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

摘要奖：DMS-0847524主要研究人员：宋健建议集中在一些关于正则度量和稳定性、几何流和复Monge-Ampere方程的项目上。这类问题是复杂分析和复杂几何中的基本问题，与偏微分方程式、代数几何和数学物理密切相关。Ricci流、多势理论和代数几何中最小模型程序的最新进展和新思想的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簇分类理论的联系，灵感来自于Perelman在Hamilton程序中的突破，以解决Ricci流的几何化猜想。特别是，PI的目的是研究Kahler-Ricci流的有限时间奇点的形成与代数几何中最小模型程序中的代数运算之间的关系。PI还打算继续他对代数簇上爱因斯坦类型的正则度量的研究，并了解这种特殊度量奇点的解析和几何方面。PI还计划研究有限维Bergman空间中的Monge-Amperegodes在无限维对称空间中的一致逼近问题。对这一问题的准确理解将使我们对Yau关于Kahler-Einstein度规与几何不变量意义上的稳定性之间的关系的猜想有新的认识。拟议的研究成果将开发新的工具，并提供对几何和宇宙结构的深刻见解和理解。提议中的问题自然源于我们试图从几何和物理上理解非线性微分方程式。这些问题的解决将对其他科学领域产生强烈的影响，如物理学和宇宙学对我们宇宙的深刻理解。分析非线性方程奇异性的方法在物理学、工程学和经济学中都有广泛的应用。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。拟议的项目将带来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立竿见影的效果。该协会还将组织和参与旨在提高国家教育水平的综合研究/教育计划和活动。

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.