Symplectic geometry is a central topic of current research in
mathematics. Indeed, symplectic methods are key ingredients in the study
of dynamical systems, differential equations, algebraic geometry,
topology, mathematical physics and representations of Lie groups. This
book is an introduction to symplectic geometry, assuming only a general
background in analysis and familiarity with linear algebra. It starts
with the basics of the geometry of symplectic vector spaces. Then,
symplectic manifolds are defined and explored. In addition to the
essential classic results, such as Darboux's theorem, more recent
results and ideas are also included here, such as symplectic capacity
and pseudoholomorphic curves. These ideas have revolutionized the
subject. The main examples of symplectic manifolds are given, including
the cotangent bundle, Kahler manifolds, and co-adjoint orbits. Further
principal ideas are carefully examined, such as Hamiltonian vector
fields, the Poisson bracket, and connections with contact manifolds.
Berndt describes some of the close connections between symplectic
geometry and mathematical physics in the last two chapters of the book.
