Analysis on manifolds and general relativity

Timeline of the universe, courtesy of the NASA/WMAP Science Team

Manifolds appear in many contexts in mathematics and physics. In general relativity, for example, spacetime is a manifold. Intuitively, manifolds should be thought of as spaces which are locally like standard Euclidean space. They are often endowed with additional structure, such as a Riemannian metric (a generalization of the ordinary Euclidean inner product) or a Lorentz metric (a generalization of the Minkowski inner product of special relativity). In physics, equations phrased in terms of the metric appear naturally; the Dirac equation and Einstein's equations of general relativity are two examples. These equations are partial differential equations on manifolds. Analyzing the behaviour of their solutions is of interest both in mathematics and physics. In terms of physics, the study of Einstein's equations, for example, corresponds to the study of black holes, stability properties of the universe, etc.