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.

Abstractaward：DMS 1308266，主要研究者：C。Robin Grahamthe研究者将进行几个研究项目，研究整形几何和渐近双曲线指标的各个方面。 其中包括研究渐近夸张指标的边界刚度，研究具有紧凑型歧管的渐近肥大空间的最小亚货物，对串联不变积分的边界项的研究以及对扭曲的迪拉克运营商对差异的毛茸茸的操作员在渐近倍增副歧管和共汇合不可分解的群体上的差异。 主要目标是进一步了解这些几何形状及其关系。 这些方法是分析性，几何和代数，在研究的这些不同方面之间具有紧密的联系。该项目将重点关注两种不同的几何结构之间的关系：一方面的保形几何形状与另一方面渐近夸张的几何形状。 共形几何形状是对仅取决于角度但不取决于距离的空间特性的研究。 双曲线几何形状涉及负曲率的空间，其中直线的类似物比通常的平坦空间更分开。 双曲几何形状的渐近结构与无穷大的下维边界上的共形几何形状有关。 几个项目将研究这些几何之间的关系。 除固有的几何兴趣外，一个动机是物理学中的AD/CFT对应关系，这是某些物理现象的拟议全息相应。 近年来，AD/CFT对应关系已用于模拟许多强烈耦合的物理理论，例如在凝结物理学中。 拟议的活动将通过调查员的以教育为导向的活动进一步使人力资源的发展，包括为研究生提供建议，指导和教学课程。 国际合作和合作伙伴关系将通过日本和法国的研究人员与研究人员之间的合作来促进。 数学与物理社区之间的联系将得到增强。 结果将通过出席和在会议和会议上以及通过文章发布和发表来有效地传播。