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 includes the proof, in collaboration with Wilhelm Klingenberg, of an 100 yearold conjecture in classical surface theory called the Caratheodory Conjecture. The complete proof has now appeared:
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 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).
Recent Research
From CT Scans to 4Manifold Topology via Neutral Geometry and here's a recent talk I gave on some of the results in SISSA, Trieste, Italy:
Below is a list of some recent papers and preprints.
4manifold Topology and the Poincaré Conjectures Over the years I have studied 4manifold topology and the 4dimensional Poincaré Conjectures in both the topological and smooth categories  the former is officially deemed proven by Michael Freedman in the 1980's, the latter is officially open. The accepted proof of the topological 4dimensional Poincaré Conjecture is but one small consequence of a far bigger topological classification of 4manifolds which follows from Freedman's "Disk Theorem". The original published proof of the Disk Theorem contains numerous errors and the subsequent monograph by Freedman and Quinn did not address the central issues. In a series of mathoverflow qestions I have probed this topic
Recently I gave a lecture on the motivation and implications of these questions:
