TOPOLOGY for undergraduates and scientists
MA 598T Spring 1999
Prof. Daniel Gottlieb
PREREQUISITE: The ability to read carefully, and some Calculus (but only for
some mathematical maturity). In a very real sense,
Elementary Topology should be taught at the Freshman level.
For those who can think carefully, or who can learn to think
carefully, the topological viewpoint should make the
following math courses much easier to learn.
Topology is the study of continuity and connectivity. These two intuitive
concepts were clarified and explored during the last one hundred years.
The resulting subject of Topology is a system of concepts and relationships
which underlies geometry and allows us to discuss with precision things like
knots (as in Knot Theory), or twistings (as in Bundle Theory), or changes in
form (as in singularity theory).
The scientists and engineers have been employing topological language more and
more as a way to describe phenomina in their fields of study. And more and more
of modern mathematics itself follows the topological paradigm. In many
universities Topology is already being taught on the undergraduate level,
resulting in students who learn to reason with words independently of
much algebraic notation and who acquire sophisticated mathematical points of
view which are aids in Analysis and Algebra.
For the economist, market equilibrium is discribed via fixed point theory;
for the biologist, knot theory is used to study DNA; for the physicist,
it is topology which is essential to the inevitabilities of
Black Holes and to discriptions of phase changes and gauge theories; for the
robotics engineer, topology can explain why robot arms must have anomolous
motions; and the computer scientist might need to know topology to help in
geometric modelling (the use of the word topology in describing computer
networks however does not correspond to the subject matter of this course)
The Mathematics Department currently teaches Topology only as a graduate
course for Mathematicians. Several engineering students and professors have
attended these courses, but our obligation to rigorously instruct our
mathematics students quickly drove these students away. It is our hope that
this course can provide a less rigorous exposure to Topology for those
scientists who come across topology in their fields.