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

爱因斯坦的引力场理论(广义相对论)不仅为天体物理和宇宙尺度上的引力物理研究提供了最准确的当前模型，也是大量数学有趣问题的来源。该奖项将支持一项专注于其中一些研究的计划，包括：1)在爱因斯坦方程的宇宙学解中研究非常接近大爆炸的引力场的行为。值得注意的是，似乎即使对于在大爆炸后的某个时间看起来非常不同的解决方案，大爆炸附近的行为也非常相似：它以一种特有的方式振荡。对于简单的解族，我们可以用数值或解析的方法来说明这一点，我们希望能更准确地说明这一点。2)构造与两个已知集合并列的引力场的新初始条件。这并不容易，因为初始条件必须满足某些非线性偏微分方程式--爱因斯坦“约束”。我们正在开发将约束的两个解决方案顺利地连接(或“粘合”成一个解决方案的技术。第一个项目，尽管它忽略了任何量子引力效应，但在我们理解非常早期的宇宙的性质方面可能非常有用。第二个问题在数学上和物理上都很有趣。对数学的兴趣源于这样一个事实，即爱因斯坦约束系统是胶合研究中应用最复杂的系统之一。物理兴趣来自于使用发展的技术建立黑洞碰撞和其他天体物理问题的初始数据的可能性，这些问题正在进行密集的数值研究，以期待来自LIGO的未来数据。

Einstein's gravitational field theory (general relativity), besidesproviding the most accurate current model for the study of gravitationalphysics on the astrophysical and cosmological scales, is also the source ofa rich collection of mathematically interesting questions. This award willsupport a program of research which focusses on some of these, includingthe following:1) Study of the behavior of the gravitational field very close to the bigbang in cosmological solutions of Einstein's equations. Remarkably, itseems that even for solutions which appear to be very different some timeafter the big bang, the behavior close to the big bang is very similar: Itoscillates in a characteristic way. We can show this, numerically oranalytically, for simple families of solutions, and we hope to show it moregenerally.2) Construction of new sets of initial conditions for the gravitationalfield which juxtapose two known sets. This is not easy because the initialconditions must satisfy certain nonlinear partial differentialequations--the Einstein "constraints". We are developing techniques forsmoothly joining ( or "gluing") two solutions of the constraints into asingle solution.The first of these projects, although it ignores any quantum gravitationaleffects, could be very useful in our drive to understand the nature of thevery early universe. The second is of interest both mathematically andphysically. The mathematical interest stems from the fact that the Einsteinconstraint system is among the most complicated to which gluing studieshave been applied. The physical interest comes from the possiblility ofusing the developed techniques to set up initial data for black holecollisions and other astrophysical problems which are being intenselystudied numerically in anticipation of future data from LIGO.