Address:	Professor D.D. Joyce,
	The Mathematical Institute,
	Andrew Wiles Building,
	Radcliffe Observatory Quarter,
	Woodstock Road,
	Oxford,
	OX2 6GG.
E-mail:	joyce@maths.ox.ac.uk

Research Interests

My research is mostly in Differential Geometry, with occasional forays into some more esoteric areas of Theoretical Physics, and more recently diversions into Algebraic Geometry and Symplectic Geometry. It is difficult to explain the ideas involved to someone who is not already mathematically literate to beyond degree level, and it\'s not easy to explain the point of it even to someone who is, but here goes.

Nowadays, geometry is not about triangles and circles and Euclid, who went out with the Ark. Instead, the up-to-date geometer is interested in manifolds. A manifold is a curved space of some dimension. For example, the surface of a sphere, and the torus (the surface of a doughnut), are both 2-dimensional manifolds.

Manifolds exist in any dimension. One branch of geometry, called manifold topology, aims to describe the shape of manifolds, using algebraic invariants. For example, the sphere and the torus are different manifolds because the torus has a \'hole\', but the sphere does not. In higher dimensions manifolds become very complicated, both to describe topologically, and to imagine in a meaningful way.

Another branch of geometry is the study of geometrical structures on manifolds. Here the manifold itself is only the background for some mathematical object defined upon it, as a canvas is the background for an oil painting. This kind of geometry, although very abstract, is closer to the real world than you might think. Einstein\'s theory of General Relativity describes the Universe - the whole of space and time - as a 4-dimensional manifold.

Space itself is not flat, but curved. The curvature of space is responsible for gravity, and at a black hole space and time are so curved they get knotted up. Everything in the universe - light, subatomic particles, pizzas, yourself - is described in terms of a geometrical structure on the space-time 4-manifold. Manifolds are used to understand the large-scale structure of the Universe in cosmology, and the theory of relativity introduced the idea of matter-energy equivalence, which led to nuclear power, and the atomic bomb.