Research Fellows Directory
Dr David Craven
University of Birmingham
My research area is representation theory of finite groups, an active field of algebra. Group theory is the study of symmetries of objects: shapes like circles and squares, equations, and even puzzles like Rubik's cube. The symmetry of an object is a powerful tool for analysing it: the easiest way to solve Rubik's cube is to understand its group of symmetries. Because of this applicability, the field of group theory has grown over the last 150 years to become one of the most widely applied areas of mathematics.
Groups can be defined in a more concrete way as symmetries of objects, or as transformations (such as rotations and reflections) of space. Constructing a group as symmetries of space is called a representation of the group. The possible representations of a group give significant information about the group itself. The theory of representations has grown into a field of mathematics on its own, with its own distinct flavour and its own difficult conjectures.
Most of these conjectures revolve around the local-global principle. This is a general idea in mathematics that understanding small pieces of an object gives you information about the whole. For example, you can calculate the circumference of the Earth (global) by measuring the angle of the Sun at two cities a few hundred miles apart (local). For groups, the local information is understanding the representation theory of a certain small subset of the symmetries in the group, and the global information is the representation theory of the whole group. The idea is, because the subset is much smaller it is easier to understand; then we apply local-global results to extract information about the global situation, the group itself. This principle, through my and others' research, is finally starting to be understood.
The applications of representation theory are substantial: molecular chemistry, wireless communications, spectroscopy, quantum physics, cryptography, even the design of voting procedures.