Carathéodory's conjecture on the umbilics of a convex surface

In 1923 Constantin Carathéodory conjectured that the number of umbilic points on a closed convex surface in $R^3$ must be at least two, one more than the topologically necessary minimum. In this talk we discuss the conjecture and its geometric reformulation in terms of complex points on Lagrangian surfaces in a complex surface. We then explain how parabolic P.D.E. methods can be used to prove the conjecture for smooth convex surfaces and how this leads to new relationships between three and four-manifolds.