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

我们的建议在于域的几何分析，数学领域中的重要问题，有关几何结构的研究使用强大的工具，从数学分析和偏微分方程。该建议包括几个组成部分，一些来自我们最近的工作，与其他专注于新的方向：**-卡尔德龙问题：卡尔德龙问题是一个反问题的重大当前研究兴趣，这是恢复几何的一个紧凑的黎曼流形的边界从知识的狄利克雷到诺依曼映射在固定的能量。这个问题的几个重要的唯一性的结果已经获得了在过去的15年的假设，使人们能够使用强大的工具，如限制Carleman权重。我们最近发现了不同的方法，一系列意想不到的非唯一性结果的情况下，光滑复曲面圆柱进行翘曲的产品指标的Calderon问题。我们将通过研究具有多个端点的流形的更一般的设置中的卡尔德龙问题来实现这个计划，该流形被赋予光滑的共形Painleve度量。我们还将考虑逆Steklov问题的稳定性的相关问题。**-Myers-Perry几何中的局部能量衰减：Myers-Perry几何是4d爱因斯坦方程的著名克尔解的n维类似物，它描述了处于平衡状态的旋转黑洞的外部时空几何。虽然克尔中狄拉克旋量的分析性质已经被广泛研究，但对于迈尔斯-佩里度规，诸如狄拉克旋量的局部能量衰减和长期行为等自然问题还没有理解到超过n=5。我们建议在外部Myers-Perry几何及其在事件视界上的解析扩展中研究这些问题。*-一个分支Cartan-Kaehler定理：Cartan-Kaehler定理是对合解析外微分系统的积分流形的主要存在定理。它有着广泛的几何应用，从局部等距浸入的存在到具有特殊完整性的度量。它的证明关键在于经典的柯西-科瓦列夫斯卡亚定理。我们将使用勒雷的分歧版本的后者，以延长Cartan-Kaehler定理的设置中的秩条件对合放松，从而产生分歧积分流形。这反过来将大大扩展Cartan-Kaehler定理的几何应用领域。齐次辛流形的辛上同调：Tseng和Yau最近研究了Hodge理论到辛环境的推广，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，并在齐次辛流形的情况下给出其李代数描述。

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.**