What's a singularity?

In high school, one studies the one-dimensional line of real numbers, the two-dimensional xy-plane, and three-dimensional xyz-space. Points in these spaces are described by a single real number, a pair of real numbers, and a triple of real numbers, respectively. Mathematically, there is no difficulty in defining higher-dimensional space; for instance, a five-dimensional space is defined by considering ordered collections of five real numbers. Of course, one can no longer picture the geometry of these higher-dimensional spaces and so one describes the geometry in terms of equations. The spaces that one obtains in this manner are called Euclidean spaces.

In high school, one also studies certain subsets of Euclidean spaces - for instance, circles, parabolas, ellipses, and hyperbolas in the xy-plane, and spheres in xyz-space. These subsets of Euclidean space can be considered as spaces in their own right. All the curves just listed in the xy-plane have the property that, near each point on the curve, the curve itself looks like the real line except that it is possibly bent a little bit. Such a space is called a one-dimensional manifold. Similarly, the sphere (the hollow sphere) in xyz-space has the property that, near each point on the sphere, the sphere looks like the xy-plane except that it is bent a little bit. Such a space is called a two-dimensional manifold. Another example of a very different-looking two-dimensional manifold is provided by thinking of an inner-tube (or a hollow doughnut); this mathematical object is called a torus. Again, a torus is a two-dimensional manifold - for, near each point on the torus, the torus looks like the xy-plane except that it is bent a little.

Just as we can describe higher-dimensional Euclidean space, without being able to picture it, so too can we mathematically define higher-dimensional manifolds. The local structure of manifolds is easy to describe: near each point, a manifold "looks like" Euclidean space except that it might be bent somewhat. Manifolds have been extensively studied by mathematicians and physicists for decades.

In singularities research, we study certain subsets of high-dimensional Euclidean spaces which are defined by very nice equations. These subsets turn out not to be manifolds; the places where these subsets do not look like bent Euclidean space are called singularities. For instance, in physics it is believed that four-dimensional space-time may have singularities at black holes.

In my research, I describe mathematically what spaces "look like" near singularities. In low dimensions, one can actually hope to visualize these singularities; you can view some pictures or some movies. However, in higher dimensions (typically, I would not work in less than six dimensions), one can not visualize the spaces under consideration; one uses the low-dimensional examples as a visual guide, but in higher dimensions one must prove that the singularities have certain properties which one thinks of as describing what the space "looks like" at a singular point.