Remarks on Symplectic Geometry

The relatively new subject, symplectic geometry, has been studied the past five decades. Symplectic geometry is the mathematical subject studying the geometry of symplectic manifolds. A symplectic manifold is an even-dimensional smooth manifold equipped with a closed non-degenerate two form. This paper surveys the progress on the study of symplectic geometry during the past five decades. It deals with a brief history of symplectic geometry and symplectic topology, Arnold’s conjecture, pseudoholomorphic curves, the convexity properties of a moment map, recent results on Hamiltonian and non-Hamiltonian symplectic group actions, the classification of symplectic group actions, recent results on the symplectic embedding problems, the theory of Gromov-Witten invariants, Lagrangian Floer homology, the Fukaya category, and mirror symmetry.