RUI: Relating quantum and classical topology and geometry

所提出的研究集中于量子拓扑学的核心结构，即空间的Kauffman skein代数。这个组合对象的定义首先考虑到了Jones多项式，因此在相应的三维流形的Witten-Reshetikhin-Turaev拓扑量子场论中起着核心作用。后来，它被实现为双曲几何，即作为PSL(2，C)-字符变体的量子化。然而，各种解释之间的关系仍然有些神秘。PI希望通过更好地理解Kauffman skein代数的代数结构，促进量子理论在3-流形理论中的进一步应用，并揭示与现有经典拓扑不变量的关系。这个项目还继续了Bonahon和PI对Kauffman括号Skein代数的表示进行分类的工作，这是一项将Skein理论论点与量子Teichmuller空间的表示理论相结合的努力。量子拓扑学从一开始就是连接数学和数学物理的桥梁。拓扑学是一个与空间的内在性质有关的数学领域，即在连续变形下保持不变的性质。这与几何学形成对比，在几何学中，空间点之间的距离有一个明确的概念，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，正如它的命名法所暗示的那样，它借鉴了量子物理学和拓扑学。这一新理论为研究开辟了令人兴奋的道路，尤其是允许许多数学定理和构造在物理、量子计算等领域找到应用。几何学和量子理论之间的深层联系也变得越来越清晰，这也是拟议研究的一个主题。事实上，主要目标是加强这三个量子理论、拓扑学和几何学之间的关系。

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.