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Monday, April 20, 2015

**Abstract:** Symplectic (and anti-symplectic) embeddings can be characterized as those embeddings that preserve the symplectic capacity (of ellipsoids). This gives rise to a proof of C^0-rigidity of symplectic embeddings, and in particular, diffeomorphisms. (There are many proofs of rigidity of symplectic diffeomorphisms, but all known proofs of rigidity of symplectic embeddings seem to use capacities.) This talk explains a C^0-characterization of symplectic embeddings via Lagrangian embeddings (of tori); the corresponding formalism is called the shape invariant (discovered by J.-C. Sikorav and Y. Eliashberg). The aforementioned rigidity is again an easy consequence. The shape invariant has two immediate advantages: it avoids the cumbersome distinction between symplectic and anti-symplectic, and the results can be adapted to contact embeddings via coisotropic embeddings (of tori). The adaptation to the contact case is (partly) work in progress. In the talk, I will give proofs of the main results and explain what makes them work (J-holomorphic disc techniques for all deep results).