Scattering Theory on Manifolds

在这个项目中，主要研究者将研究渐近欧几里得流形和渐近双曲流形上的散射和逆散射。 渐近双曲流形的例子包括双曲空间通过某些运动群的导数。黑洞外部的德西特-史瓦西模型可以被看作是一个有两个端点的渐近双曲流形，而黑洞外部的史瓦西模型可以被看作是一个有两个端点的流形，一端是渐近双曲的，另一端是渐近欧几里得的。主要研究者将研究非捕获渐近双曲流形上波动方程解的渐近行为和黑洞的史瓦西模型。在这两种情况下，波的长期行为都与共振或散射极点的分布有关，该项目也将解决这个问题。主要研究者还将研究逆散射，包括从所有能量的散射矩阵确定渐近欧几里得流形的问题。这需要证明一个支持定理，该支持定理将欧几里得空间中Radon变换的著名支持定理推广到这种设置。偏微分方程研究的一个中心问题是了解介质的几何性质如何影响波在其上传播的方式。首席研究员将研究长时间-由物理学的例子激发的空间上波动方程的解的行为。“逆问题”包括使用从在特定介质上传播的波获得的测量结果来确定介质的一些几何特性。例如，通过敲击物体的表面并倾听它如何对该信号作出响应，可以获得关于物体内部的信息;通过测量波在物体表面上的两点之间传播的时间，可以获得关于波在物体内部传播的可变速度的信息。主要研究员将在上述空间研究这类性质的问题。需要开发的技术也可能在其他领域得到应用，如断层扫描。

In this project the principal investigator will study scattering and inverse scattering on asymptotically Euclidean and asymptotically hyperbolic manifolds. Examples of asymptotically hyperbolic manifolds include quotients of hyperbolic space by certain groups of motion. The de Sitter-Schwarzschild model of the exterior of a black hole can be viewed as an asymptotically hyperbolic manifold with two ends, while the Schwarzschild model of the exterior of a black hole can be viewed as a manifold with two ends that is asymptotically hyperbolic on one end and asymptotically Euclidean on the other. The principal investigator will study the asymptotic behavior of solutions of the wave equation on nontrapping asymptotically hyperbolic manifolds and the Schwarzschild model of a black hole. In both cases the long-time behavior of waves is associated with the distribution of resonances, or scattering poles, and the project will address this question as well. The principal investigator will also study inverse scattering, including the question of determining an asymptotically Euclidean manifold from the scattering matrices at all energies. This requires the proof of a support theorem that generalizes to this setting the well-known support theorem for the Radon transform in Euclidean space.One central problem in the study of partial differential equations is to understand how geometric properties of a medium influence the way waves propagate on it. The principal investigator will study the long time-behavior of solutions to the wave equation on spaces that are motivated by examples from physics. The "inverse problem" consists of using measurements obtained from waves that propagate on a certain medium to determine some of the medium's geometric properties. For example, by knocking on the surface of an object and listening to how it responds to this signal, one can obtain information about the object's interior; by measuring the time a wave travels between two points on the surface of the object one would like to obtain information about the variable speed of propagation of the waves in the interior of the object. The principal investigator will study problems of this nature in the spaces indicated above. The techniques that need to be developed might also have applications in other areas, such as tomography.