(14 May - 13 Jul 2007)
(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  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.
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:
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.
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.
For enquiries on scientific aspects of the program, please
email A.J. Berrick at