CAREER: Equivariant topological field theories and higher cluster categories

这个建议涉及同伦高范畴理论的几个推广和应用。 我们的第一个基本目标是开发同伦更高类别的许多已知模型的等变版本，并建立它们之间的等价关系。 我们提出的第一个应用是同伦方法的发展等变扩展拓扑场论。 第二个应用是发展的拓扑集群类别所产生的表面;扩展到更高的维度应允许发展新的不变量的高维流形类似的表面。 这第二个应用程序也应该通知第一个，与高维集群类别提供新的信息拓扑场论。 第三个应用是关于霍尔代数和代数K理论之间的联系。 在一个方向上，同伦Hall代数的构造有望产生K-理论谱，这将提供有关Hall代数的新信息，特别是与量子群有关的信息。 另一方面，Hall代数结构的变体有相应的代数K-理论的变体，值得进一步研究。 广义地说，这一建议涉及到将代数信息纳入范畴和拓扑结构，目前正在使用的各种数学。 然后，我们试图将这些增强的结构应用于数学物理，流形理论和表示论。 该提案的教育部分包括一系列的四个夏季研讨会的数学专业谁是在转移到加州大学滨江的过程中。 我们的目标是帮助每年20名参与者通过介绍证明技术，更多的理论概念以及对高级数学课程主题范围的广泛概述，过渡到高级数学。 学生将提供一些后续辅导活动，包括参与本科研究的机会。

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.