Flexibility in C^0 symplectic geometry
Traditionally, objects of study in symplectic geometry are smooth - such as symplectic and Hamiltonian diffeomorphisms, Lagrangian (or more generally, isotropic and co-isotropic) submanifolds etc. However, in the course of development of the field, non-smooth objects or features still appeared - such as the Poincaré-Birkhoff theorem about fixed points of an area-preserving homeomorphism of the 2-dim annulus satisfying the twist condition, or the Eliashberg-Gromov theorem about uniform limits of symplectic diffeomorphisms.
A recent attempt for a more systematic approach led to the appearance of the subfield named C^0 symplectic geometry. C^0 symplectic geometry typically studies continuous counterparts of (smooth) symplectic objects as well as their behaviour under uniform limits. It turned out that symplectic and Hamiltonian homeomorphisms are flexible in surprising ways when we are in higher dimensions.
One of the key ingredients used for showing these flexibility phenomena, is a version of the classical h-principle of Eliashberg-Gromov.
In my talk, I will give an overview of the relevant topics from C^0 symplectic geometry.