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BRAIDS(14 May - 13 Jul 2007)
Organizing Committee
· Confirmed Visitors
· Background
· Main Theme Co-chairs
Members
(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.
References:
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
berrick@math.nus.edu.sg.
Organizing Committee
· Confirmed Visitors
· Background
· Main Theme |
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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.