The Geometry and Topology of the Jones Polynomial

这个国家科学基金会资助的项目专注于pi在量子拓扑方面的前沿研究，量子拓扑是一个涉及三维形状的数学领域，称为三流形和其中的打结圆。结在DNA的复制过程中自然出现，DNA是生命的基本组成部分。它们也出现在量子计算的理论和实践模型中，使用了琼斯和弗里德曼的开创性工作，两位数学家被授予数学最高荣誉，即菲尔兹奖章。这些量子计算模型如果实现，可能会给世界带来与电的发明相当的变化。量子拓扑的这个研究项目将量子场论与三维和四维拓扑联系起来，导致结点和三流形的强大不变量，包括琼斯多项式、三维指数、状态和和状态积分。PI和co-PI在该领域长期存在的猜想的形成和证明方面取得了实质性成果，并提出了对关键未解决问题的研究。该方案是现代的、跨学科的，并采用了拓扑学、几何、代数、数论、分析、量子场论和组合学的方法。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.