Development in symplectic Floer theory

Abstract

In the middle of the 1980s, Floer initiated a new theory, which is now called the Floer theory. Since then the theory has been developed in various ways. In this article we report some recent progress in Floer theory in symplectic geometry. For example, we give an outline of a proof of the flux conjecture, which states that the Hamiltonian diffeomorphism group is _C_1-closed in the group of symplectomorphisms for closed symplectic manifolds. We also give a brief survey on the obstruction–deformation theory for Floer theory of Lagrangian submanifolds and explain some of its applications.