Geometric and Analytic Aspects of Einstein Metrics

摘要奖：DMS 1205947，主要研究者：Michael T.安德森这个项目涉及爱因斯坦方程的几何、分析和物理方面的几项研究及其在相关领域的应用。特别感兴趣的是爱因斯坦度量的边值问题的全球性问题，包括经典微分几何的应用，如等距嵌入问题和空间形式中的最小和常数平均曲率曲面的研究。还将对广义相对论的几个专题进行研究，包括准局部质量和Bartnik静态扩展猜想的研究。与AdS/CFT对应中的重整化群流相关的研究也将在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。他们是非常重要的数学方面，因为他们是最前沿的知识和研究领域的几何分析和偏微分方程。在物理方面，它们支配着我们对大尺度物理学的理解--恒星、星系的形成以及整个宇宙的结构。作为一种具体的应用，如果不对爱因斯坦方程有一个彻底和充分的了解，全球定位系统是不可能实现的。此外，它们也是弦理论的核心，而弦理论是高能理论物理学中研究最活跃的课题。 该项目将涉及数学家和物理学家之间的合作和互动，以解开这些方程的一些谜团。此外，该项目还涉及在这些对未来基础研究很重要的领域培训研究生。

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.