Symplectic methods for sharp systolic inequalities

In this talk I would like to explain how methods from symplectic geometry can be used to obtain sharp systolic inequalities. I will focus on two applications. The first is the proof of a conjecture due to Babenko-Balacheff on the local systolic maximality of the round 2-sphere. The second is the proof of a perturbative version of Viterbo's conjecture on the systolic ratio of convex energy levels. If time permits I will also explain how to show that general systolic inequalities do not exist in contact geometry. Joint work with Abbondandolo, Bramham and Salomao.