Analysis on Metric Spaces Unit (Xiaodan Zhou)
The study of analysis and partial differential equations on general metric measure spaces has been a very active research field since the nineties. Because of the general setting of metric measure spaces, analysis on metric spaces provides a unifying framework for ideas and questions from many different fields of mathematics. Besides the pure mathematical importance of the theory, it has been widely applied to control theory, robotics and mathematical biology, etc.
Our unit focuses on partial differential equations, nonlinear potential theory and geometric function theory on various metric spaces including sub-Riemannian manifolds such as the Heisenberg group. The tools applied in our research include the first-order analysis on PI spaces, viscosity solution theory, sub-Riemannian geometry, nonlinear potential theory, control/game theory, etc. Besides the pure mathematical interest and importance of the theory and problems, we also actively seek potential collaborations on applying the theory in emerging fields like data science, machine learning, robotics engineering, etc.