Differential equations and geometric structures on manifolds

概括地说，我的研究计划是关于微分方程和弯曲空间上的几何结构之间的关系。该提案有五个组成部分，探讨这一主题的具体方面，部分通过问题的动机数学问题和acquitures未来物理和部分通过问题是一个更根本的几何性质。在这些问题上的进展应该有助于提高我们对弯曲空间上微分方程解的行为的理解，有助于进一步阐明一些现有的理论，也有助于开辟新的研究途径。我们现在列出我们建议的五个组成部分。第一个项目涉及的长期动力学的波在高维轴对称黑洞的几何形状，与AdS/CFT对应的几何形状建立在一定的cohomogeneity一个Sasaki-Einstein度量和纠缠在克尔几何。第二个是有关的情况下，谱问题的微分算子可以解决至少部分由代数方法，以及连接与正交多项式系统以及可积系统。第三个涉及流形和orbifold上的双复曲面几何的构造，部分灵感来自广义相对论中的洛伦兹类似物。第四个是尝试推广Cartan-Kaehler存在定理的存在性积分流形的对合解析外微分系统使用Leray理论的分歧柯西问题的柯西-Kovalevskaia型系统。最后第五个项目是探索行为的里奇曲率随机度量空间的三维或更高，使用技术的几何形状的游览集开创了阿德勒和泰勒。

Stated in general terms, my research proposal is concerned with the relationship between differential equations and geometric structures on curved spaces. The proposal has five components which explore specific aspects of this theme, partly through questions motivated by mathematical problems and conjectures coming Physics and partly through problems which are of a more fundamentally geometric nature. Progress on these questions should help improve our understanding of the behavior of solutions of differential equations on curved spaces, help shed further light on some existing conjectures, and also help open new avenues for research. We now list the five components of our proposal. The first project deals with the long term dynamics of waves in higher dimensional axisymmetric black hole geometries, with the AdS/CFT correspondence for the geometries built on certain cohomogeneity one Sasaki-Einstein metrics and with entanglement in Kerr geometry. The second is concerned with instances in which spectral problems for differential operators can be solved at least partially by algebraic methods, and the connections with orthogonal polynomial systems as well as integrable systems. The third deals with the construction of ambi-toric geometries on manifolds and orbifolds, inspired in part by their Lorentzian analogues in General Relativity. The fourth is an attempt to extend the Cartan-Kaehler existence theorem for the existence of integral manifolds of involutive analytic exterior differential systems by using Leray's theory of the ramified Cauchy problem for systems of Cauchy-Kovalevskaia type. Finally the fifth project is an exploration of the behavior of the Ricci curvature for random metrics on spaces of dimension three or higher, using the techniques on the geometry of excursion sets pioneered by Adler and Taylor.