Project Research 2001

# Low-Dimensional Topology in the Twenty-First Century

Workshop on "Low Dimensional Topology of Tomorrow" | |

14:00, January 7 - 12:00, January 10 | |

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Low dimensional topology Seminar | |

March 14 Alexander Fel'shtyn
March 8 Ruifeng Qiu, Masakazu Teragaito
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On Heegaard Splittings and Dehn surgeries of 3-manifolds, and topics related to them | |

June 11 - June 15 | |

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Seminars on geometric methods in low dimensional topology | |

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Workshop and Seminars on "Invariants of knots and 3-manifolds" | |

Workshop: September 17 - September 21 Seminars: In the afternoons on September 4, 6, 11, 13, 25, and 27. |

Regarding ``Analysis situs'' by H. Poincaré in 1895 as the origin of the field `Topology', interest in its `low dimensions' had already begun by the turn of the century, as we can find it mentioned in his famous conjecture of 1904. Although the Poincaré Conjecture has already been proved for higher dimensions, the original low-dimensional version still remains an open problem, and fascinates many mathematicians. Apart from the inherent attractiveness of low-dimensional topology as a field of particular mathematical complexity, it is also an enticing study because of its `visibility', that is, one can actually view and feel low-dimensional objects such as 3-dimensional manifolds and knotted circles.

For most of the twentieth century, we have studied low dimensional topology using both the direct, geometrical `cut-and-paste' methods, and the more abstract techniques of Algebraic Topology. The latter reminds us that low dimensional topology has borrowed ideas from other fields.

The situation changed dramatically with the appearance of work by V. Jones and E. Witten in the 1980s. Jones applied the theory of operator algebras to introduce a completely new and understandable link invariant (formalism later improved by L. Kauffman), and encouraged many mathematicians to generalize his invariant to an array of new invariants. Witten generalized these invariants to 3-manifold invariants and gave them novel interpretations from a physical viewpoint. (A mathematically rigorous definition was first given by N. Reshetikhin and V. Turaev.) These invariants are now called `quantum invariants'. Work by V. Vassiliev, M. Kontsevich and T. Ohtsuki has given them yet other interpretations, and new attention has been paid to the study of the `space of all invariants'. As a result, techniques from low-dimensional topology are now used in other fields including not only number theory, representation theory and integrable systems, but also theoretical physics. Thus, low-dimensional topology has developed to become a field that can mutually interact with other fields in the exchange of techniques.

On the other hand, the classical geometric cut-and-paste techniques are still a major tool used to describe various structures on manifolds. Specifically, these methods have recently appeared in the proofs of both W. Thurston's uniformization theorem and D. Gabai's theorem showing the existence of taut foliations in 3-manifolds. Other examples include the concept of thin position, introduced by Gabai in proving the `Property R' conjecture of knots (its origin can be found in H. Schubert's work classifying 2-bridge knots), which played an essential role in C. Gordon and J. Luecke's proof of the knot complement conjecture. M. Scharlemann and A. Thompson have since developed the concept further, and other applications to tunnels for knots have been obtained by Scharlemann, J. Schultens and K. Morimoto. Thus, there are many recent developments of the study using these classical techniques and others (e.g. the analysis of manifolds using Morse functions).

In this project, we will search for a direction for the centennial study of low-dimensional topology in this new century, respecting the recent developments described above. A side effect of the broadening of the field is that researchers with different interests have become rather isolated from each other. We will use this opportunity to encourage the exchange of researchers and information using low-dimensional topology as a keyword.

We hope that with the beginning of the new century, we may enjoy communication with many researchers, and share a new vision of the twenty-first century here in Kyoto, a city with a long history and tradition.

The participants will include:

H. Akiyoshi, Osaka University

D. Bachman, University of Illinois, Chicago (June 10 - June 17)

D. Bar-Natan, Hebrew University (September 11 - September 29)

R. Benedetti, University of Pisa (September 12 - September 28)

S. J. Bigelow, University of Melbourne (September 3 - September 28)

J. Scott Carter, University of South Alabama (January 4 - January 10)

N. Chbili, Monastir Faculty of Sciences (July 18 - October 13)

D.H. Choi, Korea Advanced Institute of Science and Technology
(June 10 - June 23)

S. Duzhin (Steklov Institute of Mathematics)

T. Ekholm, University of Uppsala (May 10 - June 2)

L. Funar, University of Grenoble (July 1 - September 30)

S. Garoufalidis, Georgia Institute of Technology (September 22 - September 27)

T. Gocho, University of Tokyo

H. Goda, Tokyo University of Agriculture and Technology

K. Habiro, Kyoto University

Y. Han, Liaoning Normal University (September 1 - September 30)

S. K. Hansen, University of Strasbourg (September 1 - October 5)

E. Hatakenaka, Tokyo Institute of Technology

C. Hayashi, Japan Women's University

M. Hirasawa, Gakushuin University

Ko Honda, University of Southern California (January 5 - January 10)

T. Ikeda, Kochi Medical School

J. Imai, Tokyo Metropolitan University

A. Inoue, Tokyo Institute of Technology

M.-J. Jeong, Kyungpook National University (September 16 - September 28)

G.-T. Jin, KAIST (September 17 - September 22)

S. Kamada, Osaka City University

Y. Kawahigashi (University of Tokyo)

T. Kawamura, University of Tokyo

T. Kerler, Ohio State University (September 3 - September 16)

K. Kikuchi, Tokyo Institute of Technology

E. Kin, Nara Women's University

M. Kobayashi, Assumption Junior College

T. Kohno, University of Tokyo

A. Kricker, University of Sydney (September 3 - October 2)

G. Kuperberg, University of California, Davis (September 15 - September 22)

T. Le, State University of New York, Buffalo (July 5 - October 4)

S. Lee, Korea Advanced Institute of Science and Technology
(June 10 - June 23)

F. Lei, Harbin Institute of Technology (August 1 - October 31)

C. Lescop, University of Grenoble (September 10 - September 28)

X.-S. Lin, University of California, Riverside (September 8 - September 22)

G. Masbaum, University of Paris VII (September 15 - October 12)

S. Matsumoto, Master's College (January 6- January 12)

T. Mattman, California State University, Chico (May 22 - August 22)

Y. Mizuma, Tokyo Institute of Technology

K. Morimoto, Takushoku University

H. R. Morton, University of Liverpool (September 14 - October 27)

K. Motegi, Nihon University

K. Murasugi, Toronto University (November 11 - November 18)

S. Nakabo, Kurume National College of Technology

R. Nikkuni, Tohoku University

S. Oh, Chonbok National University (June 10 - June 23)

Y. Ohyama, Nagoya Institute of Technology

N. Okuda, Tokyo Institute of Technology

C.-Y. Park, Kyungpook National University (September 16 - September 22)

M. Polyak, Tel-Aviv University (September 18 - October 12)

J. Przytycki, George Washington University (September 8 - September 21)

R. Qiu, Jilin University (January 5 - April 4)

Y. Rieck, University of Arkansas (January-December)

J. Roberts, University of California, San Diego (August 8 - September 19)

S. Satoh, Kyoto University

J. Sawon, University of Oxford (September 5 - October 3)

M. Scharlemann, University of California, Santa Barbara (April 5 - July 4)

J. Schultens, Emory University (May 1 - July 31)

E. Sedgwick, DePaul University (June 10 - June 24)

A. Shima, Tokai University

A. Sikora, University of Montreal (September 4 - September 21)

T. Stanford, New Mexico State University (September 9 - September 26)

T. Takata, Niigata University

M. Tamura, Osaka Sangyo University

T. Tanaka, Kyushu University

Z. Tao, Hangzhou Institute for Applied Engineering (September 1 - November 20)

Andrius Tamulis, Cardinal Stritch University (January 6 - January 12)

M. Teragaito, Hiroshima University

D. Thurston, Harvard University, Cambridge (September 5 - December 5)

I. Torisu, Akita University

T. Tsukamoto, State University of New York, Baffalo

V. Touraev, University of Strasbourg (September 10 - December 9)

M. Wakui, Osaka University

T. Yagi, Tokyo Institute of Technology

S. Yamada, Kyoto Sangyo University

T. Yamada, Tokyo Institute of Technology

Y. Yamada, University of Electro-Communications

T. Yashiro, University of Auckland (July 22 - July 25)

A. Yasuhara,Tokyo Gakugei University

Y. Yokota, Tokyo Metropolitan University

H. Yoshida, Nara Women's University

K. Yoshikawa, Kwansei Gakuin University

This research project is partially supported by Japan Association for Mathematical Sciences, Japan Society for the Promotion of Science, the Mitsubishi Foundation, the Sumitomo Foundation and the Inoue Foundation for Science.

Organizers: Hitoshi Murakami (Tokyo Institute of Technology)

Tsuyoshi Kobayashi (Nara Women's University)

Jun Murakami (Waseda University)

Tomotada Ohtsuki (Tokyo Institute of Technology)

Kyoji Saito (Kyoto University)

Makoto Sakuma (Osaka University)

Kouki Taniyama (Waseda University)