Monotone Lagrangian submanifolds are important objects in symplectic geometry but unfortunately we lack examples.
In complex projective spaces and products of spheres, there exists two constructions of families of monotone Lagrangian tori, the one by Chekanov and Schlenk and the one via the Lagrangian circle bundle construction of Biran. It was conjectured that these two constructions give Hamiltonian isotopic tori. I will explain why this conjecture is true in the complex projective plane and the product of two two-dimensional spheres.