Compactifications, resolution and differential equations

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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.