Higher-Order Fourier analysis and related issues

高阶傅里叶分析及相关问题

• 批准号: 1942002
• 金额: --
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2017
• 资助国家: 英国
• 起止时间: 2017 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-1942002
• 关键词: Higher; Order; Fourier; analysis; related

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.

该项目属于EPSRC数学科学研究领域。背景：高阶傅立叶分析是近20年来出现的一个数学主题，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，这些问题不适用于涉及字符的“传统”傅立叶分析。虽然已经取得了一些显著的成功，比如高尔斯对等差数列的Szemeredi定理的新证明，以及关于素数的Green-Tao定理，但仍有许多有待理解的地方。特别是，目前还不清楚为什么高阶傅立叶分析中的正确“字符”是所谓的nilsequences；虽然这在某种程度上是正确的，但现有的证明只能提供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，因为我们目前的（定性）陈述是非常自然的。有一些定理断言，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，理解我们感兴趣的函数与上面提到的nilsequences之间的相关性就足够了。nilsequence的定义有些简单和自然，并且伴随着大量的对称性（特别是，存在一个群作用）。因此，整个理论看起来是非常自然和基本的东西，但目前的严格论证是极其复杂的，缺乏任何直觉。感觉就像在不理解正交性的情况下做傅里叶分析。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。在第一种情况下，这将涉及到候选人的问题，这将使他熟悉所需的想法和工具，甚至思考上面提到的更基本的问题，而不是基本问题本身。研究方法的新颖性：这是一个年轻的课题。它已经看到了大量新颖的发展，例如遍历理论（传统上是一个非常不同的数学领域）和加性数论的思想之间的相互作用，以及更多技术层面上的大量新颖思想。对于该领域的专家来说，完全清楚的是，要正确理解这一主题，并将其以适合进一步应用的灵活形式呈现出来，就需要一种全新的观点。不幸的是，我不能更具体地说明这个新观点到底是什么，否则我自己就会写论文了。人们所能做的就是探索这一领域的新问题，以更好地理解这一现象，同时积累“高阶字符”如何组合在一起的证据。