Topics in Model Theory

该项目的重点是模型理论及其与其他数学领域的相互作用和应用。模型论是数理逻辑的一部分，研究数学对象或对象类的语言定义方式。因此，模型论发展了各种方法来度量函数或集合类的简单性和复杂性。 PI将使用模型理论方法研究的其他数学领域包括组合学，特别是图形，也被非正式地称为网络。 组合数学中的重要结果是“规则性”结果，显示了网络如何被分解为少量的片段，每个片段的行为“几乎随机”。PI将使用模型理论的方法来扩展和改进这样的正则性定理，有时在一个简单的假设下。该方案的核心部分有三个方面：第一是利用拓扑动力学的方法获得一阶理论和可定义群的新不变量。第二个关注的应用模型理论和非标准方法的算术正则性定理，以及制定新的规律性声明在一个“驯服”的环境，如稳定性理论在连续逻辑。第三个涉及制定的概念的近似子群在一般的模型理论的背景下，并给出一个分类的情况下不变的措施。 提案的其他部分包括描述p-adic半代数群和发展仿射群方案（或亲线性代数群）的模型理论。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

The focus of this project is model theory and its interactions with and applications to other areas of mathematics. Model theory is a part of mathematical logic, and studies the manner in which mathematical objects or classes of objects are defined linguistically. Consequently, model theory has developed various ways of measuring the simplicity and complexity of classes of functions or sets. Among the other areas of mathematics that the PI will study using model-theoretic methods, is combinatorics, in particular graphs, also informally called networks. Important results in combinatorics are "regularity" results, showing how networks can be decomposed into a small number of pieces, each such piece behaving "almost randomly". The PI will use model theoretic methods to extend and improve such regularity theorems, sometimes under a simplicity assumption. The core part of the proposal has three aspects: The first is to use methods from topological dynamics to obtain new invariants for first order theories and definable groups. The second concerns applications of model theory and nonstandard methods to arithmetic regularity theorems as well as the formulation of new regularity statements in a "tame" environment such as stability theory in continuous logic. The third involves the formulation of the notion of an approximate subgroup in a general model-theoretic context and giving a classification in the absence of invariant measures. Other parts of the proposal include describing p-adic semialgebraic groups and developing a model theory of affine group schemes (or pro-linear algebraic groups).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.