Studies in Categorical Algebra

分类代数研究

• 批准号: 2348833
• 负责人: Chelsea Walton
• 金额: $ 35万
• 依托单位: William Marsh Rice University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-05-01 至 2027-04-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2348833&HistoricalAwards=false
• 关键词: Studies; Categorical; Algebra

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.

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，随着类别理论的当前技术（即，研究物体及其如何运输），经典的代数结构可以升级，以提供以前不理解的自然现象的信息。这在量子物理学中产生了重要的后果。这项资助资助的工作是在monoidal范畴的框架内进行的，monoidal范畴是一种配备了组合对象和组合对象之间映射的方法的范畴。有几个项目被指定由本科生和研究生进行部分工作。此外，PI将在完成一个三卷，用户友好的量子代数教科书系列方面取得重大进展。PI也是一个积极的导师为代表性不足的群体的许多成员，特别是对于那些在PI所属的群体（妇女，非洲裔美国人，和第一代大学生）。PI将扩展域上代数的经典性质到monoidal上下文，并且还将研究仅在范畴设置中具有重要意义的性质。此外，PI将检查其他代数结构（例如，Frobenius代数）在monoidal范畴，特别是那些与拓扑量子场论（TQFT）。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monoidal类别的表示，并对应于3d-TQFT。特别感兴趣的是模张量类别的表示，PI在这里的工作将建立在最近与R的联合工作的基础上。Laugwitz和M.该奖项反映了NSF的法定使命，并被认为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来支持。