Curves and Surfaces
Module code: MA3152
Geometry is one of the oldest sciences, originally, the practical knowledge on area and shape was used in astronomy and in land surveying, but as soon as geometric figures, such as plane curves, could be represented analytically, the theory developed quickly. This module will focus on differential geometry, the part of geometry which is concerned with objects which are smooth, and in particular to the theory of curves and surfaces in space. The visual nature of this low–dimensional geometry makes the theory very accessible, and we will discuss and use various tools of visualisation to understand the shape and the curvature of a geometric object. Most results of the module have immediate extensions to higher–dimensional objects (“manifolds”) but our proofs will only use elementary methods. In particular, there will be a group project in which we will produce exhibits for a maths exhibition.
The applications of differential geometry arise in various fields, including mathematics (e.g. Perelman’s proof of the Poincare conjecture uses techniques of differential geometry), physics (after all, Einstein’s general theory of relativity is expressed in the language of differential geometry!), economics, engineering, and computer graphics.