Higher-Order Fourier analysis and related issues

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.