Model Theory, Pure and Applied

拟议的研究涉及使用的想法从数学逻辑，以提高理解数学中的一些中心对象：对称性，功能和微分方程。 数学逻辑中与所提出的研究相关的部分是模型论，它是关于数学对象或对象类的语言定义方式。 虽然数理逻辑与数学的哲学和基础有着传统的联系，但最近在数学的核心领域有许多应用，而拟议的研究属于后者的传统。 需要研究的关键问题包括：（i）对“群”（对称性的集合）作为一个好空间的对称性的方式进行分类，有时在一些模型论假设下，（ii）一些非常特殊的函数（如指数函数）满足什么样的方程。这里有一些更详细的内容，是为数学家写的。拟议的研究有两个主要部分。第一部分的一个中心主题是在没有独立性的一阶理论模型中定义的群体的分类。 这属于“驯服”模型理论的群体，推广“稳定群理论”本身概括了部分理论的代数群。 虽然动机很大程度上属于“纯”模型理论，这部分的建议也触及拓扑动力学和李群。例如，我们期望产生一个离散群的“新”不变量，并提出了许多问题。第二部分的建议涉及应用模型理论的丢番图几何的功能领域，建设早期工作的PI和其他人。一个问题是扩展的Ax-Schanuel型定理，指数和超越，家庭的semiabelian品种。 另一个问题是获得一个“透明”的帐户的积极特征莫德尔-朗猜想（由于最初Hrushovski）。

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).