My research field is differential geometry. Differential geometry is the study of manifolds; these are the spaces locally modelled on Euclidean space. A manifold is called smooth (or differentiable) if there is an intrinsically defined differential calculus for functions on the manifold. Examples include surfaces in Euclidean spaces. These objects arise naturally in science and engineering, as configuration spaces, as spaces of observables, as Einstein's model of universe, etc.
Much activity in geometry has been focused on Riemannian manifolds; these 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, Riemannian geometry has its differential aspect, closely involved with partial differential equations, and its geometric aspect, related to topology. Moreover, theorectical physics is an increasingly important source of applications and ideas for differential geometry.
My research effort has been concentrated on global Riemannian geometry. Namely, the understanding of global topological structure of the underlying manifolds. Remarkably, the study of local geometric quantities such as curvature bears important implications on the large scale structures of Riemannian manifolds, which has always fascinated me. My work on this aspect includes the study of the fundamental groups, the topological complexity at infinity (the end structure of an open Riemannian manifold), the symmetry groups, the eigenvalues, the evolution equations and the geometric structures of manifolds which collapse (i.e. to the naked eye it lookes as if it is a space of a lower dimension).
With such rich interaction with other fields such as analysis, topology, number theory, and mathematical physics, this is certainly an exciting and fertile field.
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