CAREER: Equivariant topological field theories and higher cluster categories

本文讨论了同邻高范畴理论的几个扩展和应用。我们的第一个基本目标是开发许多已知模型的等变版本，并建立它们之间的等价性。我们提出的第一个应用是发展等变扩展拓扑场理论的同调方法。第二个应用是由曲面产生的拓扑簇类的发展；扩展到高维应该允许开发类似于曲面的高维流形的新不变量。这第二个应用程序也应该告知第一个应用程序，高维簇类别提供了关于拓扑场理论的新信息。第三个应用是关于霍尔代数和代数k理论之间的联系。一方面，同调霍尔代数的构造有望产生k理论谱，这将提供关于霍尔代数的新信息，特别是那些与量子群有关的信息。另一方面，霍尔代数构造的变体有相应的代数k理论变体，值得进一步研究。从广义上讲，这一建议涉及将代数信息纳入目前广泛应用于各种数学中的分类和拓扑结构中。然后，我们寻求将这些增强的结构应用于数学物理、流形理论和表示理论。该提案的教育部分包括一系列的四个夏季研讨会，这些研讨会针对的是正在转入加州大学河滨分校的数学专业学生。目标是每年帮助20名参与者通过介绍证明技术、更多的理论概念和对高级数学课程主题范围的广泛概述，过渡到高级数学。学生将获得一些后续指导活动，包括参与本科生研究的机会。

This proposal is concerned with several extensions and applications of the theory of homotopical higher categories. Our first foundational objective is to develop equivariant versions of the many known models for homotopical higher categories and to establish equivalences between them. Our first proposed application is the development of homotopical approaches to equivariant extended topological field theories. The second application is the development of topological cluster categories arising from surfaces; extending to higher dimensions should allow for the development of new invariants of higher-dimensional manifolds analogous to ones for surfaces. This second application should also inform the first, with higher-dimensional cluster categories giving new information about topological field theories. The third application is concerned with connections between Hall algebras and algebraic K-theory. In one direction, constructions of homotopical Hall algebras are expected to give rise to K-theory spectra which should give new information about Hall algebras, especially those related to quantum groups. In another, variations of Hall algebra constructions have corresponding variants of algebraic K-theory which are worthy of further investigation. Broadly speaking, this proposal is concerned with incorporating algebraic information into categorical and topological structures which are currently being used in a wide range of kinds of mathematics. We then seek to apply these enhanced structures in mathematical physics, manifold theory, and representation theory. The educational component of this proposal consists of a series of four summer workshops for mathematics majors who are in the process of transferring to UC Riverside. The goal is to help twenty participants each year to make the transition to upper-level mathematics via introduction to proof techniques, more theoretical concepts, and a broad overview of the range of topics in higher-level mathematics courses. Students will be provided with some follow-up mentoring activities, including opportunities for participating in undergraduate research.