Classical and quantum invariants of low-dimensional manifolds

这个项目旨在促进我们对结点和三维空间的理解，这些空间自然出现在物理、计算机图形学、化学和生物等不同领域。量子场论的最新概念将物理学与3维和4维拓扑联系起来，导致了纽结和三维流形的强大不变量，其中包括琼斯多项式和3D指数。该项目旨在解决这些量子不变量与经典拓扑学和几何学相关的关键问题。该项目将对低维拓扑学产生重大影响，并导致数学和数学物理之间深刻而意想不到的联系。

This project aims to advance our understanding of knots and 3-dimensional spaces, which arise naturally in fields as diverse as physics, computer graphics, chemistry and biology. Recent ideas from quantum field theory link physics to topology in dimensions 3 and 4, leading to powerful invariants of knots and 3-dimensional manifolds that include the Jones polynomial and the 3D-index. This project aims to resolve key questions relating these quantum invariants to classical topology and geometry. The project will have a major impact in low-dimensional topology, and lead to deep and unexpected connections between mathematics and mathematical physics.