Obstructions to special Lagrangian desingularizations and the Lagrangian prescribed boundary problem

We exhibit infinitely many, explicit special Lagrangian isolated singularities that admit no asymptotically conical special Lagrangian smoothings. The existence/ nonexistence of such smoothings of special Lagrangian cones is an important compo- nent of the current efforts to understand which singular special Lagrangians arise as limits of smooth special Lagrangians. We also use soft methods from symplectic geometry (the relative version of the h–principle for Lagrangian immersions) and tools from algebraic topology to prove (both positive and negative) results about Lagrangian desingularizations of Lagrangian submanifolds with isolated singularities; we view the (Maslov-zero) Lagrangian desingularization problem as the natural soft analogue of the special Lagrangian smoothing problem.