General Relativity and Geometric Hyperbolic PDEs

广义相对论与几何双曲型偏微分方程组拟议研究的摘要伊戈尔·罗德尼安斯基本研究项目的重点是研究广义相对论中几何双曲型偏微分方程解的局部和整体行为，以及量子场论和规范理论中的其他方程。其中一个项目是研究弱引力波条件下爱因斯坦方程组的解。我们将继续我们关于L2曲率猜想的工作，断言只要爱因斯坦真空方程Ric(G)=0的曲率张量在L2中有界，则它的解，度量g可以局部扩张。我们还将研究重力与物质场耦合的模型的Minkowski空间的稳定性问题。另一个研究方向是继续对强引力场中的线性波和非线性波进行严格的数学研究。该项目还将研究与量子场论和规范理论模型相关的经典场论中拓扑孤子的稳定性问题。我们将专注于为这些模型开发稳健的分析方法，而不需要任何完全的可积性假设。我们将研究由上述模型产生的无限维动力学和在与这些方程相关的静态解的有限维模空间上诱导的测地线运动之间的联系。广义相对论的爱因斯坦方程提供了物理时空连续体演化的主要经典描述。对产生于广义相对论的数学和物理现象的研究对宇宙学具有根本的重要性。虽然对这一主题的物理理解已经取得了快速的进步，并产生了一些杰出的猜想；但到目前为止，严格的数学结果相对较少。这在很大程度上是由于爱因斯坦方程的高度非线性，以及缺乏分析它们的数学工具。这个项目将致力于开发这样的数学工具和结果，并在分析、几何和偏微分方程式的界面上涉及许多数学问题。

General Relativity and Geometric Hyperbolic PDEsAbstract of Proposed ResearchIgor Rodnianski The focus of this research project is the study of the local and global behavior of solutions of geometric hyperbolic PDEs arising in General Relativity and other equations arising in Quantum Field Theory and gauge theories. One project is to study the solutions of the full nonlinear system of Einstein equations in a weak gravitational wave regime. We will continue our work on the L2 curvature conjecture, asserting that a solution, metric g, of the Einstein vacuum equations Ric(g) = 0 can be locally extended whenever its curvature tensor is bounded in L2 . We will also study problem of stability of Minkowski space for models where gravity is coupled to matter fields. Another direction of investigation is to continue rigorous mathematical study of linear and nonlinear waves in the regime of strong gravitational fields. The project will also investigate problems of stability of topological solitons in classical field theories connected with models of Quantum Field Theory and Gauge Theory. We will concentrate on development of robust analysis methods for these models without requiring any complete integrability assumptions. We will investigate connections between infinite dimensional dynamics, generated by the above models, and geodesic motion induced on finite dimensional moduli spaces of static solutions associated with these equations. The Einstein equations of General Relativity provide the main classical description of evolution of the physical space-time continuum. The study of mathematical and physical phenomena arising in General Relativity is of fundamental importance for cosmology. While the physical understanding of the subject has made rapid advancement and generated a number of outstanding conjectures; so far there have been relatively few rigorous mathematical results. To a large extent, this is due to the highly nonlinear nature of the Einstein equations and a lack of mathematical tools for analyzing them. This project will pursue the development of such mathematical tools and results and involves many mathematical issues at the interface of Analysis, Geometry and Partial Differential Equations.