Symplectic manifolds carry a natural volume that is preserved by symplectic maps between them. Gromov proved in 1985 that one cannot embed a ball into a cylinder symplectically unless the radius of the ball is less than or equal to the radius of the cylinder. It is called the non-squeezing theorem for obvious reasons: if you squeeze a ball with a large radius into a cylinder with a small radius, you satisfy the volume constraint but the embedding cannot be symplectic.

Symplectic capacities are notions of volume more suited to the symplectic world. One example is Gromov width: the radius of the largest ball (Darboux chart) you can symplectically embed into the manifold. I will talk about some interesting features of symplectic capacities, after a brief introduction to symplectic manifolds.