Singularities of Geometric Partial Differential Equations

几何偏微分方程的奇异性

• 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

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.

这一提议涵盖了一个非常广泛的领域，从物理学中一些最基本的问题，到工程学中当前的实际问题，再到拓扑学和几何学中一些最强大的现代技术。虽然这些主题都非常不同，但很明显，每个领域的许多最大的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，我们将解决在这个拟议的研究。在上述每个主题的核心在于几何偏微分方程（PDE）。这些方程中的每一个可能是一个物理定律，或一个方程模拟一个工业过程，或更抽象，一个规则，根据该规则，一个几何对象可以被处理，以改善it.Smooth解决方案几何偏微分方程已经非常成功的应用程序的纯和应用问题，但方程一般是非线性的，因此，它是典型的奇异性将发生在解决方案。下一代的应用，具有广泛的潜在影响，需要我们改变我们对这些发展的奇点的理解。我们必须了解它们何时以及为什么发生，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。这些都是这个建议的主要挑战，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PDE的经验。在数学相对论中，人们可以看到爱因斯坦方程的解中的奇点，爱因斯坦在1915年首次写下了大尺度宇宙的基本方程。我们提出的研究挑战的进展将对该领域中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。我们还发现了几何流领域的奇点，我们指的是目前在几何，拓扑和工程应用中非常成功的“抛物”型演化方程。以及物理学和生物学中的现象建模。近年来最著名的应用是庞加莱猜想的解决，它被《科学》杂志命名为“2006年科学突破”，但许多人认为这是过去100年来数学最伟大的成就。我们提出的研究挑战是这些方程未来应用的核心，无论我们是用它们来分类具有一定曲率条件的流形，还是处理医学扫描仪的图像。与这两个主题密切相关的是极小曲面理论。这些表面在历史上被用来模拟肥皂膜，但一般理论已经发展成为一个强大的工具，应用范围从黑洞到拓扑学。在这个方向上，我们特别感兴趣的是，应用这个建议的研究挑战的进展，以解开高指数极小曲面的存在和奇点，发生在流动和变分问题，旨在找到他们之间的联系。

项目成果

A note on the index of closed minimal hypersurfaces of flat tori

关于平环面闭极小超曲面指数的注解

• 发表时间: 2018
• 期刊: Proceedings of the American Mathematical Society
• 影响因子: 1
• 作者: Ambrozio L

Bubbling analysis and geometric convergence results for free boundary minimal surfaces

自由边界最小曲面的冒泡分析和几何收敛结果

• DOI: 10.5802/jep.102
• 发表时间: 2019
• 期刊: Journal de l'École polytechnique - Mathématiques
• 作者: Ambrozio L

Asymptotic estimates and compactness of expanding gradient Ricci solitons

• 发表时间: 2014-11
• 期刊: arXiv: Differential Geometry
• 作者: Alix Deruelle

Compactness of the Space of Minimal Hypersurfaces with Bounded Volume and p-th Jacobi Eigenvalue

具有有界体积和p阶雅可比特征值的最小超曲面空间的紧性

• DOI: 10.1007/s12220-015-9640-4
• 发表时间: 2015
• 期刊: The Journal of Geometric Analysis
• 作者: Ambrozio L

On the two-systole of real projective spaces

论实射影空间的二收缩

• DOI: 10.4171/rmi/1188
• 发表时间: 2020
• 期刊: Revista Matemática Iberoamericana
• 作者: Ambrozio L