Differential equations and geometric structures on manifolds

流形上的微分方程和几何结构

• 批准号: 105490-2011
• 负责人: Kamran, Niky
• 金额: $ 3.06万
• 依托单位: McGill University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=569064
• 关键词: Differential; equations; geometric; structures; manifolds

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.

总的来说，我的研究计划是关于弯曲空间上的微分方程和几何结构之间的关系。该提案有五个组成部分，探索这一主题的具体方面，部分是通过由数学问题和来自物理学的猜想引起的问题，部分是通过更基本的几何性质的问题。这些问题的进展将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猜想，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。第一个项目涉及高维轴对称黑洞几何中的波的长期动力学，以及基于Sasaki-Einstein度量的同质性和克尔几何中的纠缠的几何的AdS/CFT对应关系。第二部分涉及微分算子的谱问题可以至少部分地用代数方法解决的实例，以及与正交多项式系统和可积系统的联系。第三章是关于在流形和轨道上构造双环几何，部分灵感来自于广义相对论中的洛伦兹类比。第四部分是利用Leray关于Cauchy- kovalevskaia型系统的分支Cauchy问题理论，尝试推广对合解析外微分系统积分流形的Cartan-Kaehler存在性定理。最后，第五个项目是探索三维或更高维度空间上随机度量的Ricci曲率的行为，使用由Adler和Taylor开创的偏移集几何技术。