Conformal Geometry and Asymptotically Hyperbolic Metrics

摘要奖：DMS-0505701主要研究者：C.罗宾格雷厄姆调查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。 这些工作包括分析Poincare-Einstein度量的Dirichlet-to-Neumann映射，研究渐近双曲度量的边界刚性问题，以及发展超越障碍物的偶数维环境度量理论。 主要目标是进一步理解这些几何及其关系。方法是解析的，几何的，代数的，这些研究的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：一方面是共形几何，另一方面是渐近双曲几何。 共形几何是研究只依赖于角度而不依赖于距离的空间性质的学科。 双曲几何涉及负曲率的空间，其中直线的类似物比通常的平坦空间分离得更多。 几个项目将研究这些几何形状之间的关系。 除了内在的几何兴趣，一个动机是物理学中的AdS/CFT对应，一种针对某些物理现象的全息对应。 拟议的活动将通过调查员的教育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。 将促进国际合作和伙伴关系。 数学界和物理界之间的联系将得到加强。 将通过出席各种会议和在会议上发言以及通过张贴和发表文章有效地传播成果。

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.