Differential Geometry and Geometric Analysis

This research topic is the natural application of analysis and differential geometry to the study of partial differential equations. Key ingredients are variational formulation, geometric flows and geometric wave equations. 

Many interesting geometric or physical structures have a variational aspect related to certain functionals. For example, the minimal surfaces and mean curvature flows are Euler-Langrange equations and gradient flows for the area functional respectively, the Willmore surfaces arise as critical points of Willmore functional, and Einstein manifolds are critical points of Einstein-Hilbert functionals in the space of manifolds with fixed volume. The central questions are existence/uniqueness of minimizers, regularity of the solutions, and fine structure and analysis of the singularities. These questions are not only interesting as a geometric problems, but they also have applications to other fields such as topology , physics and algebraic geometry. 

The geometric flows usually refer to the evolution of manifolds/submanifolds/geometric structures by nonlinear heat equations. It can be used for various applications such as classification problems (i.e. Poincare conjectures, Schoenflies conjectures) and general relativity (proving Penrose inequality using inverse mean curvature flows). The central theme in the study of geometric flows and for the purpose of application is the understanding the formation of singularities. Various tools from PDE, geometric measure theory, algebra and combinatorics are used in the singularity analysis. 

Geometric wave equations are found in many equations of physical interest. In particular, nonlinear geometric waves are hidden within the Einstein equations for gravity in relativity, leading to predictions of gravitational waves. The Centre contains experts in the analysis of geometric waves and the Einstein equations, covering topics such as well-posedness, rigidity, and asymptotic properties.