Definable sets and measures in finite, pseudofinite, and profinite structures

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Model theory, a branch of mathematical logic, tackles the interplay between mathematical structures (e.g. groups, rings, fields, graphs) and first order languages used to describe them. Definable sets (solutions sets of first order formulas in a structure) play a central role, akin to that of constructible sets in algebraic geometry. Model theory has successfully identified abstract notions of independence (e.g. non-forking) which generalise linear and algebraic independence, along with notions of dimension and measure for definable sets, and orthogonality between them. These have come from model-theoretic stability theory, but techniques from stability theory have recently been shown to apply in much wider contexts (simple theories, NIP theories, theories where part of the structure is stable, even `NTP2' theories). As a result, the techniques have had applications not just for stable structures, but in much richer mathematical contexts.The usual objects of model-theoretic study are infinite, but in this project we adapt and apply model-theoretic methods to classes of finite structures, often going via their infinite limits; these are usually ultraproducts, but sometime direct and inverse limits. The project has several facets, but at the heart is a brand new notion of a `multidimensional asymptotic class' (m.a.c.). This is a class of finite structures in which definable families of definable sets satisfy a very strong uniformity in their asymptotic sizes, which takes into account that orthogonal parts of a structure can vary independently. For example, for any positive integers d,e, the set of all groups which are direct products of at most d finite simple group of Lie rank at most e, is an m.a.c. The precise definition of `m.a.c.' is complex, but has much clearer meaning in any ultraproduct, where each definable set is assigned a value in a certain semiring, related to the `Grothendieck semiring'. We will develop the model-theoretic properties of m.a.c.s and their ultraproducts, and search for what looks like a plentiful supply of mathematically interesting examples, coming from algebra (especially group theory and representation theory) and from graph theory. In the project we aim for group-theoretic applications (e.g. to the active current topic of word maps) and to connections to related work of Hrushovski on approximate subgroups, of Gowers on quasirandom groups, and to zero-one laws in finite combinatorics. We will also develop a slightly distinct but related model theory for profinite structures, aiming, for example, to classify profinite groups which, in a 2-sorted language, have NIP theory.We approach this subject from infinite model theory, but there are connections to finite model theory, which takes its motivation from theoretical computer science and complexity theory. Our methods will give understanding of definable sets in very many classes of finite structures, some (e.g. graphs) of strong interest to finite model theory. We will actively develop links between finite and infinite model theory.