RUI: Geometry and Complexity in the Model Theory of Groups

这个项目是围绕着长期存在的“代数性猜想”有限莫利秩的群体，这自然出现在模型论，是至关重要的理解一类基本的结构。该项目通过开发一个新发现的与近似经典几何的连接来研究代数性猜想的关键剩余障碍。该项目还将把现有的部分解决方案应用于代数性猜想，以建立一组有限莫利秩可以编码多少对称性的自然限制，并增加了对极端情况进行分类的目标。此外，该项目还探索了有限对称群的“关系复杂性”概念，这是某些高度对称结构的分类理论的关键组成部分。这项研究解决了某些自然群体家族复杂性的核心问题，并开发了进一步探索的计算工具。最后，该项目为加州萨克拉门托州立大学的本科生和硕士生提供了新的机会和支持，以参与和建立在群论模型中的技能。该项目的第一个线程解决了代数性猜想的剩余障碍，其目标是利用某些小的2秩和一般定义的射影几何群体之间的联系。具体目标包括消除长期存在的病理配置，澄清第二，并显着扩展现有的技术分析组的小，但非零，2-秩。研究的第二个线索是研究有限莫利秩的置换群。重点是Borovik和Cherlin的指导问题的分类这些群体具有足够高程度的一般传递性作为一个单一的形式，自然产生的射影几何。最后一个线程研究有限置换群的关系复杂性。关系复杂性的研究目前由Cherlin的二元猜想（英语：Binary Conjecture）所锚定，它提出了复杂性2的本原群的分类。本项目旨在通过分析置换群的各种自然族，扩大关系复杂性的研究范围，重点关注作用于分区的对称和交替群的复杂性。此外，该线程进一步开发算法并改进现有代码以计算关系复杂性，并为代码和手册创建公共存储库以支持其使用。该奖项反映了NSF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而被认为值得支持。

This project is organized around the long-standing "Algebraicity Conjecture" for groups of finite Morley rank, which arise naturally in model theory and are crucial to understanding a fundamental class of structures. The project investigates key remaining obstructions to the Algebraicity Conjecture by developing a newly found connection with approximately classical geometries. The project will also apply the existing partial solution to the Algebraicity Conjecture to establish natural limits to how much symmetry a group of finite Morley rank may encode, with the added goal of classifying those at the extreme. Additionally, the project explores the notion of "relational complexity" for finite symmetry groups, a key component of a classification theory for certain highly symmetric structures. This research addresses core problems about the complexity of certain natural families of groups and develops computational tools for further exploration. Finally, the project provides new opportunities and support for undergraduate and Masters students at California State University, Sacramento, to engage with and build skill in the model theory of groups.The first thread of this project addresses remaining obstructions to the Algebraicity Conjecture with the goal of exploiting a connection between certain groups of small 2-rank and generically defined projective geometries. Specific aims include the elimination of a long-standing pathological configuration, a clarification of a second, and a significant expansion of the existing techniques for analyzing groups of small, but nonzero, 2-rank. The second thread of research studies permutation groups of finite Morley rank. The focus is on Borovik and Cherlin's guiding problem of classifying those groups with a sufficiently high degree of generic transitivity as being of a single form that arises naturally in projective geometry. The final thread investigates the relational complexity of finite permutation groups. The study of relational complexity is currently anchored by Cherlin's Binary Conjecture, which proposes a classification of the primitive groups of complexity 2. This project aims to broaden the scope of research on relational complexity by analyzing various natural families of permutation groups, focusing on the complexity of the symmetric and alternating groups acting on partitions. Moreover, this thread further develops algorithms and refines exiting code for computing relational complexity, with the additional goals of creating a public repository for the code and manual to support its use.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.