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，利用当前的范畴论技术（即对物体及其传输方式的研究），经典代数结构可以升级以提供以前未理解的自然现象的信息。这在量子物理学中产生了重大影响。这项资助资助的工作位于幺半群范畴的框架内，这些范畴配备了组合对象和组合对象之间的映射的方式。有几个项目指定由本科生和研究生进行部分工作。此外，PI 将在完成三卷、用户友好的量子代数教科书系列方面取得重大进展。 PI 也是众多代表性不足群体成员的积极导师，特别是 PI 所属群体（女性、非裔美国人和第一代大学生）。这笔赠款资助的项目的第一个研究主题是关于幺半群范畴中的代数。 PI 将把代数的经典性质扩展到幺半群环境中，并且还将研究仅在分类环境中具有重要意义的性质。此外，PI 将检查幺半群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 (TQFT) 相关的代数结构。 PI 赞助的研究工作的另一个主题是某些幺半群类别的表示，这些类别在二维共形场论 (2d-CFT) 中发挥着至关重要的作用，并且对应于 3d-TQFT。特别令人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. Laugwitz 和 M. Yakimov 的合作基础上，该工作构建了编织幺半群类别的规范表示。该奖项反映了 NSF 的法定使命，并被认为是值得的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来提供支持。