A longstanding classification programme in model theory aims to determine when a mathematical structure exhibits tame, structurally simple—as opposed to wild, intractable—behaviour. A key role is played by so-called dividing lines, i.e. properties of logical formulas (or theories) that separate these regimes. In this talk, we demonstrate how the lens of combinatorics has allowed us to gain new insight into higher-order dividing lines, drawing on examples in graphs and groups. We also explain how this perspective has led to advances in higher-order Fourier analysis and statistical learning.

 

This talk intends to be accessible to beginning graduate students in all areas of mathematics.