Curves and Surfaces

Module code: MA4152

Geometry is one of the oldest scientific disciplines. An understanding of the shape and size of objects has been, and continues to be, fundamental for advances in technologies and civilisation. This module is an introduction to the vibrant field of Differential Geometry, focusing on the geometry of curves and surfaces in 2- and 3-dimensional space. A major theme of this module is how to both represent and study properties of geometric objects analytically, a discipline which, when first developed, acted as a catalyst for the advancement of much of modern pure and applied mathematics. The subject presents a rich resource of beautiful (and deep) mathematical results, which, due to its visual nature, are some of the most accessible results in all pure mathematics. In fact, being able to make the material accessible (through visualisation, and otherwise) will form part of the module's assessment in the group project assignment, where the task is to produce two exhibits for a future mathematics exhibition. Most of the results generalise to higher dimensional abstractions (“manifolds”), however our approaches will aim to be more elementary.

The ideas presented in this course form a backbone for applications in various fields:

• In mathematics (for example, the famous proof of the Poincaré conjecture uses differential geometry);
• In physics (Einstein's theory of relativity is founded on differential geometry, and most of modern particle theory would not exist without ideas from differential geometry);
• It plays a role in wider disciplines, such as engineering, biology, and computer graphics. 

Modern differential geometry is a flourishing area of mathematics with many different branches: for example, Symplectic Geometry, Geometric Analysis, Surface Theory, and aspects of Mathematical Physics. This module will prepare you for all research active areas, but you will also discover applications in real life and links to other areas in mathematics.