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Abstract: In the last 15 to 20 years much has been learned about the interplay between symplectic geometry which is generally thought to be "flabby", and Kaehler geometry, which is much more rigid. We will discuss two specific constructions showing the interplay between the two. The first is the Grauert tube construction, which associates a complex structure to (part of) the cotangent bundle of a Riemannian manifold, the phase space of classical mechanics. The second is the symplectic reduction construction, also from classical mechanics, but here in the context of Kaehler geometry. Results discussed will include holomorphic and symplectic rigidity of tubes, metric properties of Kaehler reductions, metric versions of blowing-up and moment convexity results. The novel aspects described here represent joint work with R. Aguilar, V. Guillemin, S. Halverscheid, R. Hind, and E. Lerman.
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