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

我们的建议涉及几何分析领域，这是一个使用数学分析和偏微分方程式的强大工具来研究有关几何结构的重要问题的数学领域。该提案包括几个组成部分，其中一些源于我们最近的工作，另一些则侧重于新的方向： -Calderon问题：Calderon问题是当前研究中非常感兴趣的反问题，它是从固定能量下的Dirichlet-to-Neumann映射的知识恢复具有边界的紧致黎曼流形的几何。在过去的15年里，人们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强大的工具，如限制Carleman权。最近，我们用不同的方法发现了光滑圆柱体带有翘曲乘积度量的Calderon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流形环境中研究Calderon问题来继续这个程序，流形有几个末端，赋予光滑的共形Painleve度量。我们还将考虑Steklov逆问题的相关稳定性问题。 -Myers-Perry几何中的局部能量衰减：Myers-Perry几何是著名的4D爱因斯坦方程的Kerr解的n维类似，该方程描述了处于平衡状态的旋转黑洞的外层时空几何。虽然人们已经广泛地研究了克尔中狄拉克旋量的分析性质，但对于Myers-Perry度规，除了n=5之外，自然问题，如狄拉克旋量的局域能量衰减和长期行为，还没有被理解。我们建议在外部Myers-Perry几何和它们在视界上的解析扩张中研究这些问题。 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定理。它具有广泛的几何应用，从局部等距浸没的存在到具有特殊完整的度规的存在。它的证明依赖于经典的柯西-科瓦列夫斯卡亚定理。我们将使用后者的Leray分支版本，以便将Cartan-Kaehler定理推广到放宽对合的秩条件的情形，从而产生分支积分流形。这反过来将极大地扩展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理论在辛集上的推广，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它的李代数描述。

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.