Finite structures, combinatorial regularity, and measuring semirings

This project will explore three new directions in model theory opened up by the recent preprint `Multidimensional asymptotic classes' of the PI with Anscombe, Steinhorn and Wolf.

The framework concerns the model theory of classes of finite structures which satisfy a uniformity condition on cardinalities of definable sets, along with infinite `generalised measurable' structures which behave like limits of such classes. The motivating and already very rich example, stemming from the Lang-Weil estimates, is the class of finite fields, and the corresponding infinite pseudofinite fields (infinite fields satisfying those sentences in the language of rings which hold of all finite fields).

The three directions concern: combinatorial regularity conditions on finite structures; connections to the striking recent work of Harman and Snowden on pre-Tannakian categories and oligomorphic groups and on `linear oligomorphic' groups; and recent results by others (Evans, Marimon, Chevalier-Hrushovski) which shed new light in the generalised measurable context.

The work will be carried out in collaboration with Steinhorn, Marimon, and possibly others. This is a preliminary investigation, with goals to understand new relevant theory, tackle some concrete questions, and investigate the potential for more general results. The immediate beneficiaries are academic, but span a range of communities, including theoretical computer science, aspects of representation theory, and model theory.