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.

在过去的三十年中，我们对三维拓扑的理解的许多进步已经建立在双曲线（非欧几里得）几何形状和拓扑量子场理论中。但是，这两个数学分支长期以来一直并行演变，而没有太多的相互作用。这两个领域之间的假设桥现在开始出现。其中之一是体积猜想，通过一些计算在实验上验证，这将结的琼斯多项式的渐近生长与其补体的规范双曲线度量的体积联系在一起。该项目的目的是开发一个结合两个观点的概念框架。该项目具有2-维组件，集中在地表的量子Teichmuller空间的表示理论上，如物理学家所引入的，以模拟量子重力2+1。该项目的第二部分建立在维度2中获得的洞察力，以开发双曲线3维流形的不变性，与体积猜想中出现的物体密切相关。该项目旨在更好地理解三维几何形状，例如在空间中遇到两个打结的曲线的问题，可以使每个人置于两个打结的曲线时，可以对每个人置于另一个。解决此问题的一种传统方法涉及对与这些曲线的图形描述相关的某些多项式的代数操纵。另一种功能强大的技术，具有广泛使用的软件实现，它使用双曲线非欧盟几何形状。实验证据表明，这两种观点之间存在意想不到的联系。该项目的目的是开发技术和概念工具来确认（或反驳）这种联系，并更好地理解所涉及的现象。