Singular symplectic isotopy problems

The smooth symplectic isotopy problem asks for the classification of smooth symplectic surfaces of the complex projective plane up to isotopy. This problem remains open in degrees greater than 17. Here we will consider a singular version of the problem--classifying symplectic submanifolds of $\mathbb{CP}^2$ with singularities of specified types given by cones on torus knots. We will show that for certain rational cuspidal curves, the classification can be made completely. This is based on joint work with Marco Golla.