The Formation of Singularities in Ricci Flow and Harmonic Ricci Flow

该建议属于非线性偏微分方程（PDE）的广泛领域，这是一个广泛应用的数学领域，从工程，科学和工业的实际问题到一些最困难的几何和拓扑问题。这样的方程可以为化学或工业过程建模，是正确定义金融期权价格的规则，或者更抽象地描述几何物体的形状或演变。本建议关注的是最后提到的PDE类型。将流形的局部几何和全局拓扑联系起来是微分几何的主要目的之一。虽然纯数学的这一领域一直在稳步发展，但正是分析技术的引入——尤其是热流方法——彻底改变了这一领域，并导致了最近一些最引人注目的结果，如佩雷尔曼对庞加莱猜想和几何化猜想的解决，布伦德尔和舍恩的1/4缩微球定理，以及布伦德尔对劳森猜想的证明。因此，EPSRC纯数学研讨会2012年的报告以及国际数学科学评论2010年得出结论，英国最需要加强的几何部分是几何分析和非线性偏微分方程之间的联系，这一点也不奇怪。我建议通过世界领先的研究进一步发展英国在这一领域的研究基础设施，从分析、几何和拓扑中借鉴现代思想，并将它们结合起来，转化为全新的、强大的技术和成果。更准确地说，我们的研究包括以下几个主题：理解高维里奇流奇点，研究奇点模型的稳定性，发展任意维广义里奇流理论和三维弱里奇流理论，分析谐波里奇流奇点的形成。虽然这些主题都相互联系、交织在一起，但我努力将形式上独立的目标具体化。所提出的研究结果不仅将对几何和拓扑学产生重大影响，而且还将开辟几何流在物理和工程中的应用领域。

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.