Richard P. Stanley Seminar in Combinatorics: Nilsequences on general additive patterns

We will begin by briefly introducing the use of higher-order Fourier analysis in additive combinatorics for a general audience. In particular, we will discuss the arithmetic regularity lemma and how it identifies a certain class of arithmetically-structured functions — nilsequences — as extremal objects for problems in additive combinatorics. We will then discuss how it has recently come to light that the analysis of nilsequences on certain additive patterns — those which satisfy a certain algebraic criterion known as the flag condition — is easier than the general case, and discuss some recent ideas and developments in overcoming the difficulties that arise when the additive pattern of interest is not flag.

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