Watch this lecture by Gabriele Benedetti from VU Amsterdam on "Rigidity and flexibility of periodic Hamiltonian flows" as part of the "Systolic and Diastolic Geometry" event held at IMPA in Rio de Janeiro. The 56-minute talk explores geometric problems comparing volume of spaces with lengths/areas of curves/surfaces they contain. The event brought together specialists from Differential Geometry, Analysis, and Symplectic Geometry to discuss state-of-the-art approaches to fundamental problems like optimal inequalities for periodic orbits in convex billiards, geodesics on spheres, and Laplacian eigenvalues on surfaces. This lecture was part of a broader conference featuring presentations from fifteen invited speakers and seven short talks, organized by a scientific committee including Lucas Ambrozio, Vinicius Ramos, and Umberto Hryniewicz.