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

这个奖项的一个很好的部分是专门研究奇点的形成，无论是在爱因斯坦方程的解决方案和解决方案的里奇流。对于爱因斯坦的方程，一个关键问题是强宇宙审查（“SCC”）：霍金-彭罗斯定理预测的无处不在的测地不完备性一般涉及时空曲率（因此引力潮汐）爆破，这是真的吗？至少在某些由等距定义的时空族中，确定解表现出“AVTD”行为（奇点附近场的时间导数对空间导数的优势）是研究SCC的非常有用的工具，并且本身就很有趣。我们的一些工作涉及到一个框架的发展，证明有大量的光滑（非解析）的解决方案，在这些家庭，显示AVTD行为。我们对里奇流中奇点的研究主要集中在那些被称为“颈箍”的奇点上。我们，特别是，表明有一个离散参数化家庭的里奇流解决方案，开发“退化颈箍”，与几何渐近接近“布莱恩特孤子”在规定的速率（符合2型行为）。这些解都是旋转对称的。想要知道在这些解中看到的详细渐近行为是否也是非旋转对称解的特征，我们选择首先在平均曲率流（“MCF”）的情况下解决这个问题，因为MCF颈缩与Ricci流的颈缩共享许多特征，并且因为使用MCF更容易将非旋转对称解与旋转对称解进行比较。以及在我的研究项目中的许多其他项目中，主要目标是发展对复杂的非线性偏微分方程组的大类解的一般行为的理解。这样的方程系统，发挥了重要作用，在我们的建模中的物理现象，在地球尺度以及在天体物理/宇宙尺度，不能明确解决。然而，使用诸如在该提议中开发和使用的技术，人们可以学习关于这些模型的解的行为的大量实用信息，并且因此对由这些方程系统建模的物理系统的行为做出非常有用的预测。

A good portion of this award is devoted to the study of the formation of singularities, both in solutions of Einstein's equations and in solutions of Ricci flow. For Einstein's equations, one of the key issues is Strong Cosmic Censorship ("SCC"): Is it true that the ubiquitous geodesic incompleteness predicted by the Hawking-Penrose theorems generically involve spacetime curvature (and therefore gravitational tidal) blowup? At least in certain families of spacetimes defined by isometries, the determination that the solutions exhibit "AVTD" behavior (dominance of time derivatives of the fields over space derivatives near the singularity) is a very helpful tool for studying SCC, and is interesting in its own right. Some of our work involves the development of a framework for proving that there are large sets of smooth (non-analytic) solutions in these families which show AVTD behavior. Our studies of singularities in Ricci flow have focused on those known as "neckpinches". We have, in particular, shown that there is a discretely parametrized family of Ricci flow solutions which develop "degenerate neckpinches", with the geometry asymptotically approaching a "Bryant soliton" at a prescribed rate (consistent with Type 2 behavior). These solutions are all rotationally symmetric. Wanting to know if the detailed asymptotic behavior seen in these solutions is also characteristic of non-rotationally symmetric solutions, we have chosen to first address this issue in the case of mean curvature flow ("MCF"), since MCF neckpinches share many features with those of Ricci flow, and since it is easier with MCF to compare non-rotationally symmetric solutions with rotationally symmetric ones.In the projects discussed above, as well as in many others in my research program, the primary goal is to develop an understanding of the general behavior of large classes of solutions of complicated nonlinear systems of partial differential equations. Such equation systems, which play a major role in our modeling of physical phenomena at the terrestrial scale as well as at the astrophysical/cosmological scale, cannot be solved explicitly. However, using techniques such as those developed and used in this proposal, one can learn a tremendous amount of practical information about the behavior of solutions of these models, and therefore make very useful predictions regarding the behavior of the physical systems modeled by these equation systems.