Scattering and Traveltime Tomography

作者将研究散射理论中的问题，以及几个可以表示为边界和透镜刚性问题的非线性逆问题，以及相关的张量层析成像逆问题。具有边界的紧致黎曼流形的边界和透镜刚性是反问题，其中人们希望从两个边界点之间的距离(旅行时)或边界上的散射关系恢复流形。这些问题的线性化是从沿最大测地线的积分恢复张量场的积分几何问题。作者将研究各种系统的这类问题：黎曼流形、渐近双曲流形、洛伦兹流形和非度量哈密顿流形。我们将仔细研究焦散的情况。研究这些问题的动机既来自于纯数学：几何中的刚性问题、逆散射、双曲型方程的逆边值问题、相对论数学、张量积分几何；也来自于在地球物理、医学成像、石油勘探、无损检测、宇宙学和保形场理论等方面的应用。这个项目在散射理论部分的目标之一是研究共振的渐近分布及其与系统经典力学行为的关系。本项目中将研究的反问题在许多实际情况下用作数学模型：在医学中用于成像人体内部结构(CT、超声波、热声断层成像)；在无损材料检测中；在地球物理学中，用于从地震波的传播时间或从反射声波的传播时间获得关于地球内部结构的信息；在石油勘探中；在相对论、理论物理等领域，黎曼度规模拟各向异性介质，其中波的传播速度不仅取决于位置，而且还取决于方向，本项目的重点是各向异性介质的研究。各向异性自然发生在地球、人体等。拟议的研究将进一步发展旅行时间层析成像的数学工具，并将分析在各种情况下的稳定性。共振理论作为散射理论的一部分，对量子力学、量子化学具有重要的基础性意义。共振可以作为特定的峰值频率被观察到，本研究的目的之一是将它们与系统的特性联系起来。

The proposer will study problems in scattering theory and several non-linear inverse problems that can be formulated as boundary and lens rigidity questions and related tensor tomography inverse problems. Boundary and lens rigidity for compact Riemannian manifolds with boundary are inverse problems where one wants to recover the manifold from the distance (travel times)between each two boundary points, or the scattering relation on the boundary. A linearization of these problems is the integral geometry problem of recovering of a tensor field from integrals along maximal geodesics. The proposer will study this type of questions for various systems: Riemannian manifolds, asymptotically hyperbolic manifolds, Lorentzian manifolds, and non-metric Hamiltonians. The case of caustics will be studied carefully. The motivation to study those problems comes both from pure mathematics: rigidity questions in geometry, inverse scattering, inverse boundary value problems for hyperbolic equations, math theory of relativity, integral geometry of tensors; as well from applications to geophysics, medical imaging, oil exploration, non-destructive testing, cosmology and conformal field theory, etc. The proposer will study stable ways to recover the parameters of the system (the metric, the Hamiltonian, etc.). One of goals of this project in the scattering theory part is to study the asymptotic distribution of resonances and it relation to the classical mechanical behavior of the system.The Inverse Problems that will be studied in this project serve as mathematical models in many practical situations: in medicine for imaging the internal structure of a human body (CT, ultrasound, thermoacoustic tomography); in non-destructive material testing; in geophysics for obtaining information about the inner structure of the earth from travel times of seismic waves, or from travel times of reflected acoustic waves; in oil exploration; in theory of relativity, theoretical physics, etc. Riemannian metrics model anisotropic media, where the speed of wave propagation may depend not only on the position but also on the direction and this project's emphasis is on the study of anisotropic media. Anisotropy naturally occurs in Earth, in the human body, etc. The proposed research will develop further the mathematical tools of travel time tomography, and will analyze the stability in various situations. The theory of resonances, as a part of scattering theory, is of fundamental importance for quantum mechanics, quantum chemistry. Resonances can be observed as certain peak frequencies and one of the objectives of this research is to relate them to the properties of the system.