International Conference on Mathematical Relativity

国际数学相对论会议

• 批准号: 1856467
• 负责人: Pengzi Miao
• 金额: $ 2.99万
• 依托单位: University of Miami
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1856467&HistoricalAwards=false
• 关键词: International; Conference; Mathematical; Relativity

This award supports participation in the conference "A Celebration of Mathematical Relativity in Miami" held in Coral Gables, Florida, during December 14 - 16, 2018. The Einstein equations, which are the field equations of general relativity, describe how spacetime universe curves in the presence of matter; it is this curvature that is responsible for the effects of gravity. General relativity is a remarkably accurate theory, which describes the formation of black holes, predicts the existence of gravitational waves (now famously detected by LIGO), and governs the large-scale behavior of the entire cosmos. The aim of this conference is to gather together leading experts at the intersection of mathematical general relativity and geometric analysis to discuss recent advances and new directions of research in this area, and to expose graduate students and young mathematicians to these developments. Mathematical general relativity is a very rapidly developing area of research. Geometric analysis has played a remarkable role in this development, beginning with the singularity theorems of Hawking and Penrose. Many advances have ensued in both the elliptic and hyperbolic aspects of the theory, including developments in the theory of black holes, in understanding of Penrose's cosmic censorship conjecture, and in the study of manifolds of nonnegative scalar curvature. Geometric inequalities, such as the positive mass theorem and the Riemannian Penrose inequality, play a deep and fundamental role in general relativity. In recent years new geometric inequalities, involving physical quantities such as mass, angular momentum, charge, etc., have given rise to many interesting open questions. Studies of initial data sets for gravitational fields have proven to be fruitful areas of research. The initial data for the Cauchy problem associated to the Einstein field equations are required to satisfy the so-called constraint equations, a system of nonlinear partial differential equations, the geometric origin of which are the Gauss-Codazzi equations. Great progress has been made in developing techniques for solving the constraint equations, both by the conformal method and by gluing methods. The development of the theory of marginally outer trapped surfaces, which are general initial data versions of minimal surfaces, has led to results concerning the topology of black holes (and the region exterior to black holes), and has made possible a direct proof of the spacetime positive mass theorem in dimensions less than eight. The recent proof of the (Riemannian) positive mass theorem (in all dimensions and without spin assumption) has greatly advanced knowledge about geometric singularities and their roles in initial data sets. Efforts to obtain a localized measure of mass, which includes a contribution from the gravitational field, have led to various notions of quasi-local mass, and continues to be a very active area of research. Geometric flows have played a fundamental role in many of these developments. The conference will address all these topics. Details of the program are available at the program webpage: http://www.math.miami.edu/gg70This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该奖项支持参加2018年12月14日至16日在佛罗里达州珊瑚盖布尔斯举行的“迈阿密数学相对论庆典”会议。爱因斯坦方程是广义相对论的场方程，它描述了时空宇宙在物质存在时如何弯曲；正是这种曲率造成了重力的影响。广义相对论是一个非常精确的理论，它描述了黑洞的形成，预测了引力波的存在（现在著名的LIGO探测到了），并控制了整个宇宙的大规模行为。本次会议的目的是聚集数学广义相对论和几何分析交叉领域的顶尖专家，讨论这一领域的最新进展和新的研究方向，并让研究生和年轻数学家了解这些发展。数学广义相对论是一个发展非常迅速的研究领域。从霍金和彭罗斯的奇点定理开始，几何分析在这一发展中起了显著的作用。该理论在椭圆和双曲方面都取得了许多进展，包括黑洞理论的发展，对彭罗斯宇宙审查猜想的理解，以及对非负标量曲率流形的研究。几何不等式，如正质量定理和黎曼彭罗斯不等式，在广义相对论中起着深刻而基础的作用。近年来，新的几何不等式，涉及物理量，如质量，角动量，电荷等，引起了许多有趣的开放性问题。对引力场初始数据集的研究已被证明是富有成果的研究领域。与爱因斯坦场方程相关的柯西问题的初始数据需要满足所谓的约束方程，即非线性偏微分方程系统，其几何起源是高斯-科达齐方程。用保角法和粘接法求解约束方程的技术已经取得了很大的进展。边际外捕获面理论的发展，是最小表面的一般初始数据版本，已经导致了有关黑洞拓扑（以及黑洞外部区域）的结果，并使时空正质量定理在小于8维的直接证明成为可能。最近（黎曼）正质量定理（在所有维度和不含自旋假设）的证明大大提高了关于几何奇点及其在初始数据集中的作用的知识。对局部质量测量的努力，其中包括引力场的贡献，已经导致了准局部质量的各种概念，并且仍然是一个非常活跃的研究领域。几何流动在这些发展中发挥了重要作用。会议将讨论所有这些议题。该计划的详细信息可在计划网页上查阅：http://www.math.miami.edu/gg70This该奖项反映了美国国家科学基金会的法定使命，并通过基金会的智力价值和更广泛的影响审查标准进行评估，认为值得支持。