Surgeries on torus knots, rational balls and cabling

We consider the problem of which Dehn surgeries on a knot bound rational homology balls. After introducing various constructions we state a complete classification for positive integral surgeries on positive torus knots. We briefly explain the strategy of the proof which relies on Heegaard Floer homology as well as combinatorial arguments based on Donaldson's diagonalization theorem. Finally, we discuss the relation between our work and the related problem of classifying rational unicuspidal plane curves.  Most of the results in this talk are part of a joint project with M. Golla, K. Larson, and A. Lecuona.