One interesting fact about symplectic geometry is that for any symplectic manifold the group of diffeomorphisms that preserve the symplectic structure has infinite dimensions. (In contrast, the group of isometries of a Riemannian manifold is always finite dimensional.) Recent advances in symplectic topology have made possible a deeper understanding of this group, though there are still many open questions. This talk will attempt to describe the current state of our knowledge of its structure.

• Lecture notes.
• A preprint is available in DVI or PostScript format.

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