The top power of a symplectic form is a volume form. Thus volume is a symplectic invariant that measures the size of a symplectic manifold. Due to the famous non-squeezing theorem by Gromov we know that there are other such symplectic invariants, now called symplectic capacities. Interestingly symplectic capacities generally behave very different than volume. In this talk I will present how to compute a certain symplectic capacity for magnetically twisted tangent bundles over closed surfaces. The methods we use combine Hamiltonian dynamics and the theory of pseudo-holomorphic in a surprising way.
