Canonical Metrics, the Kahler-Ricci Flow, and Their Applica1ons

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The proposed research work focuses on a number of open problems and developing programs on canonical metrics in Kahler geometry, geometric flows, and complex Monge-Ampere equations in relation to geometry and physics. Recent progress and influx of new ideas have unraveled a deep, rich, and unifying structure among analysis, partial differential equations, complex Riemannian geometry, and algebraic geometry. The project also aims to 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 principal investigator will continue to organize and participate in the integrated research/education programs and activities that will promote the education level of the nation. Furthermore, the principal investigator plans to disseminate the exciting frontier research at the interface of analysis and geometry to a broad audience through lectures and survey papers.These projects will investigate canonical metrics of Einstein type on Kahler varieties with mild singularities. In particular, the principal investigator will study the Riemannian geometric properties of such singular metrics and analytic moduli problems for Kahler-Einstein manifolds. The PI will continue to make progress in the analytic minimal model program with Ricci flow by studying both the finite time and long time formation of singularities of the Kahler-Ricci flow on algebraic varieties. Such singularity formation is reflected by canonical geometric surgeries equivalent to birational transformations and should be understood through global and local metric uniformization. The PI willy employ new theories and techniques from L^2-theory, nonlinear PDEs and Cheeger-Colding theory. The research will develop new tools and give profound insights and understanding of topological, geometric and algebraic structures of complex spaces.