Mathematics Course
Overview
The goal is to introduce students to contemporary mainstream 20th
and 21st century mathematics.
This is not an easy task. Mathematics is a giant scaffolding which normally
needs to be constructed in one's mind before one can ascend to appreciate
the view. Calculus
and algebra learnt in college
are essential parts of this scaffolding
and are fundamental for further mathematics, but most of this material was
discovered in the 18th century.
In the course we will take a few short cuts and only use calculus at a later
stage. We will investigate some very exciting and useful modern
mathematics
and
get a feeling for "what mathematics is all about". The mathematics
which students will see in this course is usually not covered until higher
level courses in second or third year at University. Naturally, it is then
studied in much greater depth. The way we will proceed is by studying carefully
chosen parts and representative examples from various
areas of
mathematics
which illustrate important and general key concepts. In the process students
will have the opportunity to gain a real understanding and feeling for the
beauty, utility and breadth of mathematics.
Texts
The background text
is The
Heart of Mathematics [HM]
by Burger and Starbird. It is one of a small number of texts intended
to convey a feeling for the theory and applications of contemporary
mathematics
at an early stage in one's studies. However, it is directed
at a different group of students --- undergraduate students in the
United States with little mathematics background (e.g. no calculus)
who might
take no other mathematics courses in their studies.
Despite its apparently informal style, [HM] introduces a significant amount
of interesting contemporary mathematics.
The arguments are usually complete (and if not, this is indicated), correct
and well motivated. They are often done by means of studying particular
but important examples which bring out the main ideas of the general setting.
The self contained Notes An
Introduction to Contemporary Mathematics [ICM]
(preliminary version) (large file, about 8Mb) parallel and extend the
development in [HM] and provide a higher level and more extensive treatement
of the
material
there. They are the text for the course and
are also intended as a resource for teachers. The best approach would be
for students to read both [HM] and [ICM], as suggested by the teachers
involved.
