On the Behavior of Solutions of Einstein's Equations and Other Geometric Nonlinear Partial Differential Equation Systems

随着支持爱因斯坦广义相对论作为天体物理学和宇宙学中引力场卓越模型的观测证据变得越来越强大，使用最新的数学工具来仔细探索该理论的含义变得越来越重要。该奖项支持将重点放在构建时空的新方法上，以表示许多物体的相互作用，并支持新的技术，使我们能够探索爱因斯坦理论所预测的宇宙学奇点的本质。该奖项还支持研究Ricci流的各个方面的工作，Ricci流是一种分析工具，已被证明在三维空间中证明了庞加莱猜想和几何化猜想，取得了惊人的成功。要理解Ricci流动力学的细节，以及如何利用它来进一步分析几何和拓扑之间的关系，还有很多工作要做，这项建议支持这些工作，包括一些关于Ricci流稳定性的研究。一名研究生将作为博士论文的一部分参与支持的研究。

As the observational evidence supporting Einstein's theory of general relativity as the pre-eminent model for the gravitational field in astrophysics and cosmology grows stronger, it becomes increasingly important to use the latest mathematical tools to carefully explore the implications of the theory. This award supports a focus on new procedures for constructing spacetimes representing the the interaction of many bodies, and also on new techniques which allow us to probe the nature of the cosmological singularities predicted by Einstein's theory.This award also supports work which studies various aspects of Ricci flow, the analytical tool which has proven to be spectacularly successful in proving the Poincare Conjecture and the Geometrization Conjecture in three dimensions. There is much work to do in understanding the details of the dynamics of Ricci flow, and how it might be used to further analyze the relationship between geometry and topology, and this proposal supports such work, including a number of studies of Ricci flow stability.A graduate student will participate in the supported research as part of a PhD dissertation.