Geometric analysis is a branch of mathematics in which tools and techniques from the theory of differential equations, especially elliptic partial differential equations, are used to establish new results in differential geometry and differential topology.
In geometric analysis, researchers study the interplay between analytical and geometrical properties of spaces. In particular, they use tools from analysis and the theory of partial differential equations to study geometrical problems, and also study the influence of geometrical properties of manifolds, such as curvature bounds, on, for example, properties of differential operators. The most spectacular recent result in the field is the solution of the Poincaré conjecture by Perelman using Hamilton's Ricci flow.
More generally, there is currently great interest in geometric flows, where Riemannian manifolds (or sub-manifolds thereof) are allowed to evolve in a way determined by their geometrical properties. Other notable recent results include the proof of the Riemannian Penrose inequality by Huisken, Ilmanen, and Bray.
In connection with mathematical physics, many problems are motivated by the general relativity, Einstein's classical theory of gravitation. As well as studying wave equations on Lorentzian manifolds, our research includes the formation of trapped surfaces and black holes, and the regularity, stability and global structure of solutions of the Einstein equations.