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 also 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.
Update (February 2019):
Well it took ten years to prove it and now, ten years to prove we proved it! The first two parts of the Caratheodory Conjecture proof have been accepted to appear in the Transactions of the American Mathematical Society and the Annales de la Faculté des Sciences de Toulouse.
The preprints are here:
A new insight into the Conjecture (and why it is true) has also recently been provided by the construction of counter-examples in Riemannian spaces arbitrarily close to Eucliden 3-space. The details of this can be found in the paper On Isolated Umbilic Points.
The paper shows that an arbitrarily small perturbation of the Euclidean metric does not have to satisfy the Caratheodory Conjecture (or Hamburger's umbilic index bound). Here's a short video explainer: