RUI: Relating quantum and classical topology and geometry

RUI：关联量子和经典拓扑和几何

• 批准号: 1105692
• 负责人: Helen Wong
• 金额: $ 12.48万
• 依托单位: Carleton College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-06-15 至 2016-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1105692&HistoricalAwards=false
• 关键词: RUI; Relating; quantum; classical; topology

The proposed research focuses on a construction that lies at the core of quantum topology, namely the Kauffman skein algebra of a space. This combinatorial object was first defined with the Jones polynomial in mind and thus plays a central role in the corresponding Witten-Reshetikhin-Turaev topology quantum field theory for 3-manifolds. Later, it was realized in terms of hyperbolic geometry, namely as a quantization of the PSL(2,C)-character variety. However, the relationships between the various interpretations remain somewhat mysterious. By better understanding the algebraic structure of the Kauffman skein algebra, the PI hopes to facilitate further applications of quantum theory to problems in 3-manifold theory and to uncover relationships with existing classical topological invariants. This project also continues the work of Bonahon and the PI to classify representations of the Kauffman bracket skein algebra, an endeavor which combines skein theoretic arguments with the representation theory of the quantum Teichmuller space. From its inception, quantum topology has been a bridge between mathematics and mathematical physics. Topology is an area of mathematics concerned with the intrinsic properties of a space, that is, properties that are preserved under continuous deformations. This is in contrast to geometry, where there is a definite concept of distance between points in the space and deformations are not allowed. Circa 1980, researchers developed a new topological quantum field theory which, as its nomenclature suggests, drew from both quantum physics and topology. This new theory opened up exciting avenues for research, and in particular has allowed many mathematical theorems and constructions to find applications in physics, quantum computation, and beyond. Conjectured deep connections between geometry and quantum theory are too becoming clearer and is a subject of the proposed research. Indeed, the main goal is to strengthen the relationships between these three - quantum theory, topology, and geometry.

拟议的研究重点是量子拓扑核心的构造，即空间的考夫曼绞丝代数。这个组合对象首先是用琼斯多项式定义的，因此在相应的 3 流形 Witten-Reshetikhin-Turaev 拓扑量子场论中发挥着核心作用。 后来，它以双曲几何的形式实现，即作为 PSL(2,C) 字符簇的量化。 然而，各种解释之间的关系仍然有些神秘。 通过更好地理解考夫曼绞丝代数的代数结构，PI 希望促进量子理论进一步应用于三流形理论中的问题，并揭示与现有经典拓扑不变量的关系。 该项目还继续了 Bonahon 和 PI 对 Kauffman 括号绞纱代数表示进行分类的工作，这项工作将绞纱理论论证与量子 Teichmuller 空间的表示理论相结合。从一开始，量子拓扑就一直是数学和数学物理之间的桥梁。 拓扑学是一个数学领域，涉及空间的内在属性，即在连续变形下保留的属性。 这与几何相反，在几何中，空间中的点之间的距离有明确的概念，并且不允许变形。 大约 1980 年，研究人员开发了一种新的拓扑量子场论，正如其命名所示，它借鉴了量子物理学和拓扑学。 这一新理论开辟了令人兴奋的研究途径，特别是使许多数学定理和结构在物理学、量子计算等领域得到应用。 几何学和量子理论之间推测的深层联系也变得越来越清晰，并且是拟议研究的一个主题。 事实上，主要目标是加强量子理论、拓扑和几何这三者之间的关系。