FRG: Model Theory and its Applications

这个项目的重点是模型理论(数理逻辑的一个分支)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)它提供了一种用来表述结果和问题的新语言。最近的成功证明，这些方法可以为数学的许多领域提供强大的新方法，并可以作为关键问题突破的基础。因此，这个项目集中在一个定义明确和连贯的研究方向上，这是由最近的进展及时取得的，而不是单一的数学问题。模型理论方法通常涉及识别和公理化结构类的一阶理论，研究它们的非标准模型，识别结构中的可定义集的类(量词消除)，并利用紧致性定理在丰富的非标准模型中工作。有一种复杂的机器可用于理解某些一般结构类中的可定义集。例如，一旦一个具体的结构被确定为稳定的，这个机器就开始发挥作用。这个项目要强调的研究问题涉及到模型理论思想和工具在符号动力学和代数几何、微分方程组、分析和几何的代数理论中的潜在应用。这个项目的目标之一是找到新的有效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，他们对找到在自己的工作中使用模型理论的方法感兴趣。

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.