Topics in Model Theory

这个项目的主题是模型论，数学逻辑的一个分支，它是关于数学对象或对象类的语言定义方式。 模型理论发展了各种方法来测量一类函数或集合的复杂性，其中一种称为Vapnik-Chervonenkis（或VC）维，由统计学习理论的研究人员在20世纪70年代独立发现。 本项目的一部分是研究有限VC维系统的精细结构。一类集合的复杂性的相关度量称为稳定性，该项目的另一个方面是使用对稳定系统的模型理论理解来计算特殊函数（如指数函数）所满足的方程。 描述空间和系统的对称性是研究的一个普遍方面，它将其各个组成部分联系在一起。该项目有四个相互关联的方面，（i）稳定性理论和没有独立性的理论，（ii）拓扑动力学，模型理论和伪有限群，（iii）微分场的驯服理论，（iv）半交换族的Ax-Lindemann和微分伽罗瓦理论。在抽象的层次上，模型论是关于一阶理论的分类，而第（i）部分就是这种性质。在“纯”模型理论中获得的方法和结果在具体的背景下通常是数学上有意义的，导致与其他数学领域的新的相互关系。特别是在（四）PI将研究超越性质的“指数”功能相对于某些家庭的交换代数群，利用一定的伽罗瓦理论的微分方程是由建设的稳定性理论（部分模型理论）。该项目还连接和影响拓扑动力学，也可能和间接的组合学（通过研究无独立性理论中可定义集的某些措施）。

The subject of this project is model theory, a branch of mathematical logic which is about the ways in which mathematical objects or classes of objects are defined linguistically. Model theory develops various ways of measuring the complexity of a class of functions or sets, one of which is called Vapnik-Chervonenkis (or VC) dimension and was independently discovered by researchers in statistical learning theory in the 1970's. One part of the project is studying the fine structure of systems with finite VC dimension. A related measure of the complexity of a class of sets is called stability and another aspect of the project is using the model-theoretic understanding of stable systems to compute the equations satisfied by special functions such as the exponential function. Describing the symmetries of spaces and systems is a pervasive aspect of the research which ties together its various components.The project has four interrelated aspects, (i) stability theory and theories without the independence property, (ii) topological dynamics, model theory, and pseudo-finite groups, (iii) tame theories of differential fields, (iv) Ax-Lindemann for semiabelian families and differential Galois theory. At the abstract level model theory is about the classification of first order theories, and part (i) is firmly of this nature. The methods and results obtained in "pure" model theory are often mathematically meaningful in concrete contexts, leading to new interrelations with other areas of mathematics. In particular in (iv) the PI will study transcendence properties of "exponential" functions relative to certain families of commutative algebraic groups, making use of a certain Galois theory of differential equations which is given by constructions in stability-theory (part of model theory). The project also connects to and impacts on topological dynamics, and also potentially and indirectly combinatorics (via the study of certain measures on definable sets in theories without independence property).