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可以表现出来并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都有许多应用。经典上，它是关于通过矩阵表示对称的集合，但作为一个现代学科，它涉及许多更复杂的代数结构。 该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继续参与针对初中和高中学生的数学丰富计划。该项目旨在研究的具体代数结构是Tannakian和Pre-Tannakian范畴，它们是“群的表示论”的公理化和推广。 最近，PI和他的合作者安德鲁·斯诺登（Andrew Snowden）发现了前Tannakian范畴和模型论（数理逻辑的一个分支）之间的新联系。他们能够将一个前塔纳克范畴与一个寡纯群联系起来，沿着一些被称为测度的额外数值数据。 此后，这种结构产生了大量新的例子，以及对以前已知例子的新见解。此外，他们还表明，事实上，这些寡纯群在试图研究和分类前Tannakian范畴时，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调查，以构建新的和有趣的例子，前Tannakian范畴与异国情调的属性，发展一个理论，前Tannakian范畴与更广泛的一类线性寡纯群，并开发工具，更适合于构建积极的特征版本的类别先前构建。 所有这些都应被视为朝着构建和分类所有前Tannakian类别的长期最终目标迈出的一步。该奖项反映了NSF的法定使命，并被认为值得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来支持。

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.