Symplectic topology

Symplectic topology is the study of manifolds having a distinguished non-degenerate bi-linear differential 2-form. Non-degenerate means that for every vector in every tangent space there is a vector such that the result is non-zero. One of the trivial result is that a symplectic manifold must be of an even dimension. In a symplectic manifold, every differentiable function, H, defines a unique vector field such that the differential of the function, which is a 1-form, is the same as putting the vector field in the first slot of the symplectic form. This vector field is orthogonal to dH and the symplectic form, so every solution will preserve H and the symplectic form. This is a way of stating the law of conservation of energy in Hamiltonian systems, and since the n'th power of the symplectic form (where the dimension of the manifold is 2n) is a space form, it is also the law of conservation of space in [Hamiltonian system]?s.

Symplectic manifolds, unlike Riemanian? manifolds, are extremely non-rigid: they have many symplectomorphisms. Finite-dimensional subgroups of the group of symplectomorphisms are Lie groups. Representations of these Lie groups on Hilbert spaces are called "quantizations". When the Lie group is the one defined by a Hamiltonian, it is called a quantization by energy. The corresponding Lie operator from the Lie algebra to the Lie algebra of continuous linear operators is also sometimes called the quantization, and is a more common way of looking at it among physicists.