Geometric and Analytic Aspects of Einstein Metrics

摘要奖：DMS 1205947，首席研究员：迈克尔·T·安德森这个项目涉及爱因斯坦方程的几何、解析和物理方面的几项研究，以及在相关领域的应用。特别令人感兴趣的是爱因斯坦度规边值问题的整体问题，包括在经典微分几何中的应用，如等距嵌入问题和空间形式中最小和常平均曲率曲面的研究。还将对广义相对论中的几个主题进行研究，包括准局域质量和巴特尼克静态扩张猜想的研究。与ADS/CFT通信中的重整化群流相关的研究也将在一个与弦理论物理学家的联合项目中进行。爱因斯坦方程长期以来一直是数学家和物理学家研究和感兴趣的中心焦点。它们在数学方面非常重要，因为它们在几何分析和偏微分方程式领域处于知识和研究的前沿。在物理方面，它们支配着我们对大范围物理学--恒星、星系的形成和整个宇宙的结构--的理解。作为一个具体的应用，如果没有对爱因斯坦方程的透彻和充分的理解，GPS是不可能的。此外，它们是弦理论的核心，弦理论是高能理论物理中最活跃的研究课题。该项目将涉及数学家和物理学家之间的合作和互动，试图解开这些方程的一些奥秘。此外，该项目还涉及对这些领域的研究生进行培训，这些领域对基础研究的未来非常重要。

AbstractAward: DMS 1205947, Principal Investigator: Michael T. AndersonThis project concerns several studies in the geometric, analytic and physical aspects of the Einstein equations and applications to related areas. Of particular interest are global issues for boundary value problems for Einstein metrics, including applications to classical differential geometry such as the isometric embedding problem and the study of minimal and constant mean curvature surfaces in space forms. Research will also be carried out on several topics in general relativity, including the study of quasi-local mass and the Bartnik static extension conjecture. Research related to the renormalization group flow in the AdS/CFT correspondence will also be undertaken in a joint project with physicists working in string theory.The Einstein equations have long been a central focus of study and interest to mathematicians and physicists. They are very important on the mathematical side since they are at the forefront of knowledge and research in the areas of geometric analysis and partial differential equations. On the physical side, they govern our understanding of large-scale physics - the formation of stars, galaxies and the structure of the universe as a whole. As a concrete application, GPS would not be possible without a thorough and full understanding of the Einstein equations. In addition, they lie at the core of string theory - the most actively studied subject in high energy theoretical physics. The project will involve collaboration and interaction between mathematicians and physicists seeking to unravel some of the mysteries of these equations. In addition, the project involves the training of graduate students in these areas important for the future of basic research.