The first main theme of the proposed research is further exploration of the Principal Investigators (PI) re-normalized Reshetikhin-Turaev 3-manifold invariants. In particular, the PI will draw connections with these re-normalized invariants and other topological objects including Reidemeister torsion, Topological Quantum Field Theories (TQFTs) and mapping class group representations. The proposed research will also focus on continuing the PI's development and research of generalized Kashaev quantum 3-manifold invariants. These invariants are modeled on Kashaev's foundational work that led to the original formulation of the Volume Conjecture. The final topic of the proposed research is to generalize his re-normalized Reshetikhin-Turaev invariants by defining 3-manifold invariants via surgery presentations of links with flat connections in their complements. This work is both geometric and physical in nature and related to Chern-Simons theory.
One of the fundamental ways mathematics arises in nature is through geometry and more generally topology. In particular, geometric objects called manifolds are used to model space and time in general relativity. Surprisingly, several features of manifolds can be studied via their underlying topology. The discovery of the Jones polynomial by V. Jones in 1984 and its 3-dimensional quantum field theory interpretation by E. Witten in 1989 have opened the door for the use of new algebraic techniques to study topology. These developments led to a new branch of mathematics known as "quantum topology." This area has connections to theoretical physics, including quantum gravity, topological quantum field theory and quantum computing. Many established techniques in quantum topology become zero when quantum dimensions vanish. This project focuses on developing and studying new systematic strategies to re-normalize topological invariants when quantum dimensions are zero.
