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. ***
