Conformal Geometry and Asymptotically Hyperbolic Manifolds

共形几何和渐近双曲流形

• 批准号: 1308266
• 负责人: C. Robin Graham
• 金额: $ 18.9万
• 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: 2013
• 资助国家: 美国
• 起止时间: 2013-09-15 至 2017-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1308266&HistoricalAwards=false
• 关键词: Conformal; Geometry; Asymptotically; Hyperbolic; Manifolds

AbstractAward: DMS 1308266, Principal Investigator: C. Robin GrahamThe investigator will carry out several research projects studying various aspects of conformal geometry and asymptotically hyperbolic metrics. These include investigation of boundary rigidity for asymptotically hyperbolic metrics, investigation of minimal submanifolds of products of asymptotically hyperbolic spaces with compact manifolds, study of boundary terms for conformally invariant integrals, and study of twisted Dirac operators on asymptotically hyperbolic manifolds and relation to conformally invariant operators on the boundary. The main objectives are to further understanding of these geometries and their relationship. The methods are analytic, geometric, and algebraic, with an intimate connection between these different aspects of the study.This project will focus on the relationship between two different geometric structures: conformal geometry on the one hand and asymptotically hyperbolic geometry on the other. Conformal geometry is the study of properties of space which depend only on angles but not on distances. Hyperbolic geometry involves spaces of negative curvature, in which the analogues of straight lines separate more than in usual flat space. The asymptotic structure of hyperbolic geometry is related to conformal geometry on the lower dimensional boundary at infinity. Several projects will study the relationship between these geometries. Apart from the intrinsic geometric interest, one motivation is the AdS/CFT correspondence in physics, a proposed holographic correspondence for certain physical phenomena. In recent years, the AdS/CFT correspondence has been used to model many strongly coupled physical theories, for instance in condensed matter physics. The proposed activity will further enable the development of human resources through educationally oriented activities of the investigator, including advising, mentoring and teaching graduate students, and curricular development. International cooperation and partnership will be promoted through collaboration between the investigator and researchers in Japan and France. Ties between the mathematics and physics communities will be enhanced. The results will be effectively disseminated through attendance and speaking at meetings and conferences and through posting and publication of articles.

摘要奖项：DMS 1308266，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。 其中包括渐近双曲度量的边界刚性研究、渐近双曲空间与紧流形乘积的最小子流形研究、共形不变积分的边界项研究、渐近双曲流形上的扭曲狄拉克算子及其与共形不变算子的关系研究在边界上。 主要目标是进一步了解这些几何形状及其关系。 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。该项目将重点关注两种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。 共形几何是对仅取决于角度而不取决于距离的空间属性的研究。 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分开得更多。 双曲几何的渐近结构与无穷远低维边界上的共形几何有关。 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，这是针对某些物理现象提出的全息对应。 近年来，AdS/CFT 对应关系已被用来模拟许多强耦合物理理论，例如凝聚态物理。 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发展，包括为研究生提供建议、指导和教学以及课程开发。 通过日本和法国的研究者和研究人员之间的合作，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。 研究结果将通过出席会议和会议并发表讲话以及张贴和发表文章的方式得到有效传播。