I am a mathematician and my main area of research is geometry. Over the past decade I have spent most of my efforts investigating the geometry of spaces of geodesics and their applications.
This work has culminated in the proof, in collaboration with Wilhelm Klingenberg, of an 80 year-old conjecture in
classical surface theory called the Caratheodory Conjecture. More details of this can be found here.
Congratulations to my doctoral supervisor Karen Uhlenbeck (pictured on the left at my dissertation defence in 1997) who has recently been awarded the 2019 Abel Prize in Mathematics - the first woman ever to receive the award.
I recently dedicated the paper On isolated umbilic points to Karen for all of her support down the years. 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:
4-manifold Topology and the Poincaré Conjectures
Recently I gave a lecture on the motivation and implications of these questions: