Partial differential equations and geometric analysis form a central area of mathematics with deep theoretical significance and a wide range of applications, including material sciences, image processing, crystal growth, and control theory. Many natural phenomena are modeled by nonlinear PDEs whose complex behavior reflects underlying geometric or physical structures. Understanding these equations requires the development of sophisticated analytical methods and geometric insights.
The Geometric Partial Differential Equations Unit is dedicated to advancing the theory and applications of nonlinear PDEs arising in geometry and related fields. Our research is motivated by fundamental questions in analysis and geometry, such as the evolution of shapes, behavior of singularities, structure of general metric spaces. We aim to construct new analytic frameworks to study the qualitative and quantitative properties of solutions, to advance the understanding of nonlinear PDEs, and to contribute to solving complex real-world problems.
Our current major 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
