Geometric and Quantum Structures of 3-Manifolds

Michigan State University, East Lansing MI

Investigators

Abstract

The study of three-dimensional spaces, and knotted curves in them, is essential to our understanding of large- and small-scale aspects of the universe. A classification of these spaces will rest on a mathematical understanding of the possible shapes they can take and the rigidity and flexibility properties they can have. These properties are known as invariants and they come from algebraic, analytic, and geometric considerations, often with crucial input from physics. The proof of Thurston's Geometrization Conjecture established that three-dimensional spaces, called manifolds, decompose into pieces that admit explicit geometries. In the last few decades, ideas from quantum physics have led mathematicians to the discovery of a variety of subtle invariants and structures of three-manifolds and the knotted curves contained in them. 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 investigate the relations of these quantum invariants to the geometric structures arising from Thurston's picture and explore the ramifications and applications of these connections to mathematics and physics. The project also provides topics for graduate student research. The project will combine geometric and quantum topology techniques to study the interplay of geometry, topological quantum field theories (TQFT), and combinatorial structures of three-manifolds, with an eye towards developing tools to tackle open conjectures in quantum topology. One part of the project will continue work around the Turaev-Viro invariants volume conjecture, and on the geometry of quantum representations of surface mapping class groups. The goal is to understand the extent to which asymptotic features of TQFT detect or determine the existence of hyperbolic pieces in the geometric decomposition of 3-manifolds. A second part of the project will study relations between the colored Jones knot polynomials, the topology of incompressible surfaces in link complements, and hyperbolic geometry. A third part will develop methods for recognizing geometric structures on three-manifolds from purely combinatorial input and derive estimates on geometric quantities from topological data. This includes the study of low-genus incompressible surfaces in certain link complements and the understanding of how knot diagrammatic properties and constraints affect the geometric structure of link complements. 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.