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

有限、伪有限和有限结构中的可定义集合和测度

基本信息

批准号：
EP/K020692/1
负责人：
H Macpherson
金额：
$ 34.65万
依托单位：
University of Leeds
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2013
资助国家：
英国
起止时间：
2013 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FK020692%2F1
关键词：
Definable sets measures finite pseudofinite

项目摘要

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.

模型论是数学逻辑的一个分支，它解决数学结构（例如群、环、域、图）和用于描述它们的一阶语言之间的相互作用。可定义集合（结构中一阶公式的解集）起着核心作用，类似于代数几何中的可构造集合。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），它概括了线性和代数独立性，沿着可定义集的维数和测度的概念，以及它们之间的正交性。这些理论来自模型论的稳定性理论，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“NTP 2”理论）。因此，这些技术不仅可以应用于稳定结构，而且可以应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，通常通过它们的无限极限;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，但在心脏是一个全新的概念“多维渐近类”（m.a.c.）。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，所有至多d个李秩至多e的有限单群的直积的群的集合是一个m.a.c.。m.a.c.的确切定义。'是复杂的，但在任何超积中有更清晰的意义，其中每个可定义的集合被赋予一个与' Grothendieck半环'相关的特定半环中的值。我们将发展m.a.c.s及其超积的模型论性质，并从代数（特别是群论和表示论）和图论中寻找看起来丰富的数学上有趣的例子。在该项目中，我们的目标是群理论的应用（例如，当前活跃的主题词地图）和连接到相关工作的Hrushovski近似子群，高尔斯quasirandom群，和零一法律在有限组合。我们还将发展一个稍微不同但相关的模型理论profinite结构，目的是，例如，分类profinite群，在2-sorted语言中，有NIP理论。我们从无限模型理论接近这个主题，但有连接到有限模型理论，它的动机来自理论计算机科学和复杂性理论。我们的方法将给予理解的可定义集在很多类的有限结构，一些（如图）的强烈兴趣有限模型理论。我们将积极发展有限和无限模型理论之间的联系。