RUI: Relating quantum and classical topology and geometry

提出的研究重点是量子拓扑的核心结构，即空间的Kauffman skein代数。这个组合对象最初是用琼斯多项式定义的，因此在相应的3流形的Witten-Reshetikhin-Turaev拓扑量子场论中起着核心作用。后来，它在双曲几何中实现，即作为PSL(2，C)-字符变化的量化。然而，各种解释之间的关系仍然有些神秘。通过更好地理解Kauffman skein代数的代数结构，PI希望促进量子理论在3流形理论问题中的进一步应用，并揭示与现有经典拓扑不变量的关系。该项目还延续了Bonahon和PI的工作，对Kauffman括号串代数的表示进行分类，这是一项将串理论论证与量子Teichmuller空间的表示理论相结合的努力。从一开始，量子拓扑学就一直是数学和数学物理之间的桥梁。拓扑学是研究空间内在性质的数学领域，即在连续变形下保留的性质。这与几何相反，在几何中，空间中点之间的距离有一个明确的概念，并且不允许变形。大约在1980年，研究人员发展了一种新的拓扑量子场论，正如它的命名所示，它从量子物理学和拓扑学两方面得出了结论。这个新理论为研究开辟了令人兴奋的途径，特别是允许许多数学定理和结构在物理学，量子计算等领域找到应用。几何学和量子理论之间的猜想的深层联系也变得越来越清晰，并且是提议研究的一个主题。事实上，主要目标是加强这三个量子理论、拓扑和几何之间的关系。

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.