210226 Dusan Joksimovic

A folklore theorem of Eliashberg and Gromov, which states that the group of symplectic diffeomorphisms is closed in the group of all diffeomorphisms w.r.t. C^0-topology, is considered the beginning of $C^0$-symplectic geometry which roughly investigates which symplectic phenomena persist under $C^0$-limits. It also allows us to define symplectic homeomorphisms (as $C^0-limits of symplectomorphisms) and $C^0$-symplectic manifolds. So far it is not known whether there are manifolds that admit a $C^0$-symplectic, but not smooth symplectic structures. One of the central question in the field, due to Hofer, is whether the spheres admit such structure in dimension greater than 2.

In this (mostly overview) talk we will state and prove Elisahberg-Gromov's theorem using another result from $C^0$-symplectic geometry, namely, the $C^0$-rigidity of Poisson brackets due to Cardin-Viterbo, Entov-Polterovich, and Buhovski. If time allows, we will discuss Hofer's question as well.