Principal Investigator: Jian Song
The proposal focuses on a number of projects on canonical metrics and stability, geometric flows and complex Monge-Ampere equations. Such problems are fundamental in complex analysis and complex geometry, in tight relation to partial differential equations, algebraic geometry and mathematical physics. The recent progress and influx of new ideas from Ricci flow, pluripotential theory and the minimal model program in algebraic geometry have unravelled a deep, rich and unifying structure. The PI will investigate the limiting behavior of the Kahler-Ricci flow and its connection to the classification theory for algebraic varieties, inspired by Perelman's breakthrough in Hamilton's program to resolve the geometrization conjecture by Ricci flow. In particular, the PI aims to study the relation between the formation of finite time singularities of the Kahler-Ricci flow and the algebraic surgery in the minimal model program in algebraic geometry. The PI also intends to continue his study on canonical metrics of Einstein type on algebraic varieties and understand the analytic and geometric aspects of the singularities of such special metrics. The PI also plans to study the uniform approximation problem of the Monge-Ampere geodesics in infinite dimensional symmetric space by those in the finite dimensional Bergman spaces. The precise understanding of this problem will give new insight into Yau's conjecture on the relation between Kahler-Einstein metrics and certain stability in the sense of geometric invariant theory. The outcome of the proposed research will develop new tools and give profound insights and understanding of geometry and the structure of the universe.
Problems in the proposal arise naturally from our attempts to understand nonlinear differential equations from geometry and physics. The solutions to these problems will have strong impact on other fields of sciences such as physics and cosmology in the deep understanding of our universe. The method of analyzing singularities of nonlinear equations will have wide applications in physics, engineering and economics. Furthermore, the PI plans to disseminate the exciting research at the interface of geometry and analysis to a broad audience through lectures and workshops. The proposed project will bring in research and teaching innovation in mathematics from various disciplines and have an immediate beneficial effect on undergraduate and graduate students at Rutgers as well as in the regional mathematical community. The PI will also organize and participate in the integrated research/education programs and activities that will promote the education level of the nation.
