Model Theory and Definable Additive Combinatorics

这个项目的研究涉及数学逻辑的一个分支，称为模型论，它在语言层面上研究数学结构，并根据其行为的复杂性对结构进行分类，相对于固定的数学语言选择。测试数学结构是否适合分类的工具是被称为“分界线”的各种假设，这些假设禁止结构行为中的某些固定模式。这个项目的主要焦点是一个叫做vc维的分界线，它起源于机器学习，并使用图论的思想来测量数学对象的随机性。在模型理论中，这个概念出现在所谓的“NIP理论”的研究中。Conant将把这个模型理论框架应用于群论中的问题，从离散和拓扑的角度来看，也应用于算术组合的新领域，这是代数、分析和离散数学的融合。过去模型理论的大部分研究都集中在全局水平上表现出驯服性的结构上，也就是说，结构中的每个可定义关系都省略了某种固定类型的组合配置。本研究侧重于局部层面，分析一个复杂结构内部的单一驯服关系。主要目的是发展任意群中NIP公式的代数理论，继续Conant和Pillay在伪有限群情况下的先前研究。第二个目标是将这项工作扩展到NIP环境之外的伪有限群上。这个设置适合应用于有限群中的算术组合，例如Conant， Pillay， and Terry先前对驯服算术规则的研究。在拟议的研究中，Conant将使用李理论和表示理论来发展伪有限设置下的算术正则性，而不需要额外的驯服假设。目的是给出Green和Tao的算术正则性结果的模型理论解释，并将这些结果推广到非交换群。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The research in this project concerns a branch of mathematical logic called model theory, which studies mathematical structures at the linguistic level, and classifies structures according the complexity of their behavior with respect to a fixed choice of mathematical language. The tools for testing whether a mathematical structures is suitable for classification are various hypotheses called "dividing lines", which forbid some fixed pattern in the behavior of a structure. A main focus of this project is one such dividing line called VC-dimension, which originated in machine learning, and uses ideas from graph theory to measure randomness in mathematical objects. In model theory, this notion emerged in the study of so-called "NIP theories". Conant will apply this model theoretic framework to questions in group theory, from both the discrete and topological perspectives, and also in the newer field of arithmetic combinatorics, which is a fusion of algebra, analysis, and discrete math. A large part of past research in model theory has focused on structures which exhibit tameness at the global level, in the sense that every definable relation in the structure omits a combinatorial configuration of some fixed type. This research focuses on the local level, in which one analyzes a single tame relation inside an otherwise complicated structure. A major aim is to develop an algebraic theory of NIP formulas in arbitrary groups, which continues prior research of Conant and Pillay in the case of pseudofinite groups. A second aim is to extend this work on pseudofinite groups outside of the NIP environment. This setting is suitable for applications to arithmetic combinatorics in finite groups, for example previous research of Conant, Pillay, and Terry on tame arithmetic regularity. In the proposed research, Conant will use Lie theory and representation theory to develop arithmetic regularity in the pseudofinite setting without extra tameness assumptions. The goal is to give a model theoretic account of arithmetic regularity results of Green and Tao, and extend these results to non-commutative 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.