Volume 4 (2002)

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The workshop and seminars on "Invariants of Knots and 3-Manifolds" took place at the Research Institute for Mathematical Sciences (RIMS), Kyoto University, in September 2001. The workshop was held over the period September 17-21. Seminars were held on the Tuesdays, Wednesdays and Thursdays of the other weeks of September, including "Goussarov day" on September 25.

Since the interaction between geometry and mathematical physics in the 1980s, many invariants of knots and 3-manifolds have been discovered and studied: polynomial invariants such as the Jones polynomial, Vassiliev invariants, the Kontsevich invariant of knots, quantum and perturbative invariants, the LMO invariant and finite type invariants of 3-manifolds. The discovery and analysis of the enormous number of these invariants yielded a new area: the study of invariants of knots and 3-manifolds (from another viewpoint, the study of the sets of knots and 3-manifolds). There are also developing topics related to other areas such as hyperbolic geometry via the volume conjecture and the theory of operator algebras via invariants arising from 6j-symbols. On the other hand, recent works have almost completed the topological reconstruction of the invariants derived from the Chern-Simons field theory.

An aim of the workshop and seminars was to discuss future directions for this area. To discuss these matters fully, we 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. Open problems discussed there were edited and formed into a problem list, which, I hope, will clarify the present frontier of this area and assist readers when considering future directions.

The workshop and seminars were organized by T. Ohtsuki, H. Murakami, J. Murakami, and T. Kohno. This proceedings volume was edited by T. Ohtsuki, T. Kohno, T. Le, J. Murakami, J. Roberts, and V. Turaev. The workshop and seminars were partially supported by Grand-in-Aid for Scientific Research, Ministry of Education, Culture, Sports, Science and Technology, Japan: Kiban(C)13640064 (T. Ohtsuki), Kiban(B)12440014 (T. Kohno), Kiban(B)12440015 (S. Kojima), Kiban(B)12440018 (H. Murakami), and partially supported by the Inoue Foundation for Science. The workshop and seminars were held as partial activities of the 2001 RIMS project research "Low-dimensional topology in the twenty-first century", organized by H. Murakami, T. Kobayashi, J. Murakami, T. Ohtsuki, K. Saito, M. Sakuma, and K. Taniyama. The logo was designed by N. Okuda.

T. Ohtsuki, September 2002