Geodesic ray transforms and the transport equation

地球物理学的一个基本问题是根据在地表进行的测量重建地球的内部结构。医学上的一个基本问题是在不进行侵入性手术的情况下获得体内肿瘤的结构信息。两者都是三维反问题的例子。在过去的五年中，PI和他的合作者解决了关于二维逆问题的几十年的老问题。这个建议解决了剩下的关键问题，着眼于在更高维度的环境中取得进展，这是潜在应用的核心，也是分析这一分支的原始动机。这些重要的反问题的线性化，很自然地涉及到测地线射线变换，在测地线变换中，一个函数或张量场沿着黎曼度规的测地线积分。标准的x射线变换，即沿直线积分函数，对应于欧几里得度规的情况，是医学成像技术如CT和PET的基础。沿着更一般的测地线进行积分的情况出现在确定地球内部结构的地球物理成像中，因为弹性波的速度通常随深度增加而增加，从而使射线弯曲回地球表面。它也出现在超声成像，其中黎曼度量模型的各向异性折射率。在张量层析成像问题中，人们希望从测地线上的积分中确定一个不受自然阻碍的对称张量场。本文旨在通过将测地线射线变换与输运方程特解的存在性联系起来，进一步深入了解作用于对称张量的测地线射线变换的注入性。这种关系对于最近在解决二维几何逆问题方面的成功是至关重要的。

A basic problem in geophysics is to reconstruct the interior structure of the Earth from measurements carried out at the surface. A basic problem in medicine is to obtain structural information about tumours inside the body without invasive surgery. Both are examples of inverse problems in three dimensions. Over the last five years, many-decade old problems concerning inverse problems in two dimensions have been resolved by the PI and his collaborators. This proposal addresses the key remaining questions, with a view of making progress in the higher dimensional setting which are central to potential applications and which formed the original motivation for this branch of Analysis.The linearization of many of these important inverse problems takes naturally to the geodesic ray transform where one integrates a function or a tensor field along geodesics of a Riemannian metric.The standard X-ray transform, where one integrates a function along straight lines, corresponds to the case of the Euclidean metric and is the basis of medical imaging techniques such as CT and PET. The case of integration along more general geodesics arises in geophysical imaging in determining the inner structure of the Earth since the speed of elastic waves generally increases with depth, thus curving the rays back to the Earth's surface. It also arises in ultrasound imaging, where the Riemannian metric models the anisotropic index of refraction. In tensor tomography problems one would like to determine a symmetric tensor field up to natural obstruction from its integrals over geodesics.The proposal aims to get further insight into the injectivity property of the geodesic ray transform acting on symmetric tensors by relating it to the existence of special solutions to the transport equation. This relationship has been crucial for recent successes in solving geometric inverse problems in two dimensions.