\(C^0\)-characterization of symplectic and contact embeddings

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 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). I plan to also touch upon a theorem of Sikorav that characterizes subsets of the unit cotangent bundle of a torus in terms of the shape invariant (plus possible generalizations). As much as time permits, I will end the talk with a brief survey of some of my related results.