- Geometric Quantization
- Quantization of symplectic manifolds with applications to the
quantization of general classical mechanical systems and the
geometric realisation of representations of Lie groups. My recent
work has been on properties of star products. Prior to that I worked
on the quantization of Kaehler manifolds.
- Deformation Quantization
- Construction and classification of star products and comparison with other methods of quantization. Symmetries of star products.
- Geometry of Symplectic Manifolds
- Construction of symplectic connections with special properties and applications.
- Differential Geometry of Homogeneous Spaces
- The extension of methods and results from symmetric spaces to other classes of homogeneous spaces.
- Twistor Theory
- Development of twistor theories for even-dimensional manifolds, in particular symplectic manifolds.