Some problems at the interface of harmonic analysis, number theory, and combinatorics

所谓的拉姆齐理论处理的是在大而无序的集合中寻找结构的问题。在几何集合中，证明这样的集合包含给定有限集合的平移和旋转副本，或其足够大的扩张。换句话说，它是研究几何图案的发生。在过去的15年里，线性模式的研究取得了显著的进展，开发和引入了数学分析工具，通常被称为高阶傅立叶分析。主要成就之一是格林和陶的著名结果，该结果表明存在任意等距素数的长序列。本项目建立在这一发展的基础上，其主要目标之一是开发分析工具，以理解几何和算术（即由方程定义）结构在大型但其他任意集合中的出现。这些问题出现在素数格和整数格的背景下，也出现在经典欧几里得空间中。首席研究员的方法涉及离散谐波分析和数论技术的相互作用，除了一个新的成分，来自加性组合学的想法。该项目的第一个动机背景是素数：研究素数之间的非线性关系，并研究在素数坐标点之间寻找几何星座的相关问题。将素数视为整数的随机子集的基本哲学自然导致了在大整数点集和欧几里得空间的大可测量子集中的类似问题的研究。这些集合中的几何结构还没有被很好地理解。该项目旨在开发一种基于加法组合学现代观点的通用方法；也就是说，建立适当的随机性概念，以控制某个模式出现的频率，并证明非适当随机集合的结构定理。基本结构是解析的，并且与离散调和分析中研究的对象有关，例如最大算子和作用于整数格上定义的函数的Radon变换。最后，该项目旨在从这个新颖的角度研究欧几里得空间的大可测量子集的几何模式，加强加性组合学与经典调和分析之间的联系。

So-called Ramsey theory deals with the problem of finding structures in large but otherwise disorganized sets. In the geometric setting it is to show that such sets contain a translated and rotated copy of a given finite set, or of its sufficiently large dilates. In other words it is to study the occurrence of geometric patterns. Over the past fifteen years there has been a remarkable progress of the study of linear patterns, developing and introducing tools from mathematical analysis, often referred to as higher-order Fourier analysis. Among the major achievements is the celebrated result of Green and Tao, which states that there are arbitrary long sequences of equally spaced prime numbers. This project builds on this development, and one of its major objectives is to develop analytic tools to understand the occurrence of geometric and arithmetic (i.e., defined by equations) structures in large but otherwise arbitrary sets. The problems arise in the context of the prime and integer lattice and also in classical Euclidean spaces. The principal investigator's approaches involve the interplay of techniques from discrete harmonic analysis and number theory, in addition to a new ingredient, ideas from additive combinatorics.The first motivational context for the project is that of prime numbers: to study nonlinear relations among the primes and to investigate the related problem of finding geometric constellations among points with prime coordinates. The underlying philosophy of considering the primes as a random subset of the integers leads naturally to the study of analogous questions in large sets of integer points and also in large measurable subsets of Euclidean spaces. Geometric structures in such sets are not well understood. The project aims to develop a general approach based on the modern point of view of additive combinatorics; namely, to establish appropriate notions of randomness that control the frequency at which a certain pattern occurs and to prove structure theorems for sets that are not suitably random. The underlying constructs are analytic and are related to objects studied in discrete harmonic analysis such as maximal operators and Radon transforms acting on functions defined on the integer lattice. Finally, the project aims to study geometric patterns in large measurable subsets of Euclidean spaces from this novel point of view, strengthening the connections between additive combinatorics and classical harmonic analysis.