Independence in groups, graphs and the integers

数学的一个基本目标是开发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。实际上，代数学、组合学和数论中的许多基本结果和未解决的问题都可以用超图中的独立集来表述。例如，著名的Szemerédi定理的算术级数的整数可以措辞的语言独立集。在过去的几年中开发的新方法导致了许多看似无关的经典开放问题在这一领域的解决。这导致了对该理论通用的技术的驱动。该提案的根本目标是发展这种技术。这些方法将应用于解决代数，组合学，数论和概率论接口的一系列具有挑战性的问题。该项目的研究包括三个相互关联的主题。首先，该项目将研究整数的无解集;这个统一的概念封装了数论中的一系列主要主题，如算术级数，西顿集和无和集。第二，这个项目将探讨计数阿贝尔群中的自由和集与最大集合的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“鲁棒性”。将从概率的角度研究项目的这一部分。特别是在与测度论和统计物理密切相关的随机图的演化中，对锐阈值现象进行更深入的理解。以往，与此相关的代数和数论中的许多问题都是通过傅立叶分析方法解决的。这项建议的一个长期目标是提供额外的组合方法，有利于这些研究团体。例如，该项目的一个关键组成部分是开发所谓的容器结果，提供有关独立集合分布的信息;这些工具已被证明在研究一些“计数”问题中至关重要。另一个关键因素的建议是发展一个更好的理解结构的独立集在给定的代数对象。

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.