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

几何分析研究：卡尔德隆问题和流形上的微分系统

• 批准号: RGPIN-2019-04622
• 负责人: Kamran, Niky
• 金额: $ 3.64万
• 依托单位: McGill University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=715074
• 关键词: Studies; geometric; analysis; Calderon; problem

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.

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。该建议包括几个部分，其中一些来自我们最近的工作，其他部分侧重于新的方向：