In differential geometry the current research involves submanifolds, symplectic and conformal geometry, as well as affine, pseudo-Riemannian, Riemannian and complex geometry and Riemannian geometry of infinite-dimensional manifolds.
In the area of geometric topology the emphasis is on low dimensional manifold theory, Kleinian groups and related decision problems. Areas of special interest include braid theory, 3-manifolds and hyperbolic orbifolds, normal surface theory, group actions on manifolds, and applications of computational topology to computer animation, scientific visualization and engineering design.
In algebraic geometry: global and local positivity properties of numerical cycle classes and of vector bundles, asymptotic invariants.
In geometric analysis: heat kernel analysis on Lie groups and Riemannian manifolds, metric diophantine properties of the geodesic flow on a hyperbolic Riemann surface, mathematical general relativity and geometric evolution equations.