Preface
Geometry and topology are subjects generally considered to be "pure"
mathematics. Both originated in the effort to describe and quantize shape
and form in order to understand the "real" world. Both enjoy a robust and
sustained internal intellectual life, abstracted from the "reality" of their ori-
gins. Recently, some of the methods and results of geometry and topology
have found new utility in both wet-lab and theoretical science. Conversely,
science is influencing mathematics, from posing questions which call for the
construction of mathematical models to the importation of theoretical meth-
ods of attack on long-standing problems of mathematical interest.
A case in point is the subject of knot theory, which is utilized to a greater
or lesser degree in each of the six papers in this volume. Knot theory traces
its mathematical origins to the work of Gauss on computing inductance of
linked circular wires, and to the work of Kelvin and Tait on the vortex theory
of atoms. Knot theory is the study of entanglement and symmetry of elas-
tic graphs in 3-space. It has proven to be fundamental as a laboratory for
the development of algebraic topology invariants and in the understanding
of the topology of 3-manifolds. During the last decade, laboratory scientists
have become increasingly aware that the analytical techniques of geometry
and topology can be used in the interpretation and design of experiments.
Chemists have long been interested in developing techniques that will allow
them to synthesize molecules with interesting 3-dimensional structure (knots
and links). Polymer scientists study the chemical and physical ramifications
of random topological entanglement in large molecules. Models for molecu-
lar structure must be built and understood; reactions which produce specific
3-dimensional shapes must be designed; chemical proof of structure must be
produced, and these proofs often involve the use of topology to interpret data
such as NMR (Nuclear Magnetic Resonance) spectra. Molecular biologists
know that the spatial conformation of DNA and the proteins which act on
DNA is vital to their biological function; moreover, differential geometry and
knot theory can be used to describe and quantize the 3-dimensional struc-
ture of DNA and protein-DNA complexes. Biologists devise experiments on
circular DNA which elucidate 3-D molecular conformation (helical twist, su-
percoiling, etc.) and the action of various important life-sustaining enzymes
(topoisomerases and recombinases). These experiments are often performed
IX
Previous Page Next Page