Quantum topology and geometric structures in dimension 3

This project focuses on the study of three-dimensional spaces and knotted strings in them. Mathematical classification of the shapes that such spaces can take is essential for understanding the shape of the universe. A classification of these spaces requires an understanding of the rigidity and flexibility properties they have. These properties are called invariants of the spaces and are studied through a variety of mathematical considerations and tools, with crucial input from physics. In this setting, certain 3-dimensional spaces, called 3-manifolds, are important as they are known to decompose into pieces with explicit rigidity properties (geometric structures) and have a rich variety of flexibility properties (topological invariants). In the last few decades, ideas from quantum physics have led mathematicians to discover subtle invariants and structures of three-manifolds. There are open conjectures, both in physics and in mathematics, that predict far reaching relations and interplay between quantum structures and geometries of three-manifolds. The central theme of the project is the study of these relations aiming to develop tools to tackle open conjectures in topology and physics. These are fundamental problems in quantum topology and Topology Quantum Field Theory (TQFT). The project also includes topics for graduate student research and contributes to their professional development as it provides mentoring and conference travel support. The project draws methods from representation theory, hyperbolic geometry, and algebraic geometry, to establish connections between quantum and geometric structures of 3-manifolds. In one direction, the PI will study the relations between Witten-Reshetikhin-Turaev TQFTs and the Thurston norm of 3-manifolds. For instance, work in progress, aims to understand the extent to which these TQFTs determine the Seifert genus of knots. In a second direction, the PI will study skein modules of closed 3-manifolds, their relations to character varieties, and to invariants arising from analytic methods. This will give continuity to prior work on the development of tools to compute these modules, and the question of to what extent they detect incompressible surfaces in 3-manifolds. In a third direction, the project will focus on the interplay of asymptotic aspects of TQFT, with Thurston geometries of 3-manifolds, and with the geometric structures of surfaces. This line of work has applications to the 3-manifold volume conjecture, and the AMU conjecture concerning the geometric content of quantum representations of mapping class groups. The PI, with collaborators and PhD students, has made significant progress on both conjectures. In a fourth direction, the project will continue exploring relations between the colored Jones knot polynomials and essential surfaces in knot complements. This direction also has applications to some long-standing open problems about the behavior of crossing numbers under knot satellite operations. 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.