"K\"ahler-Ricci Flow with Degenerate Cohomology Limit

具有简并上同调极限的“K”ahler-Ricci 流

• 批准号: 0904760
• 负责人: Lydia Bieri
• 金额: $ 9.48万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 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: 2009-09-01 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0904760&HistoricalAwards=false
• 关键词: ahler; Ricci; Flow; Degenerate; Cohomology

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). This project is centered around the study of complex version of the very famous Ricci flow, Kahler-Ricci flow. Many closely related objects, for example, complex Monge-Ampere equation, are also discussed in order to achieve deeper understanding. The ultimate goal and original motivation would be to provide a geometric analysis point of view for algebraic geometry objects of great interests, for example, minimal model or general big line bundle. Meanwhile, using this flow in a more extensive way has been bringing up intriguing problems in geometry and analysis. One classic theory in the study of several complex variables, pluripotential theory, turns out to play a crucial role. The methods and techniques improved or invented during the process have been attracting wider attention far beyond these research fields, thus strengthening the fruitful cross-field collaboration.There are motivations and impacts of this proposal beyond the study of pure mathematics itself. The results of the research will be broadly disseminated to the scientific community using traditional channels (publications, conference talks, etc.) and electronic media (electronic preprint server, personal web-site, etc.). There have been quite some seminars, workshops and conferences on this research and related topics in and out of the United States. With the help from NSF, the exchange of ideas can be promoted to a whole new level. Similar to most fundamental research, its benefits to the society are usually not so immediate. The challenge we created for ourselves has been one of the most fundamental driving force to make this a better world. In the mean time, any research that improves our understanding of the world, being its physical models or abstract mathematical world, can potentially yield significant benefits, but it will likely take a while before these benefits eventually get materialized.

该奖项是根据2009年《美国复苏和再投资法案》(公法111-5)提供资金的。这个项目是围绕着非常著名的Ricci流--Kahler-Ricci流的复杂版本的研究展开的。文中还讨论了许多密切相关的对象，如复杂的Monge-Ampere方程，以加深对它们的理解。最终目标和最初的动机是为感兴趣的代数几何对象提供几何分析的观点，例如最小模型或一般的大线束。与此同时，以更广泛的方式使用这种流已经在几何和分析中带来了有趣的问题。在研究多个复变量的经典理论中，多势理论发挥了至关重要的作用。在这一过程中改进或发明的方法和技术吸引了更广泛的关注，远远超出了这些研究领域，从而加强了卓有成效的跨领域合作。这一提议的动机和影响超出了纯数学本身的研究。研究结果将通过传统渠道(出版物、会议演讲等)向科学界广泛传播。和电子媒体(电子预印本服务器、个人网站等)。在美国国内外，已经有相当多的关于这项研究和相关主题的研讨会、讲习班和会议。在NSF的帮助下，思想交流可以提升到一个全新的水平。与大多数基础研究类似，它对社会的好处通常不会那么立竿见影。我们为自己创造的挑战一直是让这个世界变得更美好的最根本动力之一。与此同时，任何改善我们对世界的理解的研究，无论是它的物理模型还是抽象的数学世界，都可能产生巨大的好处，但这些好处最终可能需要一段时间才能实现。