Topology related to invariants of knots and 3-manifolds

在补助金的支持下，我于2001年9月在京都大学数学科学研究所(RIMS)组织了题为“纽结和3-流形的不变量”的讲习班和研讨会，作为2001年RIMS研究项目“二十一世纪的低维拓扑学”的活动。我们有70名国内参与者和26名外国参与者。从历史上讲，这一研究领域是一个相对较新的领域，它是基于20世纪80年代从Chern-Simons场论中衍生出来的大量的纽结和3-流形的量子不变量。我认为这些不变量的拓扑重建几乎已经通过几十年来的各种工作完成了。工作坊和研讨会的目的是讨论这一领域的未来方向。为了充分讨论这些问题，我计划了一个月的活动，比往常要长一些。此外，为了鼓励参与者之间的讨论，我们在每次演讲后安排了一个简短的问题会议，并要求演讲者在那里提出他/她的开放问题。在这些问题会议上提出了许多有趣的问题，在这些问题的基础上，我们在研讨会和工作坊之间进行了有价值的讨论。我将那里讨论的开放问题编辑成一个问题列表，我希望，将澄清这一领域目前的前沿，并帮助读者考虑未来的方向。研讨会的会议记录发表在在线期刊“几何和拓扑学专著”上。问题清单也将作为会议记录的一部分发布。

By the support of the grant-in-aid I organized the workshop and seminars "Invariants of Knots and 3-Manifolds" at the Research Institute for Mathematical Sciences (RIMS), Kyoto University in September 2001, as activities of the 2001 RIMS research project "Low-Dimensional Topology in the Twenty-First Century". We had 70 domestic participants and 26 foreign participants.Historically speaking, this research area is a relatively new area based on the enormous number of invariants, called quantum invariants, of knots and 3-manifolds, which have been derived from the Chern-Simons field theory in the 1980s. I think the topological reconstruction of these invariants has almost been completed by various works of these decades.An aim of the workshop and seminars was to discuss future directions for this area.To discuss these matters fully, I planned 1 month of activities, relatively longer than usual.Further, to encourage discussions among the participants, we arranged a short problem session after each talk, and requested the speaker to give his/her open problems there.Many interesting problems were presented in these problem sessions and, based on them, we had valuable discussions in and between seminars and the workshop.I edited open problems discussed there into a problem list, which, I hope, will clarify the present frontier of this area and assist readers when considering future directions.The proceedings of the workshop was published in the online journal "Geometry and Topology Monographs".The problem list will also be published as a part of the proceedings.