Higher-Order Fourier analysis and related issues

该项目属于EPSRC数学科学研究领域。背景：高阶傅立叶分析是一个在过去20年里兴起的数学课题，但人们对它的理解仍然很少。它是一个工具，可以用来分析组合学、数论和数学分析中的某些问题，这些问题不符合涉及字符的传统的傅立叶分析。虽然已经取得了一些显著的成功，例如Gowers对Szmeredi关于算术级数的定理和关于素数的Green-Tao定理的新证明，但仍有许多需要理解的地方。尤其是目前还不清楚为什么高阶傅立叶分析中正确的“特征”是所谓的“零序列”；虽然这在某种程度上是已知的，但现有的证明给出了有限的理解和极差的定量相关性。这个话题对许多人来说是一个有趣的话题，因为我们目前拥有的(定性的)陈述是非常自然的。有一些定理断言，为了理解远远超出经典傅立叶分析范围的非常一般的方程组，理解一个人感兴趣的函数与上述零序列的相关性就足够了。零序列的定义有些简单和自然，而且还伴随着大量的对称性(具体地说，存在群体操作)。因此，整个理论看起来是非常自然和基本的东西，但目前的严谨论证极其复杂，缺乏任何直觉。感觉就像是试图在不了解正交性的情况下进行傅里叶分析。目的和目标：开发工具来更好地理解高阶傅立叶分析和相关问题。首先，这将涉及到候选人处理问题，这些问题将使他熟悉思考上述更基本的问题所需的想法和工具，而不是基础问题本身。研究方法的新奇之处：这是一个年轻的学科。它已经看到了大量的新发展，例如来自遍历理论(传统上是一个非常不同的数学领域)和加数理论的思想之间的相互作用，以及更技术层面的大量新思想。对于该领域的专家来说，完全清楚的是，需要一个全新的观点来正确理解这一主题，并以一种适合于进一步应用的灵活形式来表达它。不幸的是，我不能更具体地说明这个新观点可能是什么，否则我自己就会写下关于它的论文。人们所能做的就是探索这一领域的新问题，目的是更好地理解这些现象，同时积累证据，证明“高阶字符”是如何组合在一起的。

This project falls within the EPSRC Mathematical Sciences research area. Context: Higher-order Fourier analysis is a mathematical topic that has arisen over the last 20 years but is still poorly understood. It is a tool that may be used to analyse certain problems in combinatorics, number theory and mathematical analysis that are not amenable to "traditional" Fourier analysis involving characters. Whilst there have been some notable successes, such as Gowers' new proof of Szemeredi's theorem on arithmetic progressions and the Green-Tao theorem on primes, much remains to be understood. In particular it is not yet clear exactly why the correct "characters" in higher-order Fourier analysis are the so-called nilsequences; whilst this is known to be true on some level, the existing proofs give limited understanding and extremely poor quantitative dependences. This topic is an intriguing one to many people, since the (qualitative) statements we currently have are extremely natural. There are theorems asserting that, in order to understand very general systems of equations lying well beyond the remit of classical Fourier analysis, it is enough to understand the correlations of the functions one is interested in with the nilsequences mentioned above. The definition of nilsequence is somewhat simple and natural, and additionally comes with large amounts of symmetry (specifically, a group action is present). Therefore the whole theory has the appearance of being something very natural and basic, yet current rigorous arguments are extremely convoluted and lack in any intuition. It feels as though one is trying to do Fourier analysis without understanding orthogonality.Aims and objectives: To develop tools to better understand higher-order Fourier analysis and related issues. In the first instance this will involve the candidate working on problems that will familiarise him with the ideas and tools needed to even think about the more foundational issues mentioned above, rather than the foundational issues themselves.Novelty of the research methodology: This is a young subject. It has already seen a large number of novel developments, such as the interplay between ideas from ergodic theory (traditionally a very different mathematical area) and additive number theory, as well as a large number of novel ideas at a more technical level. It is completely clear to experts in the area that a fundamentally new viewpoint will be required to properly understand the subject, and to put it in a flexible form suitable for further applications. Unfortunately I cannot be more specific about exactly what this new viewpoint might be, else I would have written papers on it myself. All one can do is explore new problems in the area with an aim to better understanding the phenomena, all the while accumulating evidence for how the "higher-order characters" fit together.