What Is Symplectic Geometry? – Working Draft

Symplectic geometry is an even dimensional geometry. It lives on even dimensional spaces, and measures the sizes of 2-dimensional objects rather than the 1-dimensional lengths and angles that are familiar from Euclidean and Riemannian geometry. It is naturally associated with the field of complex rather than real numbers. However, it is not as rigid as complex geometry: one of its most intriguing aspects is its curious mixture of rigidity (structure) and flabbiness (lack of structure). In this talk I will try to describe some of the new kinds of structure that emerge. First of all, what is a symplectic structure? The concept arose in the study of classical mechanical systems, such as a planet orbiting the sun, an oscillating pendulum or a falling apple. The trajectory of such a system is determined if one knows its position and velocity (speed and direction of motion) at any one time. Thus for an object of unit mass moving in a given straight line one needs two pieces of information, the position q and velocity (or more correctly momentum) p := q̇. This pair of real numbers (x1, x2) := (p, q) gives a point in the plane R2. In this case the symplectic structure ω is an area form (written dp ∧ dq) in the plane. Thus it measures the area of each open region S in the plane, where we think of this region as oriented, i.e. we choose a direction in which to traverse its boundary ∂S. This means that the area is signed, i.e. as in Figure 1.1 it can be positive or negative depending on the orientation. By Stokes’ theorem, this is equivalent to measuring the integral of the action p dq round the boundary ∂S.