Skein Modules, Representations, and Quantum Invariants of Three-Manifolds

9803233弗罗曼·弗罗曼和他的学生将继续研究考夫曼括号斜切模块，作为将量子不变量和经典不变量联系起来的一种手段。他们将继续发展基于量子SL(2，C)的格点规范场理论。这将导致3-流形的Kauffman括号斜线模的一般描述为该3-流形的脊上的量子化的平坦SL(2，C)联络的不变函数。我们的目标是深入了解流形中的构型和Kauffman括号绞线模的代数结构之间的关系。它还应该导致根据流形的表示理论严格地推导出Witten的量子不变量的渐近展开。结的补边上的双曲线结构导致了外围环面上的相似结构。他们的意图是在非对易环境中进行类似的构建。他们将推导出A-多项式的非对易模拟，这将反过来允许解释A-多项式和有色琼斯多项式之间的关系。这项工作将揭示琼斯多项式中与纽结补的双曲几何有关的那部分。最后，他们将看到纽结的Kauffman括号斜线模如何在非对易环面上诱导几何。理解拓扑学的最好方法是将其作为初等几何的延伸。回想一下，如果平面的刚性运动将两个三角形连接到另一个平面，则两个三角形是全等的。刚性意味着运动将保留所有距离和角度。三角形的几何分类表明，如果两个三角形的边长相等，则它们是全等的。在拓扑学中，使用了更大的同余变换集合，特别是将近点发送到近点的平面的任何运动(尽管变换可能会扭曲距离)。有时拓扑被称为橡皮片几何，因为同余变换可以任意拉伸平面。在这种几何概念下，任何两个三角形都是全等的，实际上它们都全等同于一个圆。嵌入3-空间(结)的圆的拓扑研究要微妙得多。一个主要的问题是如何进行“拓扑”测量。也就是说，将数字与通过拓扑同余变换保持不变的节点相关联。在20世纪80年代的S，琼斯和威腾的工作预示着纽结理论的新纪元，量子不变量的引入。这些不变量来源于算符理论和量子力学中的技术。弗罗曼和他的学生将从一种观点来研究这些不变量，使它们与来自更经典技术的纽结不变量相关。这些研究将通过在每个领域的标准结构之间增加更深层次的联系来丰富拓扑学、量子力学和算子代数。***

9803233 Frohman Frohman and his students will pursue the study of the Kauffman bracket skein module as a means of relating quantum and classical invariants. They will continue to develop lattice gauge field theory based on quantum SL(2,C). This will lead to a general description of the Kauffman bracket skein module of a 3-manifold as invariant functions of the quantized flat SL(2,C) connections on a spine of the three-manifold. The goal is an intimate understanding of the relationship between configurations lying in a manifold and the algebraic structure of the Kauffman bracket skein module. It should also lead to a rigorous derivation of Witten's asymptotic expansion of quantum invariants in terms of the representation theory of the manifold. A hyperbolic structure on the complement of a knot induces a similarity structure on the peripheral torus. It is their intention to carry out an analogous construction in the noncommutative setting. They will derive a noncommutative analog of the A-polynomial, which will in turn allow the relationship between the A-polynomial and the colored Jones polynomial to be explicated. This work will uncover that part of the Jones polynomial that is related to the hyperbolic geometry of the knot complement. Finally they will see how the Kauffman bracket skein module of a knot induces a geometry on the noncommutative torus. The best way to understand topology is as an extension of elementary geometry. Recall that two triangles are congruent if there is a rigid motion of the plane that takes one to the other. Rigid means that the motion preserves all distances and angles. The geometric classification of triangles states that two triangles are congruent if their sides are of the same length. In topology, a larger collection of congruence transformations is used, specifically any motion of the plane that sends near points to near points (although the transformation may distort distances). Sometimes topology is refer red to as rubber sheet geometry, as congruence transformations may stretch the plane arbitrarily. Under this notion of geometry, any two triangles are congruent, and in fact they are all congruent to a circle. The topological study of circles embedded in 3-space (knots) is much more subtle. A major problem is how to make ``topological'' measurements. That is, associate numbers to a knot that are unchanged by topological congruence transformations. In the 1980's, work of Jones and Witten heralded a new age of knot theory with the introduction of quantum invariants. These invariants are derived from techniques in operator theory and quantum mechanics. Frohman and his students will study these invariants from a viewpoint that allows them to be related to invariants of knots coming from more classical techniques. These investigations will enrich topology, quantum mechanics, and operator algebras by adding deeper connections between standard constructions in each field. ***