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; 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结构，即空间的Kauffman绞线代数。该组合物体首先是用琼斯多项式定义的，因此在相应的witten-reshetikhin-turaev拓扑拓扑范围量子科在3个manifolds中起着核心作用。 后来，它是根据双曲线几何形状实现的，即是PSL（2，c） - 特定品种的量化。 但是，各种解释之间的关系仍然有些神秘。 通过更好地理解Kauffman Skein代数的代数结构，PI希望促进量子理论在3-manifold理论中的问题上进一步应用，并发现与现有的经典拓扑不变性的关系。 该项目还延续了Bonahon和PI的工作，以对Kauffman支架Skein代数的表示形式进行分类，这是一项将Kkein理论论证与量子Teichmuller空间的代表理论相结合的努力。从成立开始，量子拓扑一直是数学和数学物理学之间的桥梁。 拓扑是与空间的固有特性有关的数学领域，即在连续变形下保存的特性。 这与几何形状形成鲜明对比的是，在空间之间的点和变形之间存在明确的距离概念。 大约1980年，研究人员开发了一种新的拓扑量子场理论，正如其命名法所暗示的那样，它源于量子物理和拓扑。 这一新理论为研究开辟了令人兴奋的途径，尤其是允许许多数学定理和构造找到物理，量子计算及其他方面的应用。 猜想的几何形状和量子理论之间的深层联系变得越来越清晰，并且是拟议研究的主题。 确实，主要目标是加强这三个量子理论，拓扑和几何形状之间的关系。