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Abstract: After explaining the necessary background, I will discuss the significance of counting special Lagrangians in Calabi-Yau geometry and how this leads to a 3-dimensional gauge theory problem involving Fueter sections on 3-manifolds. Somewhat analogous to Heegaard Floer theory, this is expected to produce 3-manifold invariants. I then describe ongoing work with Daniel Santiago-Alvarez, where an algebro-geometric approach resolves the problem in some cases. I will highlight non-compactness and connection to Z2-harmonic 1-forms as the main analytical difficulty, based on joint work with Yang Li, and, time permitting, mention work in progress with Cliff Taubes on gluing and transversality, motivated by an Atiyah-Floer conjecture picture.