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

AbstractAward：DMS-0847524首席研究员：宋健该提案侧重于规范度量和稳定性，几何流和复杂的蒙赫-安培方程的一些项目。这类问题是复分析和复几何的基础，与偏微分方程、代数几何和数学物理有着密切的关系。最近的进展和涌入的新思想，从Ricci流，pluripotential理论和最小模型程序在algebraicgeometry已经解开了一个深刻的，丰富的和统一的结构。PI将调查的限制行为的Kahler-Ricciflow及其连接到分类理论的代数簇，启发Perelman的突破，在汉密尔顿的计划解决几何化猜想的Ricci流。特别地，PI旨在研究Kahler-Ricci流的有限时间奇点的形成与代数几何中极小模型程序中的代数手术之间的关系。PI还打算继续他的研究规范度量的爱因斯坦类型的代数品种和理解的分析和几何方面的奇异性，这种特殊的度量。PI还计划用有限维Bergman空间中的Monge-Amperegodesics来研究无穷维对称空间中Monge-Amperegodesics的一致逼近问题。 对这一问题的正确理解将使我们对Yau关于Kahler-Einstein度量与几何不变理论意义下的某种稳定性之间关系的猜想有新的认识。这项研究的成果将发展新的工具，并给予深刻的见解和理解的几何和宇宙的结构。在建议中的问题自然产生于我们试图理解非线性微分方程的几何和物理。这些问题的解决将对物理学和宇宙学等其他科学领域产生重大影响，从而加深对宇宙的认识。分析非线性方程组奇异性的方法在物理学、工程学和经济学中有着广泛的应用。此外，PI计划通过讲座和研讨会向广大受众传播几何和分析界面上令人兴奋的研究。拟议的项目将带来研究和教学创新的数学从各个学科和有一个直接的有益影响本科生和研究生在罗格斯大学以及在区域prosticalcommunity. PI还将组织和参与综合研究/教育计划和活动，以提高国家的教育水平。

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.