The Jones Polynomial and Hyperbolic Geometry of Surfaces

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The project focuses on important, fundamental problems in quantum topology and its connection to classical topology. Quantum topology studies and classifies invariants of three- and four-dimensional spaces and knotted circles in them. These objects appear naturally in nature, for example in the theory of DNA and in physics, and have many applications. Quantum topology results and methods might help in constructing theoretical and practical models of quantum computation, which, if realized, would be revolutionary for many industries, solving previously computationally impossible problems. The project is interdisciplinary and employs methods in topology, geometry, algebra, number theory, analysis, quantum field theory, and combinatorics. The project involves mentoring and training of students and postdocs, and outreach. The project has three main topics. The first is the AJ conjecture which connects the colored Jones polynomials and colored HOMFLYPT polynomials to the fundamental group of knots. The second develops and studies hyperbolic topological quantum field theory, with the ultimate goal to better understand and make progress in the volume conjecture and solve the AJ conjecture. The third studies the growth of homology in finite coverings of 3-manifolds, with connections to dynamics of pseudo-Anosov maps on surfaces.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.