CAREER: Cohomology, classification, and constructions of tensor categories

自古希腊以来，对称性的概念一直是物理学的基础。在数学上，我们通过群论使对称性的概念精确化。量子物理学的基本对象包括抽象的东西，如冯诺依曼代数和量子场论。在过去的几十年里，量子物理学中的基本对象具有对称性已经变得很清楚，这些对称性最好用某些群的推广来描述，称为张量范畴。该项目将为数学家，物理学家和其他科学家提供更深入的了解张量类别是如何构建的，张量类别的新示例以及与这些示例相关的具体数据。PI将通过分类和构造来寻找有趣的例子，这不仅对张量范畴理论本身有用，而且对数学的相关领域也有用，例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将在印第安纳州大学（IU）为初级数学家组织合作研究研讨会，并在IU为代表性不足的研究生开设为期4周的暑期课程。 更详细地说，PI将研究张量范畴的上同调，分类模范畴，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。在非半简单的设置中，PI将在上同调的研究中结合几何技术，通过支持品种，了解这些类别的结构。PI将利用李理论的技术来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类别的结构的理解，PI将专注于完美融合类别和“小”融合类别的分类程序。研究完美融合范畴将使数学家能够理解弱积分融合范畴，并将产生，例如，实验物理学家对普适任意子的检测的见解。此外，PI还将调查一些结构的影响，如测量和zesting在不同的设置。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.