We give two disparate applications of symmetry methods. The first uses a variation of the standard symplectic reduction technique to construct new simply connected Einstein manifolds in dimension seven with arbitrary second Betti number. The second studies holomorphic maps from the Riemann sphere to certain complex target spaces X with large complex automorphism groups. These large symmetry groups allow us to generalize the sheaf of meromorphic functions and principal parts sheaf on the Riemann sphere which in turn leads to topological stability theorems. Such theorems describe how topologically the holomorphic maps from the Riemann sphere to X build up the entire mapping space, namely the twofold loop space of X.

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1996-1997 Mathematics in High Performance Computing