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

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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.