Higher-Order Fourier analysis and related issues

该项目属于EPSRC数学科学研究领域的福尔斯。高阶傅立叶分析是一个数学话题，在过去的20年里出现，但仍然知之甚少。它是一种工具，可用于分析组合学，数论和数学分析中的某些问题，这些问题不适合涉及字符的“传统”傅立叶分析。虽然有一些显着的成功，如高尔斯的新证明Szemeredi定理算术级数和绿陶定理素数，还有很多有待理解。特别是它还不清楚为什么正确的"字符"在高阶傅立叶分析是所谓的零序列;虽然这是已知的是真实的在某种程度上，现有的证据给有限的理解和极差的定量依赖。这个话题对很多人来说都很有趣，因为我们现在的（定性）陈述是非常自然的。有定理断言，为了理解非常一般的方程组，远远超出了经典傅立叶分析的范围，理解函数与上面提到的零序列的相关性就足够了。nilsequence的定义有些简单和自然，并且还带有大量的对称性（特别是，存在群作用）。因此，整个理论看起来是非常自然和基本的东西，但目前严格的论点非常复杂，缺乏任何直觉。感觉就像一个人试图做傅立叶分析不理解正交性。目的和目标：开发工具，以更好地理解高阶傅立叶分析和相关问题。在第一种情况下，这将涉及候选人的工作问题，这将使他熟悉的想法和工具，甚至需要考虑更基本的问题上面提到的，而不是基本问题本身。研究方法的新奇：这是一个年轻的主题。它已经看到了大量的新的发展，如遍历理论（传统上是一个非常不同的数学领域）和加法数论的思想之间的相互作用，以及大量的新思想在更技术的水平。该领域的专家完全清楚，需要一种全新的观点来正确理解这一主题，并将其置于适合进一步应用的灵活形式中。不幸的是，我不能更具体地说明这个新观点可能是什么，否则我会自己写论文。人们所能做的就是探索该领域的新问题，以便更好地理解这些现象，同时为“高阶特征”如何组合在一起积累证据。

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.