Quantum Invariants and Representations of 3-Manifold Groups

PI将从事低维拓扑及其应用的一些项目。这些项目包括研究turaev - viroinvariant的各种性质，包括与这些不变量相关的流形基本群的表示理论的通用多项式的构造以及这些不变量从单位圆的扩展。此外，PI将考虑给定一个结的四面体分解，如何在四面体的交叉比空间上定义一个严格的路径积分来计算结的积分手术的Turaev-Viro不变量。PI还将根据特征变化的几何来研究Kauffman支架绞结模块的结构，并且更一般地说，在研究结的Dehn手术中使用量子不变量。最后，该奖项为PI的研究生提供支持，协助他进行这项研究。拓扑学是一种几何，在这种几何中，同余变换不保留距离和角度等度量属性。PI研究的对象是将多面体沿面粘合在一起的结果。要知道两个这样的对象在拓扑上什么时候是不同的，就需要进行不受拓扑同余变换影响的测量。这种测量的一个例子是曲面的欧拉特性，它是顶点的数量减去边的数量加上面的数量。如果两个曲面拓扑等价，它们具有相同的欧拉特性。最著名的几何定理之一是高斯-邦纳定理，它将任何曲面的欧拉特征与度量计算的量联系起来。三流形的量子不变量类似于欧拉特性，但更精细。它们是由“状态”组成的概率空间统计构建而成的，这些“状态”来自于对多面体如何粘合在一起形成物体的描述。PI的工作是将这些不变量与来自底层物体几何的度量量联系起来。为此，PI表明状态空间可以被在物体上进行的几何测量空间所取代。该项目的目标是了解物体的几何形状和拓扑结构是如何通过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.