Isotopies and Squeezing of Monotone Lagrangian Tori

Distinct Hamiltonian isotopy classes of monotone Lagrangian tori in $\mathbb{C} P^2$ can be associated to Markov triples. With two exceptions, each of these tori are symplectomorphic to exactly three Hamiltonian isotopy classes of tori in the ball (the affine part of $\mathbb{C} P^2$). We investigate quantitative invariants, and see these can distinguish the tori corresponding to at least one sequence of Markov triples. A similar analysis for $S^2 \times S^2$ produces symplectomorphic tori which are not Hamiltonian diffeomorphic. This is joint work with Grigory Mikhalkin and Felix Schlenk.