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

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。这个项目的核心是研究著名的Ricci流的复杂版本，Kahler-Ricci流。许多密切相关的对象，如复杂的蒙日-安培方程，也进行了讨论，以达到更深入的理解。最终的目标和最初的动机是为极大兴趣的代数几何对象提供一个几何分析的观点，例如，最小模型或一般的大线束。同时，对这种流动的更广泛的应用也引起了几何学和分析学中一些有趣的问题。在研究几个复杂变量的经典理论中，多能理论发挥了至关重要的作用。在此过程中改进或发明的方法和技术已经引起了远远超出这些研究领域的广泛关注，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）和电子媒体（电子预印本服务器、个人网站等）广泛传播给科学界。在美国国内外，关于这项研究和相关主题的研讨会、讲习班和会议相当多。在NSF的帮助下，思想的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是使这个世界变得更美好的最根本动力之一。与此同时，任何提高我们对世界的理解的研究，无论是物理模型还是抽象的数学世界，都可能产生重大的好处，但这些好处最终实现可能需要一段时间。

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.