My main research interests are computability theory, algorithmic randomness, reverse mathematics, higher recursion theory, computable model theory, and set theory.
From 2011 until 2016 I was a Rutherford Discovery fellow. From 2012 to 2015 I was a Turing Research fellow. In 2017 I was elected a Fellow of the Royal Society of New Zealand.
The proof that there are incomparable Turing degrees is really intended as a joke; there have been some misunderstandings.