CAREER: Cohomology, classification, and constructions of tensor categories

职业：张量类别的上同调、分类和构造

• 批准号: 2146392
• 负责人: Julia Plavnik
• 金额: $ 45万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2027-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2146392&HistoricalAwards=false
• 关键词: CAREER; Cohomology; classification; constructions; tensor

The concept of symmetry has been fundamental in physics since the ancient Greeks. Mathematically, we make the notion of symmetry precise via group theory. Fundamental objects in quantum physics include abstract things like von Neumann algebras and quantum field theories. In the past several decades, it has become clear that fundamental objects in quantum physics have symmetries that are best described by certain generalizations of groups, called tensor categories. This project will provide mathematicians, physicists, and other scientists invested in quantum science with a deeper understanding of how tensor categories are structured, new examples of tensor categories, and their concrete data related to these examples. The PI will look for interesting examples via classification and constructions, which would be useful not only to the theory of tensor categories itself but also to related areas of mathematics, such as topological quantum field theory and the study of vertex algebras. The educational component of this project will establish networks of diverse voices in quantum symmetries research and in the mathematical community more broadly. The PI will organize collaborative research workshops at Indiana University (IU) for junior mathematicians start a 4-week long summer program at IU for underrepresented incoming graduate students. In more detail, the PI will study the cohomology of tensor categories, classify modular categories, and find new tensor categories via constructions, which would be useful not only to the theory of tensor categories itself but also to related areas of math, such as topological quantum field theory and the study of vertex operator algebras. In the non-semisimple setting, the PI will incorporate geometric techniques in the study of the cohomology, via support varieties, to learn about the structure of these categories. The PI will utilize Lie theoretic techniques to construct Hopf algebras and Nichols algebras in some symmetric tensor categories in positive characteristic with the aim of advancing their classification. To deepen the understanding of the structure of fusion and modular categories, the PI will focus on perfect fusion categories and on the classification program for "small" fusion categories. Studying perfect fusion categories will enable mathematicians to understand weakly integral fusion categories and would yield, for example, insights into the detection of universal anyons by experimental physicists. In addition, the PI will investigate the effect of some constructions such as gauging and zesting in different settings.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.

自古希腊以来，对称概念一直是物理学的基础。在数学上，我们通过群论使对称的概念变得精确。量子物理学中的基本对象包括抽象的东西，比如冯·诺伊曼代数和量子场论。在过去的几十年里，人们已经清楚地认识到，量子物理学中的基本物体具有对称性，这种对称性可以用某些被称为张量范畴的群的一般化来最好地描述。该项目将为数学家、物理学家和其他从事量子科学的科学家提供对张量类别的结构、张量类别的新示例以及与这些示例相关的具体数据的更深入理解。PI将通过分类和构造寻找有趣的例子，这不仅对张量范畴本身的理论有用，而且对数学的相关领域也很有用，比如拓扑量子场论和顶点代数的研究。该项目的教育部分将在量子对称研究和更广泛的数学界建立不同声音的网络。PI将在印第安纳大学（IU）为初级数学家组织合作研究研讨会，为印第安纳大学未被充分代表的即将入学的研究生开始为期4周的暑期项目。更详细地说，PI将研究张量范畴的上同调，对模范畴进行分类，并通过构造找到新的张量范畴，这不仅对张量范畴本身的理论有用，而且对数学的相关领域也很有用，例如拓扑量子场论和顶点算子代数的研究。在非半简单设置中，PI将通过支持变量将几何技术纳入上同调的研究中，以了解这些类别的结构。PI将利用李论技术构造具有正特征的对称张量范畴中的Hopf代数和Nichols代数，目的是推进它们的分类。为了加深对融合和模块化分类结构的理解，PI将重点研究完美融合类别和“小”融合类别的分类方案。研究完美核聚变范畴将使数学家能够理解弱积分核聚变范畴，并将使实验物理学家对宇宙任意子的探测产生见解。此外，PI将调查一些结构的影响，如在不同的设置测量和测试。该奖项反映了美国国家科学基金会的法定使命，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，被认为值得支持。