Studies in Categorical Algebra

近两个世纪以来，代数结构一直被用来理解自然界中各种实体的行为，特别是对称性。现在，随着范畴论（即研究物体及其运输方式）的发展，经典代数结构可以得到升级，以提供以前未被理解的自然现象的信息。这在量子物理学中产生了重要的结果。这项资助的工作是在一元类别的框架下进行的，一元类别是一种配备了组合对象和组合对象之间映射的方式的类别。一些项目被指定为本科生和研究生的部分工作。此外，PI将在完成三卷本、用户友好的量子代数教科书系列方面取得重大进展。PI也是许多未被充分代表的群体成员的积极导师，特别是那些PI所属的群体（妇女、非裔美国人和第一代大学生）。本基金资助项目的第一个研究主题是一元代数范畴。PI将把代数的经典性质扩展到一元环境，并且还将研究仅在分类设置中具有重要意义的性质。此外，PI将研究一元范畴中的其他代数结构（例如，Frobenius代数），特别是与拓扑量子场论（TQFTs）相关的代数结构。PI赞助的研究工作的另一个主题是在二维共形场论（2d-CFTs）中发挥关键作用的某些单一性范畴的表示，并且与3d- tqft相对应。特别感兴趣的是模张量范畴的表示，PI在这里的工作将建立在最近与R. Laugwitz和M. Yakimov的联合工作的基础上，后者构建了编织一元范畴的规范表示。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。

Algebraic structures have been employed for nearly two centuries to understand the behavior, particularly the symmetry, of various entities in nature. Now with the current technology of category theory (i.e., the study of objects and how they are transported), classical algebraic structures can be upgraded to provide information on natural phenomena that was not previously understood. This yields significant consequences in quantum physics. The work sponsored by this grant lies in the framework of monoidal categories, which are categories that come equipped with a way of combining objects and combining maps between objects. Several projects are earmarked for partial work by undergraduate and graduate students. Moreover, the PI will make significant progress on completing a three-volume, user-friendly textbook series on quantum algebra. The PI is also an active mentor for numerous members of underrepresented groups, particularly for those in groups to which the PI belongs (women, African-Americans, and first generation college students).The first research theme of the projects sponsored by this grant is on algebras in monoidal categories. The PI will extend classical properties of algebras over a field to the monoidal context, and will also study properties that only have significant meaning in the categorical setting. In addition, the PI will examine other algebraic structures (e.g., Frobenius algebras) in monoidal categories, especially those tied to Topological Quantum Field Theories (TQFTs). Another theme of the PI's sponsored research work is on representations of certain monoidal categories that play a crucial role in 2-dimensional Conformal Field Theory (2d-CFTs), and that correspond to 3d-TQFTs. Of particular interest are representations of modular tensor categories, and the PI's work here will build on recent joint work with R. Laugwitz and M. Yakimov that constructs canonical representations of braided monoidal 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.