Behaviour of Waves on Higher Dimensional Black Hole Spacetimes

黑洞的概念是爱因斯坦的广义相对论最迷人的预言之一。从严格的数学观点理解这些物体的稳定性对于我们解释爱因斯坦的理论和天体物理观测都是至关重要的。在过去的10-15年里，偏微分方程研究中的新数学技术已经发展起来，以解决黑洞稳定性的难题。在这些发展的中心是一个数学上强大的框架来理解波在黑洞时空的传播。该研究项目的目的和目标是调查这些新技术中的哪些可以应用于高维黑洞时空的研究。(The后者表现出更复杂的时空几何，并在理论物理学中得到广泛研究，主要的例子是所谓的黑环。众所周知，并不是所有的技术都能直接进行，因此学生必须开发PDE分析的新方法来证明关于这些时空上波的全局行为的定理。除了在理论物理学中的应用（在那里各种不稳定性正在被证实），这将进一步阐明时空几何和波的色散之间的复杂关系。

The concept of a black hole is one of the most fascinating predictions of Einstein's Theory of General Relativity. Understanding the stability properties of theses objects from a rigorous mathematical point of view is crucial for our interpretation of both Einstein's theory and of astrophysical observations. New mathematical techniques in the study Partial Differential Equations have been developed in the past 10-15 years to address the difficult problem of stability of black holes. At the center of these developments is a mathematically robust framework to understand wave propagation on black hole spacetimes.The aims and objectives of the research project are to investigate which of these new techniques can be applied to the study of higher dimensional black hole spacetimes. (The latter exhibit more complicated spacetime geometries and are widely studied in theoretical physics, the main example being the so-called black ring.) It is known that not all of the techniques carry through directly, so new methods in PDE analysis will have to be developed by the student to prove theorems regarding the global behaviour of waves on these spacetimes. Besides applications in theoretical physics (where various instabilities are being conjectured) this will further elucidate the intricate relationship between spacetime geometry and the dispersion of waves.