Independence in groups, graphs and the integers

数学的一个基本目标是发展适用于不同主题的一系列问题的技术。在这个方向上，最近最令人兴奋的发展之一是“独立”作为一个统一概念的出现。事实上，代数、组合学和数论中的许多基本结果和开放问题都可以用超图中的独立集合来重新表述。例如，著名的关于整数等差数列的szemersamedi定理可以用独立集的语言来表述。近年来发展起来的新方法解决了该领域许多看似无关的经典开放问题。这导致了对理论普遍适用的技术的驱动。该提案的潜在目标就是开发这样的技术。这些方法将应用于解决一系列具有挑战性的问题，在代数，组合，数论和概率论的接口。该项目的研究包括三个相互关联的主题。首先，该项目将研究无解整数集；这个统一的概念包含了数论中的一系列主要主题，如等差数列、西顿集和无和集。其次，该项目将探索阿贝尔群中计数无和集与最大无和集大小之间的相互作用。该项目的另一个主要方面是研究组合属性的“健壮性”。项目的这一部分将从概率的角度进行研究。特别是，我们将寻求对随机图演化中的尖锐阈值现象的更深层次的理解，这是一个与测量理论和统计物理密切相关的领域。在过去，代数和数论中的许多问题都是通过傅里叶分析方法解决的。这项提议的一个长期目标是为这些研究团体提供额外的组合方法。例如，这个项目的一个关键组成部分是开发所谓的容器结果，它提供了关于独立集分布的信息；事实证明，这些工具在研究许多“计数”问题时至关重要。该建议的另一个关键要素是更好地理解给定代数对象中独立集的结构。

A fundamental aim in mathematics is to develop techniques that apply to a range of problems across different topics. One of the most exciting recent developments in this direction has been the emergence of 'independence' as a unifying concept. Indeed, many fundamental results and open problems in algebra, combinatorics and number theory can be rephrased in terms of independent sets in hypergraphs. For example, the famous Szemerédi theorem on arithmetic progressions in the integers can be phrased in the language of independent sets. Novel approaches developed in the last few years have led to the resolution of many seemingly unrelated classical open problems in this area. This has led to a drive for techniques that are universal to the theory. The underlying goal of the proposal is to develop such techniques. These methods will be applied to tackle a range of challenging problems at the interface of algebra, combinatorics, number theory and probability theory. The research in the project consists of three interconnected themes. Firstly, the project will investigate solution-free sets of integers; this unifying notion encapsulates a range of major topics in number theory such as arithmetic progressions, Sidon sets and sum-free sets. Secondly, the project will explore the interplay between counting sum-free sets in abelian groups and the size of the largest such set. Another major aspect of the project is to investigate how 'robust' a combinatorial property is. This part of the project will be studied from a probabilistic point of view. In particular, we will seek a deeper understanding of sharp threshold phenomena in the evolution of random graphs, an area which has close ties to measure theory and statistical physics.In the past, many problems in algebra and number theory related to this proposal have been tackled via Fourier analytical methods. One long-term aim of this proposal is to provide additional combinatorial approaches that are beneficial for these research communities. For example, one key component of this project is to develop so-called container results which provide information on the distribution of independent sets; such tools have proven vital in the study of a number of 'counting' problems. Another crucial element of the proposal is to develop a better understanding of the structure of independent sets in given algebraic objects.