Compactifications, resolution and differential equations

这个建议的目的是表明，分析，并在，奇异空间，可以，而且应该，以一致的方式处理，这样做将导致有用的和经常最佳的结果。这里考虑的核心几何结构是，一个紧凑的流形与角，连同光滑的映射之间的空间，以及基本操作的紧化和爆破。为了证明效用的这些想法的主要研究者提出研究从这个角度来看，以下四个问题：决议顺利行动紧凑李群和使用这种“充分决议”的拓扑结构，指数理论和分析。磁单极子模空间的紧化。爱因斯坦方程解的渐近性态。Morse型纤维化的解决方案及其在绝热极限和Kaehler度规存在性方面的应用在研究大的数学问题的解决方案时，例如爱因斯坦方程的解决方案的长时间行为，通过紧致空间使“无穷大”更接近是特别有用的。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。这个紧化过程是对偶的奇异性的解决方案的操作，通过迭代引入极坐标。这两个过程自然发生在一起，在一个系统的研究过渡行为的解析几何问题。

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.