Classical and quantum hyperbolic geometry

经典和量子双曲几何

• 批准号: 0604866
• 负责人: Francis Bonahon
• 金额: $ 51.22万
• 依托单位: University of Southern California
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0604866&HistoricalAwards=false
• 关键词: Classical; quantum; hyperbolic; geometry

In the last thirty years, much of the progress in our understanding of 3-dimensional topology has been grounded in hyperbolic (non- euclidean) geometry and in topological quantum field theory. However, these two branches of mathematics have long evolved in parallel, without much interaction with each other. Hypothetical bridges between these two fields are now beginning to emerge. One of them is the Volume Conjecture, experimentally verified by a few computations, which connects the asymptotic growth of the Jones polynomials of a knot to the volume of the canonical hyperbolic metric of its complement. The goal of the Project is to develop a conceptual framework combining the two points of view. The Project has a 2- dimensional component, focussed on the representation theory of the quantum Teichmuller space of a surface, as introduced by physicists to model quantum gravity in dimension 2+1. A second part of the Project builds on the insight gained in dimension 2 to develop invariants of hyperbolic 3-dimensional manifolds, closely related to the objects appearing in the Volume Conjecture.The Project aims at gaining a better understanding of 3-dimensional geometry, such as the problem of deciding when two knotted curves in space can be deformed to each other. One traditional approach to this problem involves the algebraic manipulation of certain polynomials associated to pictorial descriptions of these curves. Another powerful technique, with widely used software implementation, uses hyperbolic non-euclidean geometry. Experimental evidence suggests an unexpected connection between these two points of view. The goal of the Project is to develop technical and conceptual tools to confirm (or disprove) this connection, and to better understand the phenomena involved.

在过去的30年里，我们在理解 三维拓扑的基本理论基础是双曲（非欧几里德）几何和拓扑量子场论。然而，这两个数学分支长期以来一直是平行发展的，彼此之间没有太多的互动。假想桥梁 这两个领域之间的关系已经开始显现。其中之一是体积猜想，实验验证了一些计算，其中连接的渐近增长的琼斯多项式的一个结的体积的典型双曲度量的补充。该项目的目标是制定一个结合这两种观点的概念框架。该项目有一个2维组件，专注于表面的量子Teichmuller空间的表示理论，由物理学家引入以模拟2+1维的量子引力。该项目的第二部分建立在2维中获得的洞察力的基础上，开发双曲三维流形的不变量，与体积猜想中出现的对象密切相关。该项目旨在更好地理解三维几何，例如决定空间中两条打结曲线何时可以相互变形的问题。这个问题的一个传统的方法涉及代数操作的某些多项式相关的图形描述这些曲线。另一个强大的技术，广泛使用的软件实现，使用双曲非欧几里德几何。实验证据表明，这两种观点之间存在意想不到的联系。该项目的目标是开发技术和概念工具来证实（或反驳）这种联系，并更好地理解所涉及的现象。