My research interests range over a fairly broad part of modern differential geometry. It interfaces with the theory of Lie groups and both algebraic geometry and algebraic topology. Recently there have been two main threads. The first involves the study of the topology of moduli spaces of holomorphic maps, the so-called sigma-models, and of the moduli spaces of instantons.
The second uses contact geometry together with Lie group theoretical methods to give explicit constructions of Einstein manifolds of positive scalar curvature, and then to study the topology of such manifolds. Very recent work uses methods of algebraic geometry, more specifically Mori theory, to construct Einstein metrics on 5-manifolds, as well as homotopy spheres in all odd dimensions. This work appears in my monograph Sasakian Geometry with K. Galicki published by Oxford University Press.
My earlier work was also two-tiered. The first involved group representation theory applied to problems of Mathematical Physics, in particular, to the problem of separation of variables of the partial differential equations that appear in Mathematical Physics. The second used Lie group methods to study complex geometry with special applications to general relativity.