Handbook

Mathematics went through quite a revolution around the turn of the 20th century. In particular, an axiomatic approach infiltrated the mathematical paradigm, both as a tool to ensure mathematical rigour and to abstract common principles working in a variety of different settings.

First year mathematics emphasizes computation over abstraction and rigour. Later year courses (and Pure Mathematics in general) reverse this, so students need to learn some new skills and some new ways of thinking about mathematical objects.

The course consists of two halves, algebra and analysis, using lectures and tutorials for content delivery and investigation. The analysis part of the course begins with inequalities and how to bound quantities that cannot be calculated precisely. Many nice examples from geometry will be used to motivate discussions. Then the focus switches to aspects of the real numbers, such as how well one can approximate π by a rational p/q (in terms of how large q is).

The algebra half of the course looks at transformations on the plane and projective plane, studying maps such as translations, reflections, rotations in terms of groups. Topics include transformations of geometric figures that preserve some property (such as distance or angles between lines), and projective geometry. Projective transformations can change a conic section of one type to another, e.g. an ellipse to a hyperbola.