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

我们的建议属于几何分析领域，这是一个数学领域，其中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。该提案由几个部分组成，其中一些部分源于我们最近的工作，其他部分则侧重于新的方向： -卡尔德隆问题：卡尔德隆问题是当前研究中非常感兴趣的一个反问题，它是根据固定能量下的狄利克雷到诺依曼映射的知识来恢复具有边界的紧致黎曼流形的几何形状。在过去 15 年中，人们在假设下获得了针对该问题的几个重要的唯一性结果，这些假设使人们能够使用限制卡尔曼权重等强大的工具。我们最近通过不同的方法发现了在带有扭曲产品指标的光滑复曲面圆柱体的情况下卡尔德隆问题的一系列意想不到的非唯一性结果。我们将通过在更一般的多端流形设置中研究卡尔德隆问题来实现这一计划，并赋予平滑的共形 Painleve 度量。我们还将考虑逆 Steklov 问题的稳定性相关问题。 -迈尔斯-佩里几何中的局部能量衰变：迈尔斯-佩里几何是著名的 4d 爱因斯坦方程克尔解的 n 维类似物，它描述了平衡状态下旋转黑洞的外部时空几何。虽然克尔的狄拉克旋量的分析性质已被广泛研究，但对于迈尔斯-佩里度量，诸如局部能量衰减和狄拉克旋量的长期行为等自然问题尚未得到理解。我们建议研究外部迈尔斯-佩里几何中的这些问题及其跨事件视界的分析扩展。 - 分支的嘉当-凯勒定理：嘉当-凯勒定理是对合解析外微分系统积分流形的主要存在定理。它具有广泛的几何应用，从局部等距沉浸的存在到具有特殊完整度的度量。它的证明主要依赖于经典的柯西-科瓦列夫斯卡亚定理。我们将使用后者的勒雷的分支版本，以便将嘉当-凯勒定理扩展到放宽对合性的秩条件的设置，从而产生分支积分流形。这反过来将显着扩展嘉当-凯勒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 理论对辛环境的推广，并研究了相应的上同调。我们将研究这些上同调在辛约简下的行为，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.