International Conference on Mathematical Relativity

该奖项支持参加2018年12月14日至16日在佛罗里达珊瑚山墙举行的“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，它描述了在物质存在的情况下，时空宇宙如何弯曲;正是这种曲率导致了引力的影响。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（现在著名的是由LIGO探测到的），并支配着整个宇宙的大尺度行为。 这次会议的目的是聚集在数学广义相对论和几何分析的交叉点的领先专家，讨论这一领域的最新进展和新的研究方向，并使研究生和年轻的数学家接触这些发展。 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发挥了显著的作用。 在椭圆和双曲两个方面的理论，包括黑洞理论的发展，彭罗斯宇宙审查猜想的理解，以及非负标量曲率流形的研究，都有了许多进展。 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，引发了许多有趣的问题 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组，其几何起源是高斯-科达齐方程。 用保角法和胶合法求解约束方程的技术已经取得了很大的进展。 边缘外捕获面理论是极小曲面的一般初始数据版本，它的发展导致了关于黑洞（以及黑洞外部区域）拓扑的结果，并使得在小于8维的时空中直接证明正质量定理成为可能。最近证明的（黎曼）正质量定理（在所有维度和没有自旋假设）有很大的进步知识的几何奇点及其作用的初始数据集。 为了获得局部质量的测量，其中包括来自引力场的贡献，已经导致了各种准局部质量的概念，并且仍然是一个非常活跃的研究领域。几何流动在许多这些发展中发挥了根本作用。会议将讨论所有这些问题。该计划的详细信息可在该计划的网页：http://www.math.miami.edu/gg70This奖反映了NSF的法定使命，并已被认为值得通过评估使用基金会的知识价值和更广泛的影响审查标准的支持。

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.