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

模型理论是数理逻辑的一个分支，它研究数学结构(如群、环、场、图)和用来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确定了抽象的独立性概念(例如，非分叉)，它们泛化了线性和代数独立性，以及可定义集合的维度和度量的概念，以及它们之间的正交性。这些理论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，这些技术不仅应用于稳定结构，而且在更丰富的数学环境中得到应用。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适应并应用于有限结构的类，通常跨越它们的无限极限；这些通常是超积，但有时是正极限和逆极限。该项目有几个方面，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，其中可定义的可定义集族在其渐近大小上满足非常强的一致性，这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，至多d个有限单群的直积至多e的群的集合是M.A.C.‘m.a.c’的准确定义。是复杂的，但在任何超积中都有更清楚的含义，其中每个可定义的集合在某个半环中被赋值，与‘Grothendieck半环’相关。我们将发展m.a.c.s及其超积的模型理论性质，并从代数(特别是群论和表示论)和图论中寻找大量看起来像是数学上有趣的例子。在这个项目中，我们的目标是群论的应用(例如，词映射的活跃的当前主题)，以及与Hrushovski关于近似子群的相关工作、关于拟随机群的Gowers以及有限组合学中的零一定律的联系。我们还将为有限结构发展一个略有不同但相关的模型理论，目标是对在2-排序语言中具有NIP理论的有限群进行分类。我们从无限模型理论来探讨这个主题，但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类中的可定义集，一些有限模型理论非常感兴趣的(例如图)。积极发展有限模型理论与无限模型理论的联系。

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.