General Relativity and Geometric Hyperbolic PDEs

广义相对论和几何双曲偏微分方程拟议研究摘要Igor Rodnianski本研究项目的重点是研究在广义相对论中产生的几何双曲偏微分方程和量子场论和规范理论中产生的其他方程的解的局部和全局行为。其中一个项目是研究爱因斯坦方程的完整非线性系统在弱引力波区域的解。我们将继续研究L2曲率猜想，断言爱因斯坦真空方程Ric（g）= 0的解，度量g，只要其曲率张量在L2中有界，就可以局部延拓。我们还将研究闵可夫斯基空间的稳定性问题的重力耦合到物质场的模型。另一个研究方向是继续对强引力场中的线性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。我们将专注于这些模型的强大的分析方法的发展，而不需要任何完整的可积性假设。我们将研究无限维动力学之间的连接，由上述模型，测地线运动引起的有限维模空间的静态解决方案与这些方程。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。对广义相对论中出现的数学和物理现象的研究对宇宙学具有根本的重要性。虽然物理学对这一问题的理解已经取得了迅速的进展，并产生了许多杰出的成果，但迄今为止，严格的数学结果相对较少。在很大程度上，这是由于爱因斯坦方程的高度非线性性质以及缺乏分析它们的数学工具。这个项目将追求这样的数学工具和结果的发展，并涉及许多数学问题的分析，几何和偏微分方程的接口。

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.