Explore the S-duality modularity conjecture, a 40-year-old problem from string theory that predicts how partition functions encoding stable solutions to D-brane interaction equations relate to modular forms in this advanced mathematics lecture. Delve into the complex mathematical framework where surfaces deform within Calabi-Yau threefolds, creating different counting problems and corresponding versions of the conjecture. Learn about the algebro-geometric reformulation of this challenging problem and survey breakthrough results achieved over 15 years of collaborative research across various geometric settings. Discover cutting-edge approaches to the most difficult version of the conjecture, including advanced mathematical tools such as Tyurin degeneration, derived intersection theory, and the categorification of Donaldson-Thomas invariants, as presented by Artan Sheshmani from Stony Brook Mathematics.