On some applications of model theory and topological dynamics to additive combinatorics

21.03.2024 15:00 - 15:50

K. Krupiński (U Wrocławski, PL)

Model theory is a fast growing branch of mathematical logic with deep interactions with algebra, algebraic geometry, combinatorics, and, more recently, topological dynamics. I will focus on a few interactions with topological dynamics, and applications to additive combinatorics.

I will discus type-definable components of definable groups, which lead to model-theoretic descriptions of Bohr compactificatios of groups and rings, and also to so-called locally compact models of approximate subgroups and subrings which in turn are crucial to get structural or even classification results about approximate subgroups and subrings. I will discuss my result that each approximate subring has a locally compact model, and mention some structural applications. In contrast to approximate subrings, not every approximate subgroup has a locally compact model. However, Ehud Hrushovski showed that instead it has such a model in a certain generalized sense (with morphisms replaced by quasi-homomorphisms). In order to do that, he introduced and developed local logics and definability patterns. In my recent paper with Anand Pillay, we gave a shorter and simpler construction of a generalized locally compact model, based on topological dynamics methods in a model-theoretic context. I will briefly discuss it, if time permits.




HS 11, 2. OG, OMP 1