Model Theory, Pure and Applied

这项拟议的研究涉及使用数理逻辑的思想来增强对数学中一些核心对象的理解：对称、函数和微分方程。与所提出的研究相关的数理逻辑部分是模型理论，它是关于数学对象或对象类别的语言定义的方法。虽然数学逻辑与数学哲学和数学基础有着传统的联系，但最近有许多应用到数学的核心领域，拟议的研究是在后者的传统中进行的。要研究的关键问题是：(I)分类一个“群”(对称性的集合)可以作为良好空间的对称性的方式，有时是在一些模型理论假设下，以及(Ii)一些非常特殊的函数，如指数函数所满足的方程是什么。以下是为数学家撰写的更多详细信息。拟议的研究包括两个主要部分。第一部分的中心主题是可以在一阶理论的模型中定义的群的分类，而不是独立性质。这属于群的“驯服”模型理论，推广了“稳定群论”，而稳定群论本身就是部分代数群论的泛化。虽然动机在很大程度上属于“纯”模型理论，但该提案的这一部分也涉及到拓扑动力学和李群。例如，我们希望产生一个离散群的“新的”不变量，这引发了许多问题。该提案的第二部分涉及模型理论在函数域上丢番图几何中的应用，建立在PI和其他人早期工作的基础上。一个问题是关于指数和超越的Ax-Schanuel型定理推广到半交换变种。另一个问题是要“透明”地解释莫代尔-朗猜想的正特征(最初是由于赫鲁绍夫斯基的猜想)。

The proposed research concerns the use of ideas from mathematical logic to enhance the understanding of some central objects in mathematics: symmetries, functions, and differential equations. The part of mathematical logic which is relevant to the proposed research is model theory, which is about the ways in which mathematical objects or classes of objects are defined linguistically. Although mathematical logic has traditional connections with the philosophy and foundations of mathematics, there have been many recent applications to core areas of mathematics, and the proposed research is in the latter tradition. Among key problems to be studied are: (i) classifying the ways in which a "group" (collection of symmetries) can act as symmetries of a nice space, sometimes under some model theoretic assumptions, and (ii) what are the equations satisfied by some very special functions, such as exponential functions.Here are some more details, written for mathematicians. The proposed research has two main parts. A central theme of the first part is the classification of groups definable in models of a first order theory without the independence property. This belongs to the "tame" model theory of groups, generalizing "stable group theory" which itself generalizes part of the theory of algebraic groups. Although the motivation belongs largely to "pure" model theory, this part of the proposal touches also on topological dynamics and Lie groups. For example we expect to produce "new" invariants of a discrete group, with many questions raised. The second part of the proposal deals with applications of model theory to diophantine geometry over function fields, building on earlier work of the PI and others. One problem is the extension of Ax-Schanuel-type theorems, on exponentiation and transcendence, to families of semiabelian varieties. Another problem is to obtain a "transparent" account of the positive characteristic Mordell-Lang conjecture (due originally to Hrushovski).