DaLFI : Duality and Logic in the passage from the Finite to the Infinite, IRIF, Paris, 17th November 2022

Title. Asymptotic classes of finite structures and their ultralimits.

Abstract. I will discuss several notions — ‘one dimensional asymptotic classes’ and ‘multidimensional asymptotic classes’ — that are defined for classes of finite structures, and describe when the sizes of the definable sets in these structures satisfy certain asymptotic formulas. There is an infinite counterpart to these definitions: the ‘measurable’ and ‘generalised measurable’ structures. More precisely, an infinite structure $M$ is generalised measurable if there is a ‘measuring function’ from the family of definable sets in $M$ into a ‘measuring semiring’ (which are certain ordered semirings) satisfying axioms that are intended to capture aspects of the intuitive notions of measure and dimension.

The motivating examples for measurable structures as pseudofinite fields, and in this talk I will give examples of generalised measurable structures and describe what is known regarding the stability-theoretic properties of generalised measurable structures. This is joint work with Macpherson, Steinhorn, and Wolf, and it generalises the work on one-dimensional asymptotic classes and measurable structures introduced by Macpherson and Steinhorn in their 2008 paper.