CAREER: Canonical metrics, complex Monge-Ampere equations and geometric flows

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AbstractAward: DMS-0847524Principal Investigator: Jian SongThe proposal focuses on a number of projects on canonical metricsand stability, geometric flows and complex Monge-Ampereequations. Such problems are fundamental in complex analysis andcomplex geometry, in tight relation to partial differentialequations, algebraic geometry and mathematical physics. Therecent progress and influx of new ideas from Ricci flow,pluripotential theory and the minimal model program in algebraicgeometry have unravelled a deep, rich and unifying structure.The PI will investigate the limiting behavior of the Kahler-Ricciflow and its connection to the classification theory foralgebraic varieties, inspired by Perelman's breakthrough inHamilton's program to resolve the geometrization conjecture byRicci flow. In particular, the PI aims to study the relationbetween the formation of finite time singularities of theKahler-Ricci flow and the algebraic surgery in the minimal modelprogram in algebraic geometry. The PI also intends to continuehis study on canonical metrics of Einstein type on algebraicvarieties and understand the analytic and geometric aspects ofthe singularities of such special metrics. The PI also plans tostudy the uniform approximation problem of the Monge-Amperegeodesics in infinite dimensional symmetric space by those in thefinite dimensional Bergman spaces. The precise understanding ofthis problem will give new insight into Yau's conjecture on therelation between Kahler-Einstein metrics and certain stability inthe sense of geometric invariant theory. The outcome of theproposed research will develop new tools and give profoundinsights and understanding of geometry and the structure of theuniverse.Problems in the proposal arise naturally from our attempts tounderstand nonlinear differential equations from geometry andphysics. The solutions to these problems will have strong impacton other fields of sciences such as physics and cosmology in thedeep understanding of our universe. The method of analyzingsingularities of nonlinear equations will have wide applicationsin physics, engineering and economics. Furthermore, the PI plansto disseminate the exciting research at the interface of geometryand analysis to a broad audience through lectures andworkshops. The proposed project will bring in research andteaching innovation in mathematics from various disciplines andhave an immediate beneficial effect on undergraduate and graduatestudents at Rutgers as well as in the regional mathematicalcommunity. The PI will also organize and participate in theintegrated research/education programs and activities that willpromote the education level of the nation.