Differential geometry is the branch of mathematics that studies the shapes of spaces through distances and angles. Mathematically these are measured by Riemannian metrics. This research project investigates problems centering around the Ricci flow, which is in turn a natural way to evolve Riemannian metrics in their underlying space. The concept of Ricci flow was invented by R. Hamilton in 1982 and has had wide applications in mathematics since then. For instance, the Ricci flow has had a profound and far-reaching impact on 3-dimensional geometry and topology, as evidenced by G. Perelman's solution to the Poincare conjecture in 2003. The investigator will study the applications of the Ricci flow in algebraic geometry and 4-dimensional topology. These applications will deepen our understanding of several related subjects, including partial differential equations, probability, complex analysis, and theoretical physics. By organizing conferences and seminars, the PI will motivate graduate students to join this research endeavor. Moreover, the PI will present the work to undergraduate students and the general public.
The investigator will continue developing tools based on the Ricci flow. In particular, he will refine and localize the existing estimates in Ricci flow, for the purpose of applying Ricci flow in algebraic geometry and 4-dimension topology. He will generalize the pseudo-locality theorem of Perelman, and find the relationships between Ricci flow theory and the theory of metric spaces with lower bound of Ricci curvatures, i.e., the Cheeger-Colding theory. He will continue to develop the special estimates in the Kahler Ricci flow. Among other things, he will aim to combine the flow version of Chen-Lu inequality with other estimates. Based on the improved estimates, the PI will aim to understand the relationship between the minimal model program in algebraic geometry and the Kahler Ricci flow. He will also determine if some of the results in the Cheeger-Colding theory can be reproved using the Ricci flow method. Finally, the PI intends to study the expected deep connections between the Ricci flow and probability theory.