The Formation of Singularities in Ricci Flow and Harmonic Ricci Flow

这一建议属于非线性偏微分方程（PDE）的广泛领域，这是一个数学领域，从工程，科学和工业中的实际问题到几何和拓扑学中的一些最困难的问题，都有广泛的应用。这样的方程可以模拟例如化学或工业过程，是正确定义金融期权价格的规则，或者更抽象地描述几何对象的形状或演变。这是最后提到的类型的偏微分方程，这一建议的重点。有关局部几何和整体拓扑的流形构成了微分几何的主要目标之一。虽然这一领域的纯数学一直看到稳步进展，这是引进技术的分析-特别是热流方法-彻底改变了它，并导致了一些最壮观的最近的结果，如佩雷尔曼的决议庞加莱和几何猜想，1/4捏微分球定理的布伦德尔和舍恩，和布伦德尔的证明劳森猜想。因此，毫不奇怪，EPSRC纯数学研讨会2012年的报告以及2010年国际数学科学评论得出的结论是，在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。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，几何和拓扑学中借鉴现代思想，并将其结合并转化为全新的强大技术和成果。更准确地说，拟议的研究包括以下主题：理解高维Ricci流奇异性，研究奇异性模型的稳定性，发展理论的一般Ricci流在任意维和弱Ricci流在三维，并分析奇异性形成的谐波Ricci流。虽然这些主题都是相互关联和交织的，但我已经努力将正式独立的目标具体化。从所提出的研究中获得的结果不仅将对几何和拓扑学产生重大影响，而且还将为物理和工程中的应用开辟几何流领域。

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.