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

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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.