权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

RUI: Geometry and Complexity in the Model Theory of Groups

RUI：群模型论中的几何和复杂性

基本信息

批准号：
1954127
负责人：
Joshua Wiscons
金额：
$ 15.49万
依托单位：
University Enterprises, Incorporated
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-09-01 至 2024-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1954127&HistoricalAwards=false
关键词：
RUI Geometry Complexity Model Theory

项目摘要

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.

这个项目是围绕着有限Morley秩群的长期存在的“代数性猜想”来组织的，该猜想在模型理论中自然产生，对于理解一类基本的结构至关重要。该项目通过发展一个新发现的与近似经典几何的联系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应用于代数猜想，以建立一个有限Morley秩群可能编码的对称性程度的自然限制，并增加一个目标，即对那些处于极端的对称性进行分类。此外，该项目还探索了有限对称群的“关系复杂性”的概念，这是某些高度对称结构的分类理论的关键组成部分。这项研究解决了关于某些自然群族的复杂性的核心问题，并为进一步的探索开发了计算工具。最后，该项目为加州州立大学萨克拉门托分校的本科生和硕士生提供了新的机会和支持，让他们参与到群模型理论中并培养他们的技能。本项目的第一线解决了代数性猜想的剩余障碍，目的是利用某些小的二阶和一般定义的射影几何之间的联系。具体目标包括消除长期存在的病理构型，澄清第二个构型，以及显著扩展现有的分析小而非零的2-等级群体的技术。第二条线索研究了有限Morley阶的置换群。重点放在Borovik和Cherlin的指导性问题上，即将那些具有足够高的类属传递性的群归类为射影几何中自然产生的单一形式。最后一个线索研究了有限置换群的关系复杂性。关系复杂性的研究目前是基于Cherlin的二元猜想，该猜想提出了一种复杂性本原群的分类2。本项目旨在通过分析各种置换群的自然族，重点研究作用于划分的对称群和交替群的复杂性，来扩大关系复杂性的研究范围。此外，这一主题进一步开发了用于计算关系复杂性的现有代码的算法并改进了现有代码，其他目标是为代码和支持其使用的手册创建公共存储库。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。