Heating up three dimensional spaces
Dr Ben Andrews is drawing on ideas from physics to work out ways to characterise and classify different types of curved space.
“The simplest two dimensional surface is a sphere, and it’s easy to visualise it,” he says. “There is also the torus, the surface of a doughnut, which has one hole. There are surfaces with more ‘holes’, and in two dimensions, surfaces are essentially classified by the number of holes they have in them.”
However, three-dimensional spaces are more difficult to visualise. They have been studied intensively in the last century as part of the branch of mathematics called topology. Famous problems, such as the 'Poincare conjecture', have aroused interest in the field.
Andrews is working on methods to break complicated spaces down into simpler pieces amenable to mathematical treatment. He is applying analogues of the classical heat equation, a partial differential equation describing the transfer of heat through a material.
“The heat equation describes how temperature distributions smooth out as the heat diffuses,” Andrews says. “I use similar equations to smooth out the geometry of complicated spaces, to make them simpler and more easily understood. This is an idea which has developed over the last thirty years, culminating in the spectacular work of Grigory Perelman, who used equations like this to prove the Poincare conjecture and the more general geometrization conjecture, which amounts to a full classification of three-dimensional curved spaces.”
Ben is using similar ideas to attack other problems in topology and differential geometry.
“The tools I use come from very old classical ideas in physics, but applying them in the new directions of geometry and topology has been fruitful.”