Diff. Geom, Math. Phys., PDE Seminar: Robert McCann

Optimal maps in R^n to disconnected targets necessarily contain discontinuities (i.e.~tears). But how smooth are these tears? When the target components are suitably separated by hyperplanes, non-smooth versions of the implicit function theorem can be developed which show the tears are hypersurfaces given as differences of convex functions --- DC for short. If in addition the targets are convex the tears are actually C^{1,\alpha}. Similarly, under suitable affine independence assumptions, singularities of multiplicity k lie on DC rectifiable submanifolds of dimension n+1-k. These are stable with respect to W_\infty perturbations of the target measure. Moreover, there is at most one singularity of multiplicity n. This represents joint work with Jun Kitagawa.