Compactifications, resolution and differential equations

紧化、解析和微分方程

• 批准号: 1005944
• 负责人: Richard Melrose
• 金额: $ 39.2万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-06-01 至 2013-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1005944&HistoricalAwards=false
• 关键词: Compactifications; resolution; differential; equations

This proposal aims to demonstrate that analysis of, and on, singular spaces, can, and should, be dealt with in a consistent manner and that doing so will lead to useful and frequently optimal results. The core geometric structure considered here is that of a compact manifold with corners, together with the smooth maps between such spaces, and the basic operations of compactification and blow up. To demonstrate the utility of these ideas the Principal Investigator proposes to study from this point of view the following four problems: The resolution of smooth actions by compact Lie groups and the use of such `full resolutions' in topology, index theory and analysis. The compactification of moduli spaces of magnetic monopoles. The asymptotic behavior of solutions to Einstein's equation. The resolution of Morse-type fibrations with applications to adiabatic limits and the existence of Kaehler metrics.In studying solutions of mathematical problems in the large, such as the long-time behaviour of solutions to Einstein's equation, it is particularly useful to `bring infinity' closer by compactifying the space. After doing so, such `asymptotic' questions are replaced by regularity problems in a more conventional sense. This process of compactification is dual to the operation of resolution of singularities, by the iteratrive introduction of polar coordinates. These two processes naturally occur together in a systematc study of transition behaviour of analytic-geometric problems.

这项提议旨在表明，可以而且应该以一致的方式处理对奇异空间的分析，这样做将产生有用的、往往是最优的结果。这里考虑的核心几何结构是一个带角的紧致流形，以及这种空间之间的光滑映射，以及紧致和爆破的基本运算。为了证明这些想法的实用性，首席研究者建议从这个角度研究以下四个问题：紧李群对光滑作用的分解以及这种“完全分解”在拓扑学、指数理论和分析中的应用。磁单极子模空间的紧致化。爱因斯坦方程解的渐近行为。Morse型纤颤的解及其对绝热极限和Kaehler度规的存在的应用。在研究一般数学问题的解，如爱因斯坦方程解的长期行为时，通过压缩空间使无限更近是特别有用的。在这样做之后，这样的“渐近”问题被更传统意义上的正则性问题所取代。通过极坐标的迭代引入，这种紧凑化过程对奇点分解的操作是双重的。在系统地研究解析几何问题的过渡行为时，这两个过程自然地同时发生。