Independence in groups, graphs and the integers

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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.