Boundaries and duality for 3d gauge theories

3d N=4 supersymmetric gauge theory has a pair of topological twists, the A-model and B-model, the latter of which is also known as Rozansky-Witten theory. Conjecturally, boundary conditions for these TFTs ought to admit descriptions in terms of (microlocal) perverse or coherent sheaves of categories, respectively. Unfortunately, neither of these admits a general mathematical definition; nevertheless, in some cases these are well-defined 2-categories. We will survey these situations and the duality, known as 3d mirror symmetry, which relates the A- and B-models of different theories, together with its relation to the relative Langlands duality of Ben-Zvi–Sakellaridis-Venkatesh.