Singularities of Geometric Partial Differential Equations

这一建议涉及的领域非常广泛，从物理学中最基本的问题，到当前工程中的实际问题，再到拓扑学和几何学中一些最强大的现代技术。尽管这些主题都非常不同，但很明显，每个领域的许多重大未来发展都需要克服非常相似的关键研究挑战。正是这些挑战，我们将在这个拟议的研究中解决。以上每个主题的核心都是几何偏微分方程（PDE）。这些方程中的每一个都可能是一个物理定律，或者是一个模拟工业过程的方程，或者更抽象地说，是一个可以处理几何对象以改进它的规则。几何偏微分方程的光滑解在纯问题和应用问题的应用中非常成功，但方程通常是非线性的，因此解中出现奇点是典型的。下一代应用程序具有广泛的潜在影响，要求我们改变对这些发展中的奇点的理解。我们必须了解它们发生的时间和原因，它们的结构和稳定性，以及它们如何编码PDE正在做的事情。我们必须分析它们在多大程度上打破了光滑解的经典理论，以及这有什么影响。这些都是该提案的主要挑战，我们已经组建了一个团队来解决这些问题，他们在奇点分析方面具有互补的专业知识，并在数学相对论、几何流和最小曲面等学科中应用几何偏微分方程。在数学相对论中，人们可以在爱因斯坦方程的解中看到奇点。爱因斯坦方程最早是在1915年作为大尺度宇宙的基本方程写下来的。我们提出的研究挑战的进展将对该领域一些最著名的开放问题产生潜在的重大影响，例如宇宙审查猜想和黑洞稳定性问题。我们还在几何流领域发现了奇点，我们指的是“抛物线”型的演化方程，它目前在几何、拓扑和工程以及物理和生物学现象建模方面的应用非常成功。近年来最著名的应用是庞加莱猜想的解决，该猜想被《科学》杂志评为“2006年度科学突破”，但许多人认为这是过去100年来数学领域最伟大的成就。我们提出的研究挑战是这些方程未来应用的核心，无论我们是使用它们对具有特定曲率条件的流形进行分类，还是处理来自医疗扫描仪的图像。与这两个主题密切相关的是最小曲面理论。这些表面一直被用来模拟肥皂膜，但一般理论已经发展成为一个强大的工具，应用于从黑洞到拓扑学的广泛领域。在这个方向上，我们特别感兴趣的是在本提案的研究挑战中应用进展，以揭示高指数最小曲面的存在与流动中出现的奇点之间的联系，以及旨在找到它们的变分问题。

This proposal sits within a field of great scope, stretching from some of the most fundamental problems in physics, to current practical issues in engineering, to some of the most powerful modern techniques in topology and geometry. Although these topics are all very different, it has become apparent that many of the biggest future developments in each area will require overcoming key research challenges that are remarkably similar. It is these challenges that we will address in this proposed research.At the heart of each of the topics above lie Geometric Partial Differential Equations (PDE). Each of these equations could be perhaps a law of physics, or an equation modelling an industrial process, or more abstractly, a rule under which a geometric object can be processed in order to improve it. Smooth solutions to Geometric PDE have been extremely successful in applications to pure and applied problems, but the equations are generally nonlinear, and it is therefore typical that singularities will occur in solutions. The next generation of applications, with extensive potential impact, require us to transform our understanding of these singularities that develop. We must understand when and why they occur, their structure and stability, and how they encode what the PDE is doing. We must analyse to what extent they break the classical theory of smooth solutions, and what effects this has. These are the main challenges of this proposal, and we have compiled a team to address them with complementary expertise in singularity analysis and experience of applying geometric PDE across subjects such as Mathematical Relativity, Geometric Flows and Minimal Surfaces.In Mathematical Relativity, one sees singularities in solutions of the Einstein equations, first written down by Einstein in 1915 as the fundamental equations of the large-scale universe. Progress in the research challenges we propose will have potentially major impact in some of the most famous open problems in this field such as the Cosmic Censorship Conjectures, and the Black Hole Stability Problem.We also find singularities in the field of Geometric Flows, by which we mean the evolution equations of `parabolic' type that are currently being so successful in applications to geometry, topology and engineering, and in modelling phenomena in physics and biology. The most famous application in recent years has been the resolution of the Poincaré conjecture, which was named by the journal `Science' as the scientific `Breakthrough of the year, 2006,' but is considered by many to be the greatest achievement of mathematics in the past 100 years. The research challenges we propose are central to future applications of these equations, whether we are using them to classify manifolds with a certain curvature condition, or manipulate an image from a medical scanner.Intimately connected with these two subjects is the theory of Minimal Surfaces. These surfaces have been historically used to model soap films, but the general theory has developed into a powerful tool with applications to a wide range of subjects from black holes to topology. In this direction, we are particularly interested in applying progress on the research challenges of this proposal to unravel the connection between the existence of higher-index minimal surfaces and the singularities that occur in flows and variational problems that are designed to find them.