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

9803233 Frohman Frohman和他的学生将继续研究Kauffman托架绞结模块作为一种联系量子不变量和经典不变量的方法。他们将继续发展基于量子SL（2，C）的晶格规范场理论。这将导致将3流形的考夫曼支架串模一般描述为3流形的脊柱上的量子化平坦SL（2，C）连接的不变函数。我们的目标是深入了解流形中的构型与考夫曼支架串模的代数结构之间的关系。根据流形的表示理论，它也应该导致Witten的量子不变量渐近展开的严格推导。在结的补上的双曲结构在周围环面上引起相似结构。他们的意图是在非交换设置中进行类似的结构。他们将推导出a -多项式的非交换类比，这将反过来允许a -多项式和有色琼斯多项式之间的关系得到解释。这项工作将揭示部分琼斯多项式，这是有关的双曲几何的结补。最后，他们将看到结的Kauffman支架绞结模如何在非交换环面上诱导出几何形状。理解拓扑学的最好方法是将其作为初等几何的扩展。回想一下，如果平面上有一个从一个到另一个的刚体运动，两个三角形是相等的。刚性意味着运动保持所有的距离和角度。三角形的几何分类表明，如果两个三角形的边长相同，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运动（尽管变换可能会扭曲距离）。有时拓扑学被称为橡胶板几何，因为同余变换可以任意拉伸平面。在这个几何概念下，任意两个三角形都是全等的，事实上它们都是一个圆的全等。嵌入在三维空间（结点）中的圆的拓扑研究要微妙得多。一个主要问题是如何进行“拓扑”测量。也就是说，将数关联到拓扑同余变换不变的结上。在20世纪80年代，Jones和Witten的工作随着量子不变量的引入预示着一个新的结理论时代的到来。这些不变量是从算符理论和量子力学中推导出来的。Frohman和他的学生将从一种观点来研究这些不变量，这种观点允许它们与来自更经典技术的结的不变量相关联。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，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. ***