Model Theory and Definable Additive Combinatorics

这个项目的研究涉及数学逻辑的一个分支，称为模型理论，它在语言水平上研究数学结构，并根据结构相对于固定选择的数学语言的行为的复杂性对结构进行分类。测试数学结构是否适合分类的工具是各种称为“分界线”的假设，这些假设禁止结构行为中的某些固定模式。这个项目的一个主要关注点是这样一条分界线，称为VC维，它起源于机器学习，使用图论的思想来衡量数学对象中的随机性。在模型理论中，这一概念出现在对所谓的“NIP理论”的研究中。柯南特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，以及算术组合学的新领域，算术组合学是代数、分析和离散数学的融合。过去在模型理论中的大部分研究都集中在在全局水平上表现出驯服的结构上，即结构中的每一个可定义的关系都忽略了某种固定类型的组合构型。这项研究集中在地方层面，在这个层面上，人们分析了一个原本复杂的结构中的单一驯服关系。主要目的是发展任意群中NIP公式的代数理论，该理论继承了Conant和Pillay在伪有限群情况下的前人研究。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。这种设置适合应用于有限群中的算术组合学，例如Conant、Pillay和Terry关于驯服算术正则性的先前研究。在提出的研究中，Conant将使用Lie理论和表征理论在没有额外驯服假设的伪有限环境中发展算术正则性。其目的是给格林和陶的算术正则性结果一个模型理论解释，并将这些结果推广到非对易群体。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.