Course description for MATH 517A, Group Theory, Fall 2004.
Numerous problems of the natural and technical sciences possess
certain symmetry properties. If we manage to understand them, then a
mathematical treatment adjusted to the symmetry properties lead to
a considerable simplification.
In this course on group theory, the main emphasis will be on this
connection. We will develop the basics of group theory and then apply
them to study a variety of applications. The most important tool that
allows us to study symmetries via group theory is covered by the
theory of realizations of groups as matrices, also called
representation theory of groups.
We will therefore start with studying representation theory using the
book ``Groups and characters`` by L. Grove. Moreover, we will look at
applications from A. Terras book on Fourier analysis on finite
groups. The examples discussed there cover applications of groups to
graph theory, coding theory, statistics and probability theory. If
time permits we are also going to look at problems from combinatorics,
algebraic topology, number theory and other areas. Since examples are
an important part of any mathematical theory, I will offer
introductions on how to use various computer algebra systems such as
GAP, Maple, and others, to study symmetries in concrete situations.
The main prerequisite for this course is the core algebra course
511A/B.