 Symplectic topology

Symplectic topology studies global properties of symplectic manifolds and maps preserving their symplectic forms. Local coordinates put every symplectic form in the same standard form, by the Symplectic Darboux theorem; global embedding and deformation problems can nevertheless have rigid obstructions.

A striking example is the Gromov non-squeezing theorem. In ${\mathbb{R}}^{2n}$ with its standard symplectic form, the ball $B^{2n}(R)$ embeds symplectically into the cylinder $B^2(r)\times {\mathbb{R}}^{2n-2}$ only if $R\le r$. For $n\ge 2$ the cylinder has infinite volume, so ordinary volume comparison cannot explain the obstruction. The selected two-dimensional plane is a conjugate coordinate plane. Tools such as J-holomorphic curves connect this global rigidity to topology and Hamiltonian mechanics.^[ref]

Symplectic topology is a branch of differential topology and geometry that studies symplectic manifolds and their properties. A symplectic manifold is a smooth, even-dimensional manifold equipped with a closed, non-degenerate 2-form, called the symplectic form. This symplectic form captures essential geometric and topological information about the manifold.