Analysis & Applied Math

A theorem of Ballantine (1968) implies that every orientation-preserving linear automorphism of $\mathbb{R}^n$ is a composition of five Monge maps for the Euclidean quadratic cost between nondegenerate centered Gaussian distributions.

In this talk, I will present a nonlinear analogue: every isotopic-to-the-identity diffeomorphism of a compact connected Riemannian manifold without boundary admits a Monge factorization, i.e., can be written as a finite composition of diffeomorphisms that are Monge maps for the squared Riemannian distance cost. The proof provides an explicit upper bound on the number of factors required to generate an open neighborhood of the identity. I will conclude with a discussion of related open problems.