Families of non-product minimal surfaces with cylindrical tangent cones

The study of singularities of minimal submanifolds has a long history, with isolated singularities being the best understood case. The next simplest case is that of minimal submanifolds with families with singularities locally modeled on the product of an isolated conical singularity and a Euclidean space — such submanifolds are said to have cylindrical tangent cones at these singularities. Despite work in many contexts on minimal submanifolds with such singularities, the only known explicit examples at present are global products. In this talk, I will describe a method for constructing infinite-dimensional families of non-product minimal submanifolds whose singular set is itself an analytic submanifold. The construction uses techniques from the analysis of singular elliptic operators and Nash-Moser theory.