Topological Quantum Field Theory and Geometric Structures in Low Dimensional Topology

The study of three-dimensional spaces, and knotted strings in them, is essential for our understanding of several aspects about the shape of the universe. To classify such spaces we need to mathematically understand the possible shapes they can take as well as their rigidity and flexibility properties. Such properties are called invariants of the spaces and they arise from a variety of mathematical considerations, often with crucial input from physics. The proof of Thurston's Geometrization Conjecture established that certain three-dimensional spaces, called manifolds, decompose into pieces with nice geometric properties. In the last few decades, ideas from quantum physics have led mathematicians to discover a variety of subtle invariants and structures of three-manifolds.There are several open conjectures, both in physics and in mathematics, that predict deep relations between quantum structures and geometries of three-manifolds. This project will study the relations of these quantum invariants to the geometric structures arising from Thurston's picture, with an eye towards developing tools to tackle open conjectures in topology and physics. The project includes topics for graduate student research and contributes to training and professional development by providing critical support through mentoring and conference travel. One direction of the project will study relations between asymptotic aspects of Topological Quantum Field Theories, the coarse geometry of Teichmuller spaces and volumes of fibered 3-manifolds.Another direction will study the asymptotic growth of the Turaev-Viro three-manifold invariants aiming to prove that it detects the existence of hyperbolic pieces in the geometric decompositions of three-manifolds. A third direction will continue the study of relations between the colored Jones knot polynomials and incompressible surfaces in link complements. The PI will further develop this framework and, in particular, study the strong slope conjecture, and its applications, within it. The PI will also use quantum invariants to understand classical invariants such as crossing numbers and crosscap numbers of knots. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.