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General relativity theory represents gravity as curvature of the geometry of spacetime. Unlike the rigid structure of flat Euclidean space, curved geometries can in principle have a wide range of global topological structures. Little is known, however, about the properties of the solutions to Einstein's equation on manifolds having non-trivial topologies. This talk will discuss recent work on developing flexible practical methods for solving Einstein's equation numerically on manifolds
with arbitrary spatial topologies. Examples will be given to illustrate the use of these methods for solving simple elliptic and hyperbolic differential equations numerically on manifolds with non-trivial spatial topologies. Extending these methods to general relativity also required the development of a new fully covariant symmetric-hyperbolic representation of the Einstein equation. This new representation will be discussed, along with some simple numerical tests of Einstein evolutions on manifolds with non-trivial spatial topologies. | 
