Conformal Geometry and Asymptotically Hyperbolic Metrics

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。研究员将进行几个研究项目，研究共形几何和渐近双曲度量的各个方面。 其中包括分析渐近双曲度量的边界刚性，研究渐近双曲流形上的扭曲Dirac算子与无穷远处边界上的共形不变对象之间的关系，以及分析由重整化体积系数定义的完全非线性方程. 主要目的是进一步了解这些几何形状及其关系。 该方法是分析，几何和代数，与这些不同方面的研究之间的密切联系。这个项目将集中在两个不同的geometricstructures之间的关系：保形几何一方面和渐近双曲几何另一方面。 共形几何是研究空间性质的学科，它只依赖于角度而不依赖于距离。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离得更多。 双曲几何的渐近结构与无穷远处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，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，包括向研究生提供咨询、指导和教学以及课程编制，进一步促进人力资源的开发。 将通过调查员与日本和法国的研究人员之间的协作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上发言以及通过张贴和发表文章来有效传播成果。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The investigator will carry out several research projects studying various aspects of conformal geometry and asymptotically hyperbolic metrics. These include analyzing boundary rigidity of asymptotically hyperbolic metrics, studying the relation between twisted Dirac operators on asymptotically hyperbolic manifolds and conformally invariant objects on their boundaries at infinity, and analyzing fully nonlinear equations defined by renormalized volume coefficients. 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 geometricstructures: 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. 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.