Conformal Geometry and Asymptotically Hyperbolic Metrics

共形几何和渐近双曲度量

• 批准号: 0505701
• 负责人: C. Robin Graham
• 金额: --
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-06-01 至 2009-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0505701&HistoricalAwards=false
• 关键词: Conformal; Geometry; Asymptotically; Hyperbolic; Metrics

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.

摘要奖项：DMS-0505701 首席研究员：C. Robin Graham 该研究员将开展多个研究项目，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-爱因斯坦度量的狄利克雷到诺依曼映射，研究渐近双曲度量的边界刚性问题，以及发展障碍物之外偶维的环境度量理论。 主要目标是进一步理解这些几何及其关系。这些方法包括解析、几何和代数，这些研究的不同方面之间有着密切的联系。该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。 共形几何是对仅依赖于角度而不依赖于距离的空间特性的研究。 双曲几何涉及负曲率空间，其中直线的类似物比通常的平坦空间中的间隔更大。 有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之外，一个动机是物理学中的 AdS/CFT 对应关系，这是针对某些物理现象提出的全息对应关系。 拟议的活动将通过研究者以教育为导向的活动进一步促进人力资源的开发，包括建议、指导和教学研究生以及课程开发。 将促进国际合作和伙伴关系。 数学界和物理学界之间的联系将得到加强。 结果将通过出席会议和会议并发表演讲以及张贴和发表文章的方式得到有效传播。