The proposed research focuses on a construction that lies at the core of quantum topology, namely the Kauffman skein algebra of a space. This combinatorial object was first defined with the Jones polynomial in mind and thus plays a central role in the corresponding Witten-Reshetikhin-Turaev topology quantum field theory for 3-manifolds. Later, it was realized in terms of hyperbolic geometry, namely as a quantization of the PSL(2,C)-character variety. However, the relationships between the various interpretations remain somewhat mysterious. By better understanding the algebraic structure of the Kauffman skein algebra, the PI hopes to facilitate further applications of quantum theory to problems in 3-manifold theory and to uncover relationships with existing classical topological invariants. This project also continues the work of Bonahon and the PI to classify representations of the Kauffman bracket skein algebra, an endeavor which combines skein theoretic arguments with the representation theory of the quantum Teichmuller space.
From its inception, quantum topology has been a bridge between mathematics and mathematical physics. Topology is an area of mathematics concerned with the intrinsic properties of a space, that is, properties that are preserved under continuous deformations. This is in contrast to geometry, where there is a definite concept of distance between points in the space and deformations are not allowed. Circa 1980, researchers developed a new topological quantum field theory which, as its nomenclature suggests, drew from both quantum physics and topology. This new theory opened up exciting avenues for research, and in particular has allowed many mathematical theorems and constructions to find applications in physics, quantum computation, and beyond. Conjectured deep connections between geometry and quantum theory are too becoming clearer and is a subject of the proposed research. Indeed, the main goal is to strengthen the relationships between these three - quantum theory, topology, and geometry.