Model Theory and Definable Additive Combinatorics

该项目的研究涉及数理逻辑的一个分支，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。测试一个数学结构是否适合分类的工具是各种各样的被称为“分界线”的假设，它们禁止一个结构的行为中的某些固定模式。这个项目的一个主要焦点是这样一条分界线，称为VC维，它起源于机器学习，并使用图论的思想来衡量数学对象的随机性。在模型论中，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，也应用于算术组合学的较新领域，这是代数，分析和离散数学的融合。过去模型理论研究的很大一部分集中在全局水平上表现出驯服的结构上，在这个意义上，结构中的每一个可定义的关系都省略了某种固定类型的组合配置。本研究的重点是局部层面，其中一个分析一个单一的驯服关系内，否则复杂的结构。一个主要的目标是发展一个代数理论的NIP公式在任意群，继续先前的研究的情况下，伪有限群的科南特和皮莱。第二个目标是扩展这项工作的伪有限群以外的NIP环境。这种设置适用于有限群中的算术组合学，例如科南特、皮莱和特里以前对驯服算术正则性的研究。在拟议的研究中，科南特将使用李理论和表示理论来开发伪有限设置中的算术正则性，而无需额外的驯服假设。其目标是给一个模型理论帐户的算术规律性结果的绿色和陶，并将这些结果扩展到非交换groups.This奖项反映了NSF的法定使命，并已被认为是值得支持的，通过评估使用基金会的智力价值和更广泛的影响审查标准。

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.