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.

代数结构已被近两个世纪以来，以了解自然界各种实体的行为，尤其是对称性。现在，随着类别理论的当前技术（即对物体的研究及其运输方式），可以升级经典的代数结构，以提供有关自然现象的信息，这些信息先前尚未理解。这会在量子物理学中产生重大影响。该赠款赞助的工作在于单型类别的框架，这些类别是配备了将对象组合和组合对象之间的地图结合的方法。本科生和研究生专门为部分工作指定了一些项目。此外，PI将在完成有关量子代数的三卷，用户友好的教科书系列方面取得重大进展。 PI还是代表性不足的群体的众多成员的积极导师，特别是对于PI所属的组（妇女，非裔美国人和第一代大学生）的团体。 PI将将代数的经典特性扩展到字段上，并将研究仅在分类环境中具有显着含义的属性。此外，PI将检查单素类别中的其他代数结构（例如Frobenius代数），尤其是与拓扑量子场理论（TQFTS）相关的类别。 PI赞助的研究工作的另一个主题是在某些单型类别的表示中，这些类别在二维形式的保形场理论（2D-CFTS）中起着至关重要的作用，并且对应于3D-TQFTS。特别令人感兴趣的是模块化张量类别的表示，PI的工作将基于与R. Laugwitz和M. Yakimov的最新联合合作，该合作构建了编织的单型单类类别的规范表示。该奖项反映了NSF的法定任务，并被认为是通过基金会的知识优点和广泛的crietia的评估来通过评估来进行评估，这是值得的。