Singularities of Geometric Partial Differential Equations

几何偏微分方程的奇异性

基本信息

批准号：
EP/K00865X/1
负责人：
Peter Topping
金额：
$ 197.63万
依托单位：
University of Warwick
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2013
资助国家：
英国
起止时间：
2013 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FK00865X%2F1
关键词：
Singularities Geometric Partial Differential Equations

项目摘要

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.

该建议位于一个范围内的领域，从物理学中一些最根本的问题到工程学的当前实际问题，再到拓扑和几何学上一些最强大的现代技术。尽管这些主题都大不相同，但很明显，每个领域的许多最大未来发展都需要克服非常相似的关键研究挑战。我们将在这项拟议的研究中解决这些挑战。在上面的每个主题的核心中，几何偏微分方程（PDE）。这些方程中的每个方程都可能是物理定律，或者是对工业过程进行建模的方程，或更抽象地，可以处理几何对象以改善几何对象。几何PDE的平滑解决方案在纯和应用问题的应用中非常成功，但是方程通常是非线性的，因此典型的奇异性会在解决方案中发生。具有广泛潜在影响的下一代应用程序要求我们改变对这些奇异性的理解。我们必须了解它们何时以及为什么发生，结构和稳定性以及它们如何编码PDE的所作所为。我们必须分析它们在多大程度上打破了平滑解决方案的经典理论，以及这种影响。这些是该提案的主要挑战，我们已经编译了一个团队，以奇异性分析和在数学相对性，几何流动和最小的表面上应用几何PDE的完整专业知识来解决他们。宇宙。 Progress in the research challenges we proposal 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.近年来，著名的应用是庞加莱猜想的解决，该猜想是由《科学》杂志命名为“ 2006年度最佳突破”，但许多人认为是过去100年来数学的最大成就。我们提出的研究挑战对于这些方程式的未来应用是至关重要的，无论我们是利用它们对具有一定曲率条件的流形进行分类，还是从医疗扫描仪中操纵图像。与这两个主题有关，都是最小表面的理论。这些表面历史上一直用于建模肥皂膜，但是一般理论已发展为一个强大的工具，并将其应用于从黑洞到拓扑的广泛主题。在这个方向上，我们特别有兴趣将进步应用于该提案的研究挑战，以揭示较高索引最小表面的存在与流量和旨在找到它们的变异问题的奇异性之间的联系。