Geometric model theory, pseudofinite combinatorics, and compressible types

• Applicant: Dr. Martin Bays
• Subject Area: Mathematics
• Term: from 2020 to 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 440345213

This project continues the previous project. The primary aim is to deepen the model theoretic understanding of pseudofinite sets in fields. In classical combinatorial terms, this means understanding asymptotic bounds on sizes of finite configurations satisfying polynomial constraints, a context which includes well-known problems and results such as Erdős's discrete distances problem, the orchard problem, the sum-product phenomenon, and the Breuillard-Green-Tao characterisation of approximate subgroups of algebraic groups. Hrushovski exposed deep analogies between these combinatorial phenomena and pre-existing results of pure model theory, by turning combinatorial conditions on asymptotic cardinalities into structure which can be analysed using the tools of geometric stability theory. With this project, we will deepen these analogies and explore new aspects of this structure.More specifically, we will consider the geometric implications on a choice of polynomial constraints of the existence of finite configurations which satisfy the constraints and achieve cardinality bounds implied by the algebraic dimensions of those constraints. Previous results, in particular results of Elekes-Szabó and Bays-Breuillard, have found that under a certain extra assumption that the co-ordinates have no "internal structure" interacting with the solutions, commutative algebraic groups explain any such configurations. The analogy with geometric stability theory makes it natural to allow internal structure, and this opens up a large class of more interesting potential structures, such as iteratively compounded nilpotent group schemes. This project aims to explore and precisely classify the possibilities, starting with the natural concrete case of the orchard problem on a cubic surface.Another aim is to use ideas from the model theory of NIP structures to obtain incidence bounds, and hence Elekes-Szabó results, in positive characteristic. Bays-Martin did this for function fields over finite fields by exploiting distality within corresponding valued fields. Distality within NIP can be approached via compressible types, which Bays-Kaplan-Simon showed to be plentiful. We aim both to advance this abstract theory of compressibility, and to use it to obtain new incidence bounds.

• DFG Programme: Research Grants