Canonical Metrics, Geometric Flows and Formation of Singularities

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Recent progress and influx of new ideas have unraveled a deep, rich and unifying structure among analysis, Riemannian geometry, pluripotential theory, classical several complex variables and the minimal model program in algebraic geometry. The proposed research work focuses on a number of open problems and developing programs on canonical metrics, geometric flows and complex Monge-Ampere equations arising from geometry and physics. The proposed project also aims to bring in research and teaching innovation in mathematics from various disciplines and have an immediate beneficial effect on graduate and undergraduate students at Rutgers as well as in the regional community of mathematics. The PI will also organize and participate in the integrated research/education programs and activities that will promote the education level of the nation. Furthermore, the PI plans to disseminate the exciting frontier research at the interface of geometry, analysis and algebra to a broad audience through lectures and survey papers. The PI will investigate and continue to make progress in the analytic minimal model program with Ricci flow. In particular, the PI will study 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/analytic surgeries equivalent to birational transformations and should be understood through global and local metric uniformization. The PI will also investigate the canonical metrics of Einstein type on singular varieties, in particular, the Riemannian geometric properties of such singular metrics and related moduli problems with applications in string theory such as geometric transitions and mirror symmetry. The PI willy employ new theories and techniques from L^2-theory, nonlinear PDEs, Perelman's works and Cheeger-Colding theory. The outcome of the proposed research will develop new tools and give profound insights of the structure of the universe as well as many other applied sciences.