Quantum Invariants and Representations of 3-Manifold Groups

DMS-0207030Charles D.Frohman PI将在低维拓扑及其应用方面开展多个项目。这些项目包括研究Turaev-Viro不变量的各种性质，包括构造将流形的基本群的表示理论与这些不变量联系起来的泛多项式，以及这些不变量在单位圆之外的推广。此外，PI将考虑，给定纽结的四面体分解，如何在四面体的交叉比空间上定义严格路径积分，以计算纽结上的积分运算的Turaev-Viro不变量。PI还将根据特征簇的几何来研究Kauffman括号绞线模块的结构，更一般地，在纽结上的Dehn手术的研究中使用量子不变量。最后，这个奖项为PI的研究生提供了支持，以帮助他进行这项研究。拓扑是一种几何，在这种几何中，同余变换不保持距离和角度等度量属性。PI研究的对象是沿面粘合多面体的结果。要知道两个这样的对象何时在拓扑上不同，需要进行通过拓扑同余变换保持不变的测量。这种测量的一个例子是曲面的欧拉特性，它是顶点数减去边数再加上面数。如果两个曲面在拓扑上等价，则它们具有相同的欧拉特征。最著名的几何定理之一是Gauss-Bonnet定理，它将任何曲面的欧拉特性与按度规计算的量联系起来。三种流形的量子不变量类似于欧拉特性，但更微妙。它们是从一个概率空间中统计地构造出来的，该空间由多面体如何粘合在一起形成对象的描述中产生的“状态”组成。PI的工作是将这些不变量与源自底层物体几何的度量值联系起来。为此，PI表明状态空间可以被对对象进行几何测量的空间所取代。该项目的目标是了解对象的几何和拓扑是如何通过其多面体的组合描述来确定的。

DMS-0207030Charles D. FrohmanThe PI will work on a number of projects in low dimensionaltopology and its applications. These projects include investigating various properties of the Turaev-Viroinvariants including the construction of universal polynomials relating the representation theory of the fundamental groups of manifolds to these invariants and the extension of these invariants away from the unit circle. In addition, the PI will consider, given a tetrahedral decomposition of a knot, how one may define a rigorouspath integral over the space of cross ratios of the tetrahedra to compute the Turaev-Viro invariant of an integral surgery on the knot. The PI will also investigate the structure of the Kauffman bracket skein module in terms of the geometry of character varieties,and, more generally, use quantum invariants in the study of Dehn surgery on knots. Finally, this award provides support for the PI's graduate students to assist him in this research. Topology is a kind of geometry where the congruence transformations do not preserve metric properties such as distance and angle. The objects the PI studies are given as the result of gluing together polyhedra along faces. To know when two such objects are different topologically one needs to make measurements that are unchanged by topological congruence transformations. An example of such a measurement is the Euler characteristic of a surface, which is the number of vertices minus thenumber of edges plus the number of faces. If two surfaces are topologically equivalent they have the same Euler characteristic. One of the most celebrated theorems of geometry is the Gauss-Bonnet theorem which relates the Euler characteristic of any surface to a quantity computed metrically. Quantum invariants of three manifolds are like Euler characteristic, but more delicate. They are constructed statistically from a probability space made up of "states" which come from the description of how polyhedra are glued together to form the object. The PI's work is about relating these invariants to metric quantities derived from the geometry of theunderlying object. To this end, the PI shows that the space of states can be replaced by spaces of geometric measurements made on the object. The goal of the project is to see how the geometry and topology of the object are determined by its combinatorial description in terms of polyhedra.