In algebra abstract methods are used to study what are known as discrete structures. Algebra has a long history of important applications throughout mathematics, science and engineering, and has also been studied for its intrinsic beauty. With the advent of computers, discrete structures have taken on a new importance, and algebraic methods have become increasingly useful.
A major aim of our algebra courses is to give students an appreciation of some of these abstract methods which have allowed for spectacular advances over the past 150 years, in a variety of areas. These advances occurred in mathematical problems, such as in number theory and geometry; and also in related areas, such as the theory of computing, encryption, and quantum and relativistic physics. A notable example of the former is Fermat's Last Theorem.
Our algebra courses concentrate on basic ideas and examples, and give samples of applications at a level that does not demand a background in the areas mentioned above. The main idea is to study algebraic laws satisfied by familiar objects such as numbers, polynomials, matrices and permutations in a new and abstract way, thereby simplifying our understanding of the connections between these laws, and allowing us to apply the laws to quite interesting, new entities.
Algebra is an immense area of mathematics. Some of the areas of interest at Waterloo are algebraic geometry, field theory, group theory, Lie theory, linear algebra and ring theory. Famous people who have made major contributions to algebra include Gauss, Descartes, Noether, and Von Neumann.