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

模型理论是数学逻辑的一个分支，它处理数学结构（如群、环、域、图）和用于描述它们的一阶语言之间的相互作用。可定义集（结构中一阶公式的解集）起着中心作用，类似于代数几何中的可构造集。模型理论已经成功地识别了抽象的独立性概念（例如，非分叉），它概括了线性和代数独立性，以及可定义集合的维数和度量概念，以及它们之间的正交性。这些都来自模型理论稳定性理论，但稳定性理论的技术最近被证明适用于更广泛的背景（简单理论，NIP理论，部分结构稳定的理论，甚至“NTP2”理论）。因此，这些技术不仅适用于稳定结构，还适用于更丰富的数学环境。模型理论研究的通常对象是无限的，但在这个项目中，我们将模型理论方法应用于有限结构类，经常通过它们的无限极限；这些通常是超产品，但有时是直接和反向限制。这个项目有几个方面，但核心是一个全新的“多维渐近类”（m.a.c）概念。这是一类有限结构，其中可定义集合的可定义族在其渐近大小上满足很强的均匀性，它考虑到结构的正交部分可以独立变化。例如，对于任意正整数d，e，最多d个李秩最多e的有限单群的直积的所有群的集合是一个m.a.c。’是复杂的，但在任何超积中都有更清晰的含义，其中每个可定义集在特定的半环中被赋予一个值，与“Grothendieck半环”相关。我们将发展m.a.c.s及其超产物的模型论性质，并从代数（特别是群论和表示论）和图论中寻找看起来大量的数学上有趣的例子。在这个项目中，我们的目标是群论的应用（例如，当前活跃的词映射主题），并与赫鲁晓夫斯基关于近似子群的相关工作，高尔斯关于准随机群的相关工作，以及有限组合中的0 - 1定律联系起来。我们还将为无限结构开发一个稍微不同但相关的模型理论，例如，旨在对无限群进行分类，这些群在2排序语言中具有NIP理论。我们从无限模型理论来研究这个问题，但它与有限模型理论有联系，有限模型理论的动机来自理论计算机科学和复杂性理论。我们的方法将使我们理解许多有限结构中的可定义集，其中一些（例如图）对有限模型理论非常感兴趣。我们将积极发展有限和无限模型理论之间的联系。

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.