"Symplectic geometry is the geometry of a closed skew-symmetric form. It turns out to be very different from the with which we are familiar. One important difference is that, although all its concepts are initially expressed in the smooth category (for example, in terms of differential forms), in some intrinsic way they do not involve derivatives. Thus symplectic geometry is essentially topological in nature. Indeed, one often talks about symplectic topology. Another important feature is that it is a 2-dimensional geometry that measures the area of complex curves instead of the length of real curves."