Studies in geometric analysis: the Calderon problem and differential systems on manifolds

我们的建议位于几何分析领域，这是一个数学领域，使用数学分析和偏微分方程的强大工具来研究有关几何结构的重要问题。该建议包括几个组成部分，一些源于我们最近的工作，与其他集中在新的方向：******-卡尔德隆问题：卡尔德隆问题是一个重要的逆问题，目前的研究兴趣，这是恢复几何紧凑黎曼流形的边界从狄利克雷到诺伊曼映射的知识在固定能量。在过去的15年里，在一些假设下得到了这个问题的几个重要的唯一性结果，这些假设使得人们能够使用诸如限制Carleman权等强大的工具。最近，我们用不同的方法发现了带有翘曲积指标的光滑环面圆柱体的Calderon问题的一系列意想不到的非唯一性结果。我们将通过研究具有光滑保形Painleve度量的具有多个端点的流形的更一般的设置中的Calderon问题来继续这个程序。我们还将考虑逆Steklov问题的相关稳定性问题。******-迈尔斯-佩里几何中的局部能量衰减：迈尔斯-佩里几何是著名的四维爱因斯坦方程克尔解的n维类似物，它描述了处于平衡状态的旋转黑洞的外层时空几何。虽然Kerr中狄拉克旋量的分析性质已经得到了广泛的研究，但诸如狄拉克旋量的局部能量衰减和长期行为等自然问题尚未被理解为n=5以外的Myers-Perry度量。我们建议在外部迈尔斯-佩里几何及其跨视界的解析扩展中研究这些问题。*** ***-派生的Cartan-Kaehler定理：Cartan-Kaehler定理是对合解析外微分系统积分流形的主要存在性定理。它具有广泛的几何应用，从局部等距浸入的存在到具有特殊完整度的度量。它的证明关键依赖于经典的柯西-科瓦列夫定理。为了将Cartan-Kaehler定理推广到对合性的秩条件松弛的情况下，我们将使用Leray定理的分支版本，从而得到分支积分流形。这将进一步扩展卡坦-凯勒定理在几何领域的应用。******-齐次辛流形的辛上同调：Tseng和Yau最近研究了Hodge理论在辛环境中的推广，并研究了相应的上同调。我们将研究这些上同调在辛约化下的行为，并给出齐次辛流形的李代数描述

Our proposal lies in the domain of geometric analysis, an area of mathematics in which important questions concerning geometric structures are investigated using powerful tools from mathematical analysis and partial differential equations. The proposal comprises several components, some stemming from our recent work, with others focusing on new directions:******-The Calderon problem: The Calderon problem is an inverse problem of significant current interest in research, which is to recover the geometry of a compact Riemannian manifold with boundary from the knowledge of the Dirichlet-to-Neumann map at fixed energy. Several important uniqueness results for this problem have been obtained in the last 15 years under hypotheses that make enable one to use powerful tools such as limiting Carleman weights. We have recently discovered by different methods a series of unexpected non-uniqueness results for the Calderon problem in the case of smooth toric cylinders carrying warped product metrics. We shall pursue this program by studying the Calderon problem in the much more general setting of manifolds with several ends, endowed with smooth conformally Painleve metrics. We shall also consider the related question of stability for the inverse Steklov problem. ******-Local energy decay in Myers-Perry geometries: The Myers-Perry geometries are the n-dimensional analogues of the well-known Kerr solution of the 4d Einstein equations, which describes the outer space-time geometry of a rotating black hole in equilibrium. While the analytical properties of Dirac spinors in Kerr have been extensively studied, natural questions such as the local energy decay and long-term behaviour of Dirac spinors have not yet understood beyond n=5 for the Myers-Perry metrics. We propose to investigate these problems in the exterior Myers-Perry geometries and their analytic extensions across the event horizon.*** ***- A ramified Cartan-Kaehler Theorem: The Cartan-Kaehler Theorem is the main existence theorem for integral manifolds of involutive analytic exterior differential systems. It has a wide range of geometric applications, from the existence of local isometric immersions to that of metrics with exceptional holonomy. Its proof rests crucially on the classical Cauchy-Kovalevskaia Theorem. We shall use Leray's ramified version of the latter in order to extend the Cartan-Kaehler Theorem to a setting in which the rank conditions for involutivity are relaxed, giving rise to ramified integral manifolds. This will in turn significantly expand the realm of geometric applications of the Cartan-Kaehler Theorem.******- Symplectic cohomologies for homogeneous symplectic manifolds: Tseng and Yau have recently studied a generalization of Hodge theory to the symplectic setting and have investigated the corresponding cohomologies. We shall study the behaviour of these cohomologies under symplectic reduction, and give a Lie-algebraic description thereof in the case of homogeneous symplectic manifolds.**