Geometric Analysis

Geometric analysis is a mathematical discipline at the interface of differential geometry and differential equations. It includes both the use of geometrical methods in the study of partial differential equations (when it is also known as "geometric PDE"), and the application of the theory of partial differential equations to geometry. It incorporates problems involving curves and surfaces, or domains with curved boundaries, but also the study of Riemannian manifolds in arbitrary dimension. The calculus of variations is sometimes regarded as part of geometric analysis, because differential equations arising from variational principles have a strong geometric content. Geometric analysis also includes global analysis, which concerns the study of differential equations on manifolds, and the relationship between differential equations and topology
Geometric analysis is a very typical and effective way to study geometry and topology,and it has become one of the most popular branch in mathematics especially in geometry.The tool which Perelman use to solve Poincare conjecture is Ricci flow,which is just from geometric analysis.Geometric analysis really has a high position in geometry and more and more people study it to get to know geometry better.