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
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