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格局发生了戏剧性的变化。通过Jones多项式、量子不变量和Vassiliev不变量的出现，揭示了纽结、3-流形和其他低维拓扑对象可以外在地作为组合对象来研究，拓扑等价是一种对称。研究人员认识到这些拓扑对象可能扮演的更广泛的角色，这一新的观点反映在这个项目中研究的许多主题中：有限域上同调三维球体的基本群的表示变化及其与同调三维球体的Ohtsuki不变量的关系；与量子不变量密切相关的某些弦链接不变量的概率解释；高斯积分的积分和射影几何，勒让德结的分类，关于纽结空间上几何结构的各种能量泛函的研究，以及高斯积分水平集的拓扑；以及使用弦链接作为工具计算(交替的)多个Zeta数之间的关系。结是令人着迷的物体。在系绳子时，在人类历史上很早就认识到打结和“打滑结”(可以通过拉动解开的打结)之间的区别。然而，对结的数学研究在十九世纪末才开始，并在最近几年终于成熟为一种有用的科学工具。目前，纽结理论中的基本概念对于DNA生物学和基础物理学来说是至关重要的，它们涉及到数学中的许多其他领域。另一方面，纽结理论中容易描绘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，即不同种类纽结的多样性和对称性在其分类中所起的作用，可以很好地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. ***