Joint IAS/Princeton/Montreal/Paris/Tel-Aviv Symplectic Geometry Zoominar

It is conjectured that every Reeb flow on a closed three-manifold has either two, or infinitely many, simple periodic orbits. I will survey what is currently known about this conjecture. Then, I will try to explain some of the key ideas behind recent joint work proving it, as long as the associated contact structure has torsion Chern class. Then, I will state some related open questions.