The Geometry and Topology of the Jones Polynomial

琼斯多项式的几何和拓扑

• 批准号: 1811114
• 负责人: Thang Le
• 金额: $ 44万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-05-15 至 2024-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1811114&HistoricalAwards=false
• 关键词: Geometry; Topology; Jones; Polynomial

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.

这个国家科学基金会资助的项目的重点是PIS在量子拓扑领域的尖端研究，这是一个数学领域，其中涉及三维形状，这些形状称为三个manifolds和其中的圆圈。结在DNA复制过程中自然而然地出现，这是生活的基本部分。它们还使用琼斯和弗里德曼（Freedman）的开创性工作，这是两位数学家，在数学中最高荣誉装饰，即是田野勋章。量子计算的那些模型，如果实现的话，可能会为世界带来与电力的发明相媲美的变化。量子拓扑的研究项目量子场理论将量子场理论与拓扑结合在第三和四个方面，从而导致结的强大不变性和三个manifords，其中包括琼斯琼斯多一项，状态，状态，状态 - 犹太人和状态及其状态和状态。 PI和COPI在该领域的长期猜想的制定和证明方面取得了重大结果，并建议研究关键的未解决问题。该提案是现代的，跨学科的，并采用拓扑，几何，代数，数字理论，分析，量子场理论和组合学方面的方法。该奖项反映了NSF的法定任务，并被认为是通过基金会的知识分子优点和更广泛的审查标准来通过评估来通过评估来支持的。

项目成果

Triangular decomposition of skein algebras

绞纱代数的三角分解

• DOI: 10.4171/qt/115
• 发表时间: 2018
• 期刊: Quantum Topology
• 影响因子: 1.1
• 作者: Lê, Thang T.

The slope conjecture for Montesinos knots

• DOI: 10.1142/s0129167x20500561
• 发表时间: 2018-07
• 期刊: International Journal of Mathematics
• 影响因子: 0.6
• 作者: S. Garoufalidis;C. Lee;Roland van der Veen

Graph Complexes and the Sympletic Character of the Torelli Group

图复形和 Torelli 群的辛特征

• 发表时间: 2017
• 期刊: arXiv.org
• 作者: Garoufalidis, S;Getzler, E.
• 通讯作者: Getzler, E.

Asymptotics of Nahm sums at roots of unity

• DOI: 10.1007/s11139-020-00266-x
• 发表时间: 2020-07-21
• 期刊: RAMANUJAN JOURNAL
• 影响因子: 0.7
• 作者: Garoufalidis, Stavros;Zagier, Don
• 通讯作者: Zagier, Don

Stated skein algebras of surfaces

规定的表面绞纱代数

• DOI: 10.4171/jems/1167
• 发表时间: 2022
• 期刊: Journal of the European Mathematical Society
• 影响因子: 2.6
• 作者: Costantino, Francesco;Lê, Thang T.
• 通讯作者: Lê, Thang T.