FRG: Model Theory and its Applications

这个项目的重点是模型理论（数学逻辑的一个分支）在数学中的应用。 模型理论以两种方式出现：（a）它提供了一套新的工具;（B）它提供了一种新的语言，在这种语言中，可以公式化结果和问题。 最近的成功表明，这些可以为许多数学领域提供强大的新方法，并可以成为关键问题突破的基础。 因此，本项目的重点是一个明确的和连贯的研究方向，及时取得了最新的进展，而不是在一个单一的数学问题。模型理论方法通常涉及识别和公理化的一阶理论类的结构，研究他们的非标准模型，识别类的可定义集的结构（量词消除），并利用紧性定理丰富的非标准模型。有一个复杂的机制（通过希拉和其他人）可以理解某些一般类的结构中的可定义集合。例如，一旦一个具体的结构被确定为稳定的，这种机制开始发挥作用。在这个项目中要强调的研究问题涉及模型理论的思想和工具在符号动力学和代数几何，微分方程的代数理论，分析，这个项目的目标之一是找到新的有效的方法来帮助数学家学习如何使用模型理论。 特别是，这个项目选择支持的研究人员通常是在模型理论之外工作的数学家，他们有兴趣找到在自己的工作中使用模型理论的方法。

The focus of this project is on the application of model theory (a branch of mathematical logic) within mathematics. Model theory comes into the picture in two ways: (a) it provides a new set of tools; (b) it provides a new language within which to formulate results and problems. Recent successes demonstrate that these can provide powerful new methods for many areas of mathematics and can be the basis for breakthroughs on critical problems. This project focuses, therefore, on a well defined and coherent research direction, made timely by recent advances, not on a single mathematical problem.Model-theoretic methods typically involve identifying and axiomatizing the first order theories of classes of structures, studying their nonstandard models, identifying the class of definable sets in a structure (quantifier-elimination), and making use of the compactness theorem to work within rich nonstandard models. There is a sophisticated machinery available (via Shelah and others) for understanding definable sets in certain general classes of structures. For example, once a concrete structure is identified as being stable, this machinery comes into play.The research problems to be emphasized in this project involve potential application of model theoretic ideas and tools within symbolic dynamics and algebraic geometry, the algebraic theory of differential equations, analysis, and geometry.One of the goals of this project to find new and effective ways to help mathematicians learn how to use model theory. In particular, the researchers selected for support by this project will often be mathematicians who work outside model theory, and who are interested in finding ways to use it in their own work.