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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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