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

摘要/ abstract摘要：项目负责人：宋健，主要研究方向为典型度量与稳定性、几何流动和复杂蒙日-安培方程。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。近年来，代数几何中的利玛西流、多能理论和最小模型程序等新思想的发展和涌入，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变量分类理论的联系，灵感来自Perelman在hamilton程序中解决ricci流的几何化猜想的突破。特别地，PI旨在研究代数几何中最小模型程序中kahler - ricci流的有限时间奇点的形成与代数手术之间的关系。PI还打算继续他在代数变异体上的爱因斯坦型规范度量的研究，并了解这种特殊度量的奇异性的解析和几何方面。PI还计划研究无限维对称空间中Monge-Amperegeodesics在有限维Bergman空间中的一致逼近问题。对这一问题的精确理解将对丘德威关于几何不变量理论意义上的Kahler-Einstein度量与一定稳定性之间关系的猜想提供新的见解。拟议研究的结果将开发新的工具，并对几何和宇宙结构提供深刻的见解和理解。由于我们试图从几何和物理学的角度来理解非线性微分方程，这个建议中的问题自然就产生了。这些问题的解决方案将对物理学和宇宙学等其他科学领域产生强烈的影响，从而加深对宇宙的理解。分析非线性方程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观众传播几何学和分析界面的令人兴奋的研究。该计划将带来来自不同学科的数学研究和教学创新，并对罗格斯大学的本科生和研究生以及地区数学界产生直接的有益影响。PI还将组织和参与提高国家教育水平的综合研究/教育项目和活动。

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.