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，有了当前的范式论技术(即研究物体及其如何运输)，经典的代数结构可以升级，以提供关于以前不被理解的自然现象的信息。这在量子物理学中产生了重大后果。由这项拨款赞助的工作是在一元范畴的框架内进行的，这是一种配备了组合对象和对象之间组合地图的方法的类别。有几个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系列教科书方面取得重大进展。国际学生联合会也是许多代表不足的群体成员的积极导师，特别是对国际学生联合会所属群体(女性、非裔美国人和第一代大学生)的指导。这项拨款资助的项目的第一个研究主题是一元化范畴中的代数。PI将把域上代数的经典性质推广到么半群环境中，还将研究仅在范畴环境中有重要意义的性质。此外，PI还将检查么半群范畴中的其他代数结构(例如，Frobenius代数)，特别是那些与拓扑量子场论(TQFT)相关的结构。PI赞助的研究工作的另一个主题是关于在二维共形场论(2D-CFT)中起关键作用的某些么半范畴的表示，以及对应于3D-TQFT的表示。特别令人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.Lauwitz和M.Yakimov共同构建编织一元式范畴的规范表示的基础上。该奖项反映了NSF的法定使命，并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，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.