I work in noncommutative algebra both in pure mathematics and in mathematical physics, including quantum gravity. In algebra, I introduced one of the two main classes quantum groups or Hopf algebras, and have also developed noncommutative differential geometry or `quantum geometry', braided algebra, a Lie theory for finite groups, and a new way of thinking about nonassociative algebra such as the octonions. In quantum gravity I introduced the most studied model of quantum spacetime including the prediction that the speed of light in such models depends on energy. Some of my best recent papers are:
1. `Poisson-Riemannian geometry', J. Geom. Physics. 114 (2017) 450-491 (with E.J. Beggs)
We look at the classical data needed to `quantise' geometry to a noncommutative one. At the level of the coordinate algebra this is well-known as a Poisson structure but to quantise also the differential calculus of spacetime needs a new `gauge field' of physics, the carrier of the differential structure. This needs to be compatible with the metric if we want to quantise that leading to a new field of Poisson-Riemannian geometry. Here the quantisation parameter is the Planck scale so the new paradigm of physics has the same relation to quantum gravity as classical mechanics had in the 1920's to quantum mechanics; we call it classical-quantum-gravity.
2. `Reconstruction and quantization of Riemannian structures' preprint http://arxiv.org/abs/1307.2778
I show that Riemannian geometry, possibly with degenerate metric, emerges out of the nothing but the product rule for differentials, namely as the `cocycle' extension data to go from classical differential forms on a manifold to its extension by one extra direction. This gives a new way of thinking about Riemannian geometry in general as well as a mechanism for how and why geometry emerges out of algebra in quantum gravity.
3. `Cosmological constant from quantum spacetime', Phys. Rev. D, 91 (2015) 124028 (12pp) (with W.-Q. Tao)
We show that the requirements of the product rule for differentiation in the simplest quantum spacetime model force the geometry to be curved and for there to be a cosmological constant. The algebra here has relations [x,t]=x and had a unique form of differential calculus that admits a quantum metric. We show that this and spherical symmetry forces the metric to be a quantisation of the Bertotti-Robinson solution of the Einstein-Maxwell equations.
4. `Lie theory and coverings of finite groups', J. Algebra, 389 (2013) 137-150 (with K. Rietsch)
We introduce the notion of a locally skew IP quandle as a refinement of the notion of a `Lie algebra' of a finite group. We prove that this applies to many types of groups including Coxeter reflection groups and we show that when group a group is generated by such a `Lie algebra' is noncommutative de Rham cohomology has trivial classical part.
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