种技术研究其他加法问题，如加法基数、基本分量和指数和。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。

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.