Mainly differential geometry, but branching into algebraic geometry and topology. Emphasis on the study of manifolds, typically Riemannian or almost complex ones, often defined with reference to the action of a Lie group. Main focus at present is on the construction of Ricci-flat metrics, working within the Simons Collaboration on Special Holonomy in Geometry, Analysis, and Physics.

Exceptional holonomy; twistor spaces and projective geometry; almost complex manifolds; quaternionic geometry; harmonic maps; spinors, differential operators and index theory; nilmanifolds; combinatorial geometry; mathematical software and visualization.