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BRAIDS

(14 May - 13 Jul 2007)

Organizing Committee Confirmed Visitors · Background · Main Theme
Activities Membership Application

 Organizing Committee

Co-chairs

Members

 Confirmed Visitors

 Background

(The pictures below are taken from Mitch Berger's website, where they are discussed in more detail.)

The notion of a braid as "anything plaited, interwoven, or entwined" goes back many centuries, and braids have been used universally for decoration, art and fastening purposes. Only recently have mathematicians tried to describe braids by means of abstract theory. Fortuitously, as the theory has developed, it has enabled applications to outstanding problems in physics, chemistry and biology.

In topology, braid theory is an abstract geometric theory studying the everyday braid concept, and some generalizations. The idea is that braids can be organized into groups, in which the group operation is "do the first braid on a set of strings, and then follow it with a second on the twisted strings". Such groups may be described by explicit presentations, as was shown by E. Artin in 1925. A braid with n strands can also be thought of as paths of n distinct particles moving through time, and which do not collide (with variations involving particles which can collide). Braids may also be given a deeper mathematical interpretation: as the fundamental group of certain configuration spaces, comprising configurations of distinct points in a manifold X.

When X is the plane, the braid can be closed, that is, corresponding ends can be connected in pairs, to form a link, a possibly intertwined union of possibly knotted loops in three dimensions. The number of components of the link depends on the permutation of strands determined by the link. J.W. Alexander (1928) observed that every link can be obtained in this way from a braid (see also work by Markov). Different braids can give rise to the same link, just as different crossing diagrams can give rise to the same knot. The Jones polynomial of a link (1987) is defined, a priori, as a braid invariant and then shown to depend only on the class of the closed braid.

Until recently, the study of braids has been regarded as a topic within knot theory, a major branch of low-dimensional topology. However, work largely done at NUS [2] has shown that the study of Brunnian braids has application to longstanding problems in homotopy theory, and in particular the fundamental question of the homotopy groups of spheres. (Brunnian braids [Brunn, 1892] are those which reduce to the trivial, untwisted braid when any strand is removed. The familiar Borromean rings are the link obtained by closing up one such Brunnian braid.)

To date, most mathematical interest in braids has come from algebraists, topologists and mathematical physicists. As well, braids are also engaging the attention of computer scientists, as a basis for public-key cryptosystems. Probabilistic algorithms are being employed to search for solutions to word problems in the braid group. Relevance to robotics, cryptography and to magnetohydrodynamics is also to be explored during the program.

References:

  1. D. Bensimon, G. Charvin & V. Croquette: DNA unbraiding by a single type II topoisomerase, PHYS 196 [636703], http://www.hackberry.chem.trinity.edu/PHYS/fall2003.html
  2. A.J. Berrick, F.R. Cohen, Y.-L. Wong & J. Wu: Configurations, braids and homotopy groups, J. Amer. Math. Soc., to appear.
  3. C.O. Dietrich-Buchecker & J.-P. Sauvage: Interlocked and knotted rings in biology and chemistry, in Bioorganic Chemistry Frontiers, Springer (Berlin, 1991) Vol. 2, 195 - 248.
  4.  D. Garber, S. Kaplan, M. Teicher, B. Tsaban and U. Vishne: Probabilistic solutions of equations in the braid group, preprint (2004).
  5. R. Ghrist: Configuration spaces and braid groups on graphs in robotics, in Braids, Links, and Mapping Class Groups: the Proceedings of Joan Birman's 70th Birthday, AMS/IP Studies in Mathematics volume 24 (2001), 29 - 40. 
  6. K. H. Ko, S. J. Lee, J. H. Cheon, J. W. Han, S. J. Kang and C. S. Park, New Public-key cryptosystem using braid groups, CRYPTO 2000, LNCS 1880 (2000), 166 - 183.
  7. P. Pieranski et al.: Physical Review Letters, 19 August 1996.
  8. P.F. Schewe & B. Stein: Braids plaited by magnetic holes, The American Institute of Physics Bulletin of Physics News 285 (Story #2), September 9, 1996.
  9. Michael D. Stone, Zev Bryant, Nancy J. Crisona, Steven B. Smith, Alexander Vologodskii, Carlos Bustamante & Nicholas R. Cozzarelli: Chirality sensing by Escherichia coli topoisomerase IV and the mechanism of type II topoisomerases, Proc. Nat. Acad. Sci. 100 no. 15, (July 22, 2003), 8654 - 59.
  10. A. V. Vologodskii, W. Zhang, V. V. Rybenkov, A. A. Podtelezhnikov, D. Subramanian, J. D. Griffith & N. R. Cozzarelli: Mechanism of toplogy simplification by type II DNA topoisomerases, Proc. Natl. Acad. Sci. USA 98 (2001), 3045 - 3049.
  11. D.M. Walba, T.C. Homan, R.M. Richards & R.C. Haltiwanger, New J. Chem. 17 (1993), 661.
  12. Wikipedia, http://www.sciencedaily.com/encyclopedia/Braid_theory

 Main Theme

The main theme of the program is the mathematical structure of the braid group, together with applications arising from this structure both within mathematics, and outside of mathematics such as (a) magnetohydrodynamics, (b) robotics and (c) cryptography.

It is proposed to invite workers in these different areas with the intention of cross-fertilization.

The interests of the organizers lie mostly in topology. Therefore it is likely that most long-term visitors will be from that area. Reflecting the theme of the program, it is intended to have tutorials that would:

  1. introduce outsiders (e.g. graduate students) to the mathematics of braid theory
  2. facilitate communication between those working in mathematical theory of braids and those who apply braids elsewhere, specifically in magnetohydrodynamics, robotics and cryptography.

 Activities

IMS Membership is not required for participation in above activities. For attendance at these activities, please complete the online registration form.

If you are an IMS member or are applying for IMS membership, you do not need to register for these activities.

 Membership Application

The Institute for Mathematical Sciences invites applications for membership for participation in the above program. Limited funds to cover travel and living expenses are available to young scientists. Applications should be received at least three (3) months before the commencement of membership. Application form is available in (MSWord|PDF|PS) format for download. Membership application deadline is on the 10 March 2007.

More information is available by writing to:
Secretary
Institute for Mathematical Sciences
National University of Singapore
3 Prince George's Park
Singapore 118402
Republic of Singapore
or email to imssec@nus.edu.sg.

For enquiries on scientific aspects of the program, please email A.J. Berrick at berrick@math.nus.edu.sg.

 

Organizing Committee ·  Confirmed Visitors · Background · Main Theme
Activities Membership Application