The Formation of Singularities in Ricci Flow and Harmonic Ricci Flow

里奇流和谐波里奇流奇点的形成

基本信息

批准号：
EP/M011224/1
负责人：
Reto Buzano
金额：
$ 12.81万
依托单位：
Queen Mary University of London
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2015
资助国家：
英国
起止时间：
2015 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FM011224%2F1
关键词：
Formation Singularities Ricci Flow Harmonic

项目摘要

This proposal sits within the broad field of nonlinear partial differential equations (PDE), an area of mathematics with wide-ranging applications from practical issues in engineering, science and industry to some of the most difficult problems in geometry and topology. Such an equation could model for example a chemical or industrial process, be a rule to correctly define the price of a financial option, or more abstractly describe the shape or the evolution of a geometric object. It is the last mentioned type of PDE that this proposal focuses on.Relating the local geometry and global topology of manifolds constitutes one of the main aims of differential geometry. While this area of pure mathematics has always seen steady progress, it was the introduction of techniques from analysis - and in particular heat flow methods - that revolutionised it completely and led to some of the most spectacular recent results such as Perelman's resolution of the Poincaré and Geometrisation Conjectures, the 1/4-pinched Differentiable Sphere Theorem of Brendle and Schoen, and Brendle's proof of the Lawson Conjecture. It therefore comes as no surprise that the report of the EPSRC Pure Mathematics Workshop 2012 as well as the International Review of Mathematical Sciences 2010 come to the conclusion that the part of geometry that needs most strengthening in the UK is the connection between geometric analysis and nonlinear partial differential equations. I propose to further develop the UK's research infrastructure in this field through world-leading research that borrows modern ideas from analysis, geometry and topology and unites and transforms them into completely new and powerful techniques and results. More precisely, the proposed research consists of the following themes: understanding higher-dimensional Ricci Flow singularities, investigating stability properties of singularity models, developing theories of generic Ricci Flow in arbitrary dimensions and of weak Ricci Flow in dimension three, and analysing the singularity formation in the Harmonic Ricci Flow. While these themes are all connected and intertwined, I have made an effort to crystallise out formally independent objectives. The results obtained from the proposed research will not only have a major impact on geometry and topology, but also open up the field of geometric flows for applications in physics and engineering.

该建议位于非线性偏微分方程（PDE）的广泛领域，这是一个数学领域，具有广泛应用程序，从工程，科学和工业的实际问题到几何和拓扑中一些最困难的问题。这样的方程式可以建模，例如化学或工业过程，是正确定义财务选择价格的规则，或更抽象地描述几何对象的形状或演变。这是该提案重点关注的最后提到的PDE类型。将局部几何形状和歧管的全球拓扑结合起来构成了差异几何的主要目的之一。虽然这一纯数学领域一直保持稳定的进步，但正是从分析（特别是热流方法）引入技术的引入，这完全彻底改变了它，并导致了一些最壮观的结果，例如佩雷尔曼（Perelman）解决了彭康（Perelman）和几何构度的猜想，即1/4固定的跨越构造的统治和劳伦（Schoen）和劳伦（Schoen）和布伦德尔（Brendle）的统治，并构成了布伦德尔（Schoen），布伦德尔（Schoen）和布伦德尔（Brendle）的规定。因此，EPSRC纯数学研讨会的报告以及2010年的国际数学科学评论得出的结论是，英国需要加强最大的几何形状的一部分是几何分析与非线性部分偏微分方程之间的联系，这不足为奇。我建议通过世界领先的研究进一步发展英国的研究基础设施，从分析，几何学和拓扑和单位借用现代思想，并将其转变为全新，有力的技术和结果。更确切地说，拟议的研究包括以下主题：了解较高维度的RICCI流动奇异性，研究奇异性模型的稳定性特性，在任意维度中开发了通用RICCI流动的理论，而Dimension中的RICCI流量较弱，并在第三维度中发展了RICCI流动，并分析了谐波RICCI流动中的奇异性。尽管这些主题都是连接和交织在一起的，但我还是努力将正式独立的目标结晶。从拟议的研究中获得的结果不仅会对几何和拓扑产生重大影响，而且还为物理和工程应用程序开辟了几何流量。