The Geometry and Topology of the Jones Polynomial

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This National Science Foundation funded project is focused on PIs' cutting-edge research in Quantum Topology, an area of mathematics that involves three-dimensional shapes that are called three-manifolds and knotted circles within them. Knots appear naturally in the process of replication of DNA, a fundamental part of life. They also appear in theoretical and practical models of quantum computation, using pioneering work of Jones and Freedman, two mathematicians decorated by the highest honor in Mathematics, namely a Fields Medal. Those models of quantum computation, if realized, may bring changes to the world comparable to the invention of electricity.This research project in Quantum Topology links quantum field theory to topology in dimensions three and four, leading to powerful invariants of knots and three-manifolds that include the Jones polynomial, the 3D-index, state-sums and state-integrals. The PI and the co-PI have had substantial results on the formulation and proof of long-standing conjectures in this area, and propose to study key unsolved problems. The proposal is modern, interdisciplinary and employs methods in topology, geometry, algebra, number theory, analysis, quantum field theory and combinatorics.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.