Constructing and Classifying Pre-Tannakian Categories

前坦纳克阶范畴的构建和分类

• 批准号: 2401515
• 负责人: Nate Harman
• 金额: $ 15.5万
• 依托单位: University of Georgia Research Foundation Inc
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-06-01 至 2027-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2401515&HistoricalAwards=false
• 关键词: Constructing; Classifying; Pre; Tannakian; Categories

This award funds research related to the representation theory of groups, which is the study of symmetry and the different ways symmetry can manifest itself and influence mathematical objects. It is an area of classical interest which has numerous applications to number theory, mathematical physics, algebraic geometry, topology, functional analysis, and many more areas of math. Classically, it is about representing collections of symmetries via matrices, but as a modern subject, it involves a number of more sophisticated algebraic structures. Broader impacts of this project include research training opportunities for undergraduate and graduate students, as well as the PI’s continued involvement in mathematical enrichment programs aimed at middle and high school students.The specific algebraic structures this project aims to study are Tannakian and Pre-Tannakian categories, which are axiomatizations and generalizations of what is meant by “the representation theory of a group.” Recently, the PI and his collaborator, Andrew Snowden, found a new connection between pre-Tannakian categories and model theory, a branch of mathematical logic. They were able to associate a pre-Tannakian category to an oligomorphic group, along with some additional numerical data known as a measure. This construction has since led to a slew of new examples as well as new insights into previously known examples. Moreover, they have shown that, in fact, these oligomorphic groups are, in a sense, unavoidable when trying to study and classify pre-Tannakian categories and need to be a part of any classification story. This project aims to continue these investigations to construct new and interesting examples of pre-Tannakian categories with exotic properties, to develop a theory for pre-Tannakian categories associated with a wider class of linear-oligomorphic groups, and to develop tools that are better suited for constructing positive characteristic versions of the categories previously constructed. All of these should be considered steps toward a long-term eventual goal of constructing and classifying all pre-Tannakian categories.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.

该奖项资助了与群体表示理论有关的研究，即对对称性的研究以及对称性可以表现出来并影响数学对象的不同方式。它是一个具有古典感兴趣的领域，在数字理论，数学物理学，代数几何，拓扑，功能分析以及更多数学领域都有许多应用。从经典上讲，这是关于通过物品代表对称性的集合，但作为现代主题，它涉及许多更复杂的代数结构。 Broader impacts of this project include research training opportunities for undergraduate and graduate students, as well as the PI’s continued involvement in mathematical enrichment programs aimed at middle and high school students.The specific algebraic structures this project aims to study are Tannakian and Pre-Tannakian categories, which are axiomatizations and generalizations of what is meant by “the Representation theory of a group."最近，PI和他的合作者Andrew Snowden在Tannakian类别和模型理论（数学逻辑的一个分支）之间找到了新的联系。他们能够将Tannakian前类别与纯态群相关联，以及一些称为度量的其他数值数据。此后，这种结构导致了许多新示例以及对先前已知示例的新见解。此外，他们已经表明，实际上，在尝试研究和分类Tannakian类别时，这些纯态群体在某种意义上是不可避免的，并且需要成为任何分类故事的一部分。该项目旨在继续这些研究，以构建具有异国特性的Tannakian前类别的新的有趣的例子，以开发与与更广泛的线性 - 寡素类群相关的Tannakian类别的理论，并开发更适合于以前构建类别的积极特征版本的工具。所有这些都应被视为朝着建造和分类所有塔纳基人类别的长期目标的步骤。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响评估标准，通过评估被认为是宝贵的支持。