Geometric Partial Differential Equations (Qing Liu)

Analysis of partial differential equations (PDEs) is a very rich mathematics subject, which is broadly applied in a large variety of fields of science. It is particularly important to study nonlinear PDEs that arise in geometry and many related areas. The Geometric Partial Differential Equations Unit aims to develop new analytic methods to understand behavior of solutions to various geometric evolutions and explore solvability of nonlinear equations in general geometric settings such as sub-Riemannian manifolds and metric spaces. Our research is motivated by numerous applications in material sciences, crystal growth, image processing and is also closely connected with topics in optimal control, game theory and machine learning, etc.

Our current research topics include the following:

• Viscosity solution theory for fully nonlinear PDEs
• Motion by curvature and more general surface evolution equations
• Convexity and other geometric properties of solutions to nonlinear PDEs
• Analysis on sub-Riemannian manifolds and general metric spaces