The Carathéodory conjecture, dating from the early 1920's, states that any closed convex surface in 3-dimensional Euclidean space must have
at least 2 umbilic points (points where the surface curves equally in all directions). This has remained a conjecture for 80 years, with fitful, occasionally intense, work on the special case where the surface is real analytic.
Background and further references can be found on Wikipedia.
In fact, with Wilhelm Klingenberg (University of Durham, England), we have proved the Carathéodory conjecture utilizing PDE in a geometric setting that we have spent the last decade exploring. While our proof has been posted on the internet since September 2008, it will take a while for the proof of such a long-standing conjecture to be accepted. Following feedback, a new version of proof of the global conjecture has been posted here.
Direct applications of the conjecture are thin on the ground, although the methods we use may prove to be useful in a wide variety of
mathematical settings. Our main innovations have been:
Recently, we have extended our proof from the global conjecture to a local index bound for umbilics on smooth convex surfaces. To help explain our methods I have put together a couple of expository youtube video clips that goes through the proof. Below is the introduction video:
In an interesting twist, the smooth bound obtained, which we claim is sharp, is weaker than Hamburger's famous result in the real analytic case. Thus, we predict the existence of "exotic" umbilic points of index 3/2, which are contained on smooth but non-real analytic surfaces.
I recently gave the Perspectives in Geometry Lecture Series, at the University of Texas at Austin. The videos of the four lectures can be found by clicking here.