The Formation of Singularities in Ricci Flow and Harmonic Ricci Flow

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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.