Much of my early work involved classical general relativity. For instance, my dissertation confirmed the existence of solutions of Einstein's equations containing gravitational radiation and satisfying known "physical" boundary conditions — formulated in terms of a conformal structure on spacetime.
I also published a paper on a rotating version of the famous twin paradox, illustrating at an elementary pedagogical level the subtleties involved in working with rotating observers.
I then became interested in algebraic computing in relativity, in which computer algebra is used to classify exact solutions of Einstein's equations. I have always been intrigued by this use of computers to derive equations which are then solved by hand, not vice versa.
I have also done purely mathematical work in differential geometry. For instance, together with my student, Stuart Boersma, I introduced the concept of parametric manifolds, a generalization of the idea of surfaces. This geometric structure is well-suited to the study of rotation, and should have applications to quantum field theory as described by rotating observers.
I have been fortunate over the years to have collaborated with a number of world-class scientists. Foremost among these is Professor Gerard 't Hooft, recipient of the 1999 Nobel Prize in Physics and one of the most often cited physicists ever, with whom I was a postdoc for two years. Our first paper, describing the first 2-body solution in general relativity, has been cited more than 150 times.
I have also had the chance to work with Professor Paul Davies, recipient of the 1995 Templeton Prize and prolific author of popular science books, on quantum field theory in curved space, the attempt to generalize quantum theory to relativity. Together with Corinne Manogue, we resolved the apparent paradox that rotating particle detectors see particles in "the" vacuum; the analogous problem for linearly accelerating detectors is well-understood, and is related to the Hawking effect, in which quantum black holes create particles with a thermal spectrum.
In the early 1990s, I was part of a collaboration that proposed considering signature-changing spacetimes, which contain Euclidean regions. This idea was proposed simultaneously, in the context of early-universe cosmology, by a group led by Professor George Ellis, the recipient of the 2004 Templeton Prize. The Euclidean region is a possible model for the Big Bang, and a comparison of our approaches led to a long and fruitful collaboration between the two groups.
For my work in mathematical relativity, I was elected a Fellow of the American Physical Society in 2010.
More recently, again together with Corinne Manogue, I have been studying the octonions with a view to describing the physics of fundamental particles. Intriguing results have been obtained regarding the eigenvalues of $3\times3$ Hermitian octonionic matrices, notably that they admit 6, rather than 3, real eigenvalues. Using division algebras to do Clifford algebra manipulations in suitable dimensions provides an elegant mathematical framework that, among other things, shows why superstring theory only works in certain dimensions. This approach has already led us to new insights in particle physics, based on our eigenvalue results. Notable among these is a dimensional reduction scheme which suggests that the octonionic Dirac equation may lead to 3 generations of leptons with single-helicity neutrinos, observed properties of nature which remain unexplained by current theories. In our most recent work, we establish a close relationship between the exceptional Lie algebra $\mathfrak{e}_8$ and the Standard Model of particle physics.