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.