The Jones Polynomial and Hyperbolic Geometry of Surfaces

该项目侧重于量子拓扑中的重要基础问题及其与经典拓扑的联系。量子拓扑学研究并分类了三维和四维空间的不变量以及其中的打结圆。这些物体在自然界中自然出现，例如在DNA理论和物理学中，并且有许多应用。量子拓扑的结果和方法可能有助于构建量子计算的理论和实践模型，如果实现，将对许多行业具有革命性意义，解决以前在计算上不可能解决的问题。该项目是跨学科的，采用了拓扑、几何、代数、数论、分析、量子场论和组合学等方法。该项目包括对学生和博士后的指导和培训，以及外展。该项目有三个主要主题。第一个是AJ猜想，它将有色的琼斯多项式和有色的HOMFLYPT多项式与结的基本群联系起来。二是发展和研究双曲拓扑量子场论，最终目的是更好地理解和推进体积猜想和解决AJ猜想。第三章研究了3流形有限覆盖上的同调生长，并与曲面上伪anosov映射的动力学联系起来。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。

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.