Almost 80 years ago Constantin Caratheodory conjectured that the number of umbilic points on a closed convex surface in Euclidean $3$-space must be at least two. In this talk we outline the proof of this conjecture by the speaker and Wilhelm Klingenberg, which uses a codimension $2$ parabolic flow with boundary conditions analogous to the classical capillary problem. While the Conjecture lies firmly in classical differential geometry, the techniques required for its proof span a number of modern PDE developments, including holomorphic curves and mean curvature flow. We will also indicate how the ideas developed in the proof are opening a new set of geometric relationships between $3$- and $4$-manifolds.
