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

职业：规范度量、复杂的 Monge-Ampere 方程和几何流

• 批准号: 0847524
• 负责人: Jian Song
• 金额: $ 42.7万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-15 至 2015-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0847524&HistoricalAwards=false
• 关键词: CAREER; Canonical; metrics; complex; Monge

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.

摘要奖项：DMS-0847524首席研究员：Jian Song该提案重点关注一些关于规范度量和稳定性、几何流和复杂蒙日-安培方程的项目。这些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密切相关。 Ricci 流、多能理论和代数几何中的最小模型程序的最新进展和新思想的涌入已经揭示了一个深刻、丰富和统一的结构。PI 将研究 Kahler-Ricci 流的极限行为及其与代数分类理论的联系品种，受到佩雷尔曼在汉密尔顿方案中的突破的启发，解决了里奇流的几何化猜想。特别是，PI旨在研究Kahler-Ricci流有限时间奇点的形成与代数几何最小模型程序中的代数手术之间的关系。 PI 还打算继续他对代数簇的爱因斯坦型规范度量的研究，并了解此类特殊度量奇点的分析和几何方面。 PI还计划通过有限维Bergman空间中的问题来研究无限维对称空间中Monge-Ampergeodesics的一致逼近问题。 对这个问题的准确理解，将为丘关于卡勒-爱因斯坦度量与几何不变量理论意义上的一定稳定性之间关系的猜想提供新的见解。拟议研究的成果将开发新的工具，并为几何和宇宙结构提供深刻的见解和理解。提案中的问题自然产生于我们试图从几何和物理学理解非线性微分方程。这些问题的解决将对物理学和宇宙学等其他科学领域对我们宇宙的深入理解产生重大影响。分析非线性方程奇异性的方法将在物理、工程和经济学中具有广泛的应用。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成果。拟议的项目将带来各个学科的数学研究和教学创新，并对罗格斯大学以及该地区数学界的本科生和研究生产生直接的有益影响。 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。