I will discuss a famous 40-year-old conjecture from string theory known as the S-duality modularity conjecture. It predicts that a certain partition function encoding the count of stable solutions to the partial differential equations describing D-brane interactions, supported on complex surfaces deforming inside a Calabi-Yau threefolds is given by a modular form. Depending on how these surfaces deform in the ambient Calabi-Yau threefold, one obtains different counting problems and correspondingly different versions of the S-duality conjecture. I will explain an algebro-geometric reformulation of this problem and survey a series of results obtained with collaborators over the past 15 years toward proving the conjecture in various geometric settings. Finally, I will describe ongoing work on the most difficult version of the conjecture, which involves tools such as Tyurin degeneration, derived intersection theory, and the categorification of Donaldson-Thomas invariants.