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本概念。在数学上，我们通过群论使对称性的概念变得精确。量子物理学的基本对象包括冯·诺依曼代数和量子场论等抽象事物。在过去的几十年里，量子物理中的基本物体具有对称性，这一点已经变得很清楚，这种对称性最适合用群的某些推广来描述，即张量范畴。这个项目将为数学家、物理学家和其他致力于量子科学的科学家提供更深入的理解，了解张量范畴是如何构成的，张量范畴的新例子，以及他们与这些例子相关的具体数据。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，如拓扑量子场论和顶点代数的研究也是有用的。该项目的教育部分将在量子对称性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，在印第安纳大学为未被充分代表的即将到来的研究生开始为期4周的暑期计划。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，并通过构造寻找新的张量范畴，这不仅对张量范畴理论本身有用，而且对数学的相关领域，如拓扑量子场论和顶点算子代数的研究都是有用的。在非半简单的情况下，PI将通过支持变种将几何技术融入上同调的研究中，以了解这些范畴的结构。PI将利用Lie理论技巧在一些具有正特征的对称张量范畴中构造Hopf代数和Nichols代数，目的是促进它们的分类。为了加深对融合和模块范畴结构的理解，PI将重点放在完美融合范畴上，并将重点放在对“小”融合范畴的分类方案上。研究完美融合范畴将使数学家能够理解弱积分融合范畴，并将产生例如对实验物理学家探测万能任意子的见解。此外，PI将调查一些建筑的影响，如在不同环境下的测量和激励。这一裁决反映了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。

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.