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 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).
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.
4-manifold Topology and the Poincaré Conjectures
Over the years I have studied 4-manifold topology and the 4-dimensional 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 4-dimensional Poincaré Conjecture is but one small consequence of a far bigger topological classification of 4-manifolds 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: