This directory contains the outlines of an ongoing research project wherein we apply our results on Kantorovich's contravariant singularity functor to Alexandrov spaces. Our main goal is to obtain proofs and constructions of souls in singular Alexandrov spaces.
Alexandrov geometry is very difficult subject, of which I am only a novice. But all the same, the Alexandrov spaces is a side project of mine where I think our Reduction-to-Singularity methods from our thesis could find ready application, especially in singular Alexandrov spaces. Indeed Alexandrov spaces with curvature bounded from below are really the best behaved nonsmooth spaces available. What we lack is sufficient experience with more involved Alexandrov spaces. But all our questions begin with the space of ends on an Alexandrov space, and from there constructing appropriate optimal semicoupling programs and their singularities.