Conformal Geometry and Asymptotically Hyperbolic Metrics

摘要奖：DMS-0505701首席研究员：C·罗宾·格拉汉姆该研究员将开展几个研究项目，研究共形几何和渐近双曲度量的各个方面。其中包括分析庞加莱-爱因斯坦度规的Dirichlet-to-Neumann映射，研究渐近双曲度规的边界刚性问题，以及发展障碍物以外偶数维的环境度规理论。主要目的是为了进一步了解这些几何及其关系。这些方法是解析、几何和代数的，研究的这些不同方面之间有着密切的联系。这个项目将研究不同几何结构之间的关系：一方面是保角几何，另一方面是渐近双曲几何。保形几何研究的是空间的性质，它只依赖于角，而不依赖于距离。双曲几何涉及负曲率的空间，其中直线的模拟比通常的平坦空间中的距离更大。有几个项目将研究这些几何图形之间的关系。除了内在的几何兴趣，一个动机是物理中的ADS/CFT对应，这是对某些物理现象提出的全息对应。拟议的活动将通过调查员以教育为导向的活动进一步促进人力资源的开发，包括为研究生提供咨询、指导和教学以及课程开发。促进国际合作与伙伴关系。数学界和物理界之间的联系将得到加强。将通过出席和在各种会议上发言以及张贴和发表文章来有效地传播成果。

AbstractAward: DMS-0505701Principal Investigator: C. Robin GrahamThe investigator will carry out several research projectsstudying various aspects of conformal geometry and asymptoticallyhyperbolic metrics. These include analyzing aDirichlet-to-Neumann map for Poincare-Einstein metrics, studyingthe boundary rigidity problem for asymptotically hyperbolicmetrics, and developing the theory of the ambient metric in evendimensions beyond the obstruction. The main objectives are tofurther understanding of these geometries and their relationship.The methods are analytic, geometric, and algebraic, with anintimate connection between these different aspects of the study.This project will study the relationship between differentgeometric structures: conformal geometry on the one hand andasymptotically hyperbolic geometry on the other. Conformalgeometry is the study of properties of space which depend only onangles but not on distances. Hyperbolic geometry involves spacesof negative curvature, in which the analogues of straight linesseparate more than in usual flat space. Several projects willstudy the relationship between these geometries. Apart from theintrinsic geometric interest, one motivation is the AdS/CFTcorrespondence in Physics, a proposed holographic correspondencefor certain physical phenomena. The proposed activity willfurther enable the development of human resources througheducationally oriented activities of the investigator, includingadvising, mentoring and teaching graduate students, andcurricular development. International cooperation andpartnership will be promoted. Ties between the mathematics andphysics communities will be enhanced. The results will beeffectively disseminated through attendance and speaking atmeetings and conferences and through posting and publication ofarticles.