Constructing and Classifying Pre-Tannakian Categories

该奖项资助与群的表示理论相关的研究，即研究对称性以及对称性表现自身和影响数学对象的不同方式。它是一个经典的兴趣领域，在数论、数学物理、代数几何、拓扑学、泛函分析和更多的数学领域有许多应用。在传统上，它是关于通过矩阵表示对称集合的，但作为一门现代学科，它涉及到许多更复杂的代数结构。这个项目的更广泛的影响包括为本科生和研究生提供研究培训机会，以及PI继续参与针对初中生的数学丰富课程。这个项目旨在研究的具体代数结构是Tannakian和Pre-Tannakian范畴，它们是对什么是“群的表示理论”的公理化和泛化。最近，PI和他的合作者安德鲁·斯诺登在前Tannakian范畴和数理逻辑的一个分支模型理论之间发现了一种新的联系。他们能够将前Tannakian范畴与一个寡态群联系起来，以及一些被称为度量的额外数字数据。自那以后，这种结构导致了许多新的例子，以及对以前已知的例子的新见解。此外，他们已经证明，事实上，在某种意义上，当试图研究和分类Tannakian之前的范畴时，这些寡态群是不可避免的，并且需要成为任何分类故事的一部分。这个项目的目的是继续这些研究，以构造具有奇异性质的前Tannakian范畴的新的和有趣的例子，发展与更广泛的一类线性-寡晶群相关的前Tannakian范畴的理论，并开发更适合于构造先前构造的范畴的正特征版本的工具。所有这些都应该被认为是朝着构建和分类所有Tannakian之前的类别的长期最终目标迈出的一步。这一奖项反映了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。

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.