My research field is in differential geometry and geometric analysis. Differential geometry is the study of manifolds---spaces that are locally modelled on Euclidean space. These objects arise naturally in science and engineering, as configuration spaces, as spaces of observables, as Einstein's model of universe, etc. Riemannian manifolds are smooth manifolds equipped with a Riemannian metric, i.e., a notion of length for tangent vectors. From these one derives various notions of curvature, which measures how the space is curved. In a Riemannian manifold one can measure such quantities as lengths, areas, and volumes, and when the manifold models some physical system, certain geometric
quantities can be interpreted as energy, mass, and so on.
Thus, dfferential geometry has its differential aspect, closely involved with partial differential equations, and its geometric aspect, intimately related to topology. Moreover, theorectical physics is an increasingly important source of applications and ideas for differential geometry.
My current research has been concentrated on Atiyah-Singer index theorems and related geometric invariants. Atiyah-Singer index theorem is truely one of the great landmarks of twentieth century mathmatics, a grand unification of the classical Gauss-Bonnet formula, the Riemann-Roch formula and Hirzebruch's signature formula, with broad applications in differential geometry, topology, algebraic geometry, representation theory, number theory, and physics.