CAREER: The geometry and physics of non-semi-simple quantum topology

纯数学的世界植根于文字和思想，而不是数字和代数方程。 乍一看，这些词语和观念似乎是外来的，与自然世界无关。 然而，纯数学是现代物理学的语言。 从广义上讲，这个建议的主要智力价值是创造新的数学，这将增加物理学的语言。 更确切地说，与李代数相关的量子群理论在低维拓扑中得到了广泛而富有成效的应用，特别是在量子不变量的创建方面。 在此背景下，PI和他的合作者给出了新的系统策略来定义从非半简单范畴产生的重新归一化的量子不变量。 该建议的重点是描述和探索这些重新规范化的不变量的几何和物理意义，同时使用这些不变量的独特属性，以获得有关低维拓扑中的知名问题的新信息。 这一建议的更广泛影响可分为两大类：辅导和外联。 PI将为犹他州州立大学数学系的学生创建一个指导系统，旨在为学生提供技能，帮助他们在未来的学术追求中取得成功。 该补助金的推广部分是组织一个摩押拓扑会议，这将使学生和博士后参加并介绍他们的研究，同时也与领先的专家互动。建议的研究的第一个主题是进一步探索PI的重新规范化的Reshetikhin-Turaev 3-流形不变量的体积猜想的背景下。 特别是，他建议开发一种理论，细化重新规范化的3-流形不变量，导致新的不变量，这是可计算的，并捕捉新的几何特征。 拟议的研究还将集中在继续PI的创建和研究新的拓扑量子场论（TQFT）所产生的重新规范化的不变量。 建议的研究的最后一个主题是扩大PI的发展和检查的莱文-温模型。 他建议继续使用已知的数学技术来解释以前在L-W模型中没有发现的物理性质。

The world of pure mathematics is rooted in words and ideas rather than numbers and algebraic equations. At first glance these words and ideas seem foreign and unrelated to the natural world. However, pure mathematics is the language of modern physics. Broadly speaking the main intellectual merit of this proposal is to create new mathematics which will add to the language of physics. More precisely, the theory of quantum groups associated to Lie algebras is widely and productively used in low-dimensional topology, in particular with the creation of quantum invariants. Within this context, the PI and his collaborators have given new systematic strategies to define re-normalized quantum invariants arising from non-semi-simple categories. The focus of this proposal is to describe and explore the geometric and physical meanings of these re-normalized invariants, while concurrently using the unique properties of these invariants to gain new information about well-known problems in low-dimensional topology. The broader impacts of this proposal can be arranged in two main categories: mentoring and outreach. The PI will create a mentoring system for the students of the Mathematics Department at Utah State University, with the aim of providing students skills to help them succeed in future academic pursuits. The outreach component of this grant is to organize a Moab Topology Conference which will allow students and postdocs to attend and present their research, while also interacting with leading experts. The first main theme of the proposed research is further exploration of the PI's re-normalized Reshetikhin-Turaev 3-manifold invariants in the context of the Volume Conjecture. In particular, he proposes to develop a theory of refined re-normalized 3-manifold invariants, leading to new invariants which are computable and catch novel geometric features. The proposed research will also focus on continuing the PI's creation and study of new Topological Quantum Field Theories (TQFTs) arising from re-normalized invariants. A final theme of the proposed research is to widen the PI's development and inspection of the Levin-Wen models. He proposes to continue to use known mathematical techniques to interpret physical properties which were not previously detected in the L-W models.