Remarks on Symplectic Geometry

Abstract

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. In this paper, we survey the progress on the study of symplectic geometry the past five decades. We deal with the 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.

54 pages