Research in Knots and 3-Manifolds

9704726林 在过去的十年里，纽结和三维流形的研究领域发生了巨大的变化。 随着Jones多项式、量子不变量和Vassiliev不变量的出现，人们发现，纽结、三维流形和其他低维拓扑对象可以作为组合对象来研究，拓扑等价作为一种对称。 研究者认识到这些拓扑对象可能发挥的更广泛的作用，这一新的观点反映在本项目研究的许多主题中：有限域上同调3-球面基本群的表示变体及其与Ohtsuki同调3-球面不变量的关系;与量子不变量密切相关的某些弦链不变量的概率解释;高斯积分的积分和射影几何，勒让德结的分类，各种能量泛函相对于结空间的几何结构的研究，高斯积分的水平集的拓扑结构;和计算（交替）多个zeta数之间的关系，使用字符串链接作为设备。 绳结是很有意思的东西。 当系紧绳子时，结和“滑结”（一种可以通过拉来解开的结）之间的区别在人类历史上一定很早就被认识到了。 然而，对绳结的数学研究大约在世纪末才开始，近年来终于成熟为一种有用的科学工具。 纽结理论中的基本概念对DNA生物学和基础物理学至关重要，并且与数学中的许多其他领域相关。 另一方面，纽结理论中容易描述但困难的问题仍然超出了数学技术的范围。 纽结理论的两个基本特征，即不同类型纽结的多样性和对称性在纽结分类中所起的作用，很可能为许多不同类型的科学研究提供线索和工具。 ***

9704726 Lin The landscape of the study of knots and 3-manifolds has undergone a dramatic change in the past decade. Through the advent of the Jones polynomial, quantum invariants and Vassiliev invariants, it transpired that knots, 3-manifolds and other lower dimensional topological objects could be studied extrinsically as combinatorial objects, with topological equivalence as a kind of symmetry. The investigator recognized the broader roles these topological objects may play, and this fresh point of view is reflected in the many topics studied in this project: the representation varieties of the fundamental groups of homology 3-spheres over finite fields and their relations with Ohtsuki's invariants of homology 3-spheres; the probabilistic interpretation of certain string link invariants closely related to quantum invariants; integral and projective geometry of Gauss integrals, the classification of Legendrian knots, the study of various energy functionals with respect to the geometric structures on knot spaces, and the topology of level sets of Gauss integrals; and calculation of relations among (alternating) multiple zeta numbers, using string links as devices. Knots are fascinating objects. When fastening a rope, the distinction between a knot and a "slip-knot" (one that can be undone by pulling) must have been recognized very early in human history. However, a mathematical study of knots was started only around the end of the nineteenth century and has finally in recent years matured into a useful scientific tool. Nowadays, basic concepts in knot theory are crucial for DNA biology and foundational physics, and they are related to many other fields in mathematics. On the other hand, easily pictured but difficult questions in knot theory still remain beyond the reach of mathematical techniques. Two basic features of knot theory, the diversity of different kinds of knots and the role played by symmetry in their classification, may well provide hin ts and tools for many different kinds of scientific research. ***