Propagation Phenomena for Waves and Scattering

本研究的目的是研究不同语境下的传播现象。提案中的主题涵盖了不同的领域，如研究边上波动方程的绕射现象，以及弯曲时空中波的渐近行为，如广义相对论。该项目将采用几何微局部或相空间技术来探索这些领域的新问题。其中一些技术是建设性的，例如在一类渐近反de-Sitter(ADS)空间上构造正向基本解和散射算子，而另一些技术则依赖于相空间能量估计，例如微局部正交换子估计，有望用于研究弯曲时空(如描述旋转黑洞的Kerr-de Sitter空间)上波动方程的渐近行为，以整体条件代替初始条件的波动方程，以及具有渐近边缘和尖点的弯曲时空上的波传播。即使是对应于静止(无传播)问题的椭圆问题，有时也可以从不同的角度受益：最近关于渐近双曲空间上拉普拉斯预解式的亚纯延拓及其高能行为的工作就是一个例子。波的传播在物理世界中无处不在，电磁波(如光波)是最普遍的例子之一。一个典型的问题是波是如何与边缘等结构相互作用的，这是众所周知的镜面反射定律的推广。散射理论是另一门受传播现象支配的学科。例如，散射现象进入了对大尺度量子波的描述，在数学上被建模为“无穷大”。由于传播现象在物理世界中无处不在，因此对其更精确的理解具有许多潜在的应用前景。一个预计会引起人们兴趣的具体物理应用是开发散射理论(即，在弯曲的时空背景下研究光波或量子粒子的全球行为，这给出了广义相对论框架下对我们宇宙的描述)。所涉及的一些技术预计也将有助于理解成像问题，例如，在地球物理背景下。

The object of the proposed research is the study of propagation phenomena in different contexts. Topics in the proposal encompass such diverse areas as the study of diffractive phenomena for the wave equation at edges, as well as the asymptotic behavior of waves on curved space-times, such as in general relativity. The project will employ geometric microlocal, or phase space, techniques to explore new problems in these areas. Some of these techniques are constructive, such as the proposed construction of the forward fundamental solution and the scattering operator on a class of asymptotically Anti-de Sitter (AdS) spaces, while others rely on phase-space energy estimates, such as the microlocal positive commutator estimates that are expected to be of use in the study of the asymptotic behavior of the wave equation on curved space-times (such as Kerr-de Sitter space, describing rotating black holes), of wave equations with global conditions replacing initial conditions, and in wave propagation on curved space-times with asymptotic edges and cusps. Even elliptic problems, which correspond to stationary (no propagation) problems, can sometimes benefit from a different point of view: recent work on the meromorphic continuation of the resolvent of the Laplacian on asymptotically hyperbolic spaces, and on its high energy behavior, provides an example.Wave propagation is ubiquitous in the physical world, with electromagnetic waves (e.g., light waves) being one of the most prevalent examples. A typical question is how waves interact with structures such as edges, generalizing the well-known law of reflection from mirrors. Scattering theory is another subject governed by propagation phenomena. For instance, scattering phenomena enter into the description of quantum waves on large scales, mathematically modelled as "infinity." As propagation phenomena are ubiquitous in the physical world, their more precise understanding has many potential applications. A concrete physical application that is expected to be of interest is developing scattering theory (i.e., investigating how light waves or quantum particles behave globally, in the context of curved space-times, which give a description of our universe in the framework of general relativity). Some of the techniques involved are also expected to benefit the understanding of imaging problems, for instance, in the geophysical context.