Advances in Mean Curvature Flow: Theory and Applications

This project aims to develop the theoretical framework of the singularity formation of the Mean Curvature Flow. The Mean Curvature Flow is a geometric flow that describes the motion of a surface. It was introduced by Mullins as a model for the formation of grain boundaries in annealing metals. It also appears as the flow to equilibrium of soap films, the motion of embedded branes in approximations of the renormalisation group flow in theoretical physics, boundaries of Ginzburg-Landau equations of simplified superconductivity and as a method of denoising in image processing. The results of this project, as well as the methods pioneered, will enable the next generation of applications.

This proposal lies in the intersection of the EPSRC research areas Mathematical Analysis and Geometry and Topology with applications to Mathematical Physics and Algebra. It has underpinning relevance ranging from fundamental problems in theoretical physics to current issues in engineering. At the heart of these problems is a single system of geometric Partial Differential Equations (PDE). Such equations have had a tremendous impact in mathematics: they have been extremely successful with applications over diverse areas such as topology (Poincaré Conjecture, Geometrisation conjecture), Kähler geometry (minimal model problem), gravitation (Penrose inequality), image processing and material science (Martensite, nonlinear plate models).

Geometric PDE are inherently nonlinear and therefore singularities are expected to occur. In fact, these singularities turn out to be useful - they tell us something about the underlying geometry of our object. This project will develop our understanding of singularities of systems of geometric PDE, an extremely important area in geometry, analysis and PDE theory which is relatively poorly understood. In principle, the singularities of systems of nonlinear partial differential equations may be unstructured, but due to their geometric origins, the singularities of systems of geometric PDE display a surprising order. This project seeks to characterise the mechanisms of the formation of singularities, obtain classifications of the singularity models, and to develop a geometric hierarchy of singularity models and quantitatively analyse their stability. Modern applications of geometric flows require a detailed understanding of singularities using an integrated approach combining algebra, analysis, geometry and topology.

More precisely, the proposed research consists of the following themes: understanding the singularity formation of the Mean Curvature Flow in high codimension and in curved background spaces, developing new concentration compactness results to analyse the singularities and surgery procedures to geometrically undo the singularity formation, and finally exploring applications of the new theory to various fields of mathematics. The results pioneered in this project will have a direct and significant impact on geometry, analysis, and topology. Furthermore, the methodologies and techniques developed in this project can also be applied to a number of outstanding problems in physics, biology, engineering and computer imaging.

Nonlinear Partial Differential Equations (PDE) constitute one of the most important areas of research in mathematics. Such equations, which form the heart of this research proposal, have widespread impact from academic science to engineering, industry, finance and society.

A substantial part of the economic and societal impact of this project will be generated via other academic work based on the research in this proposal, in particular through PDE theorists, physicists, biologists, material and computer scientists, and engineers. This research focuses on constructing cutting-edge methodologies and knowledge that will be applicable to a number of different areas of research thus improving the effectiveness and sustainability of UK universities. Another area of academic impact is through the building and maintenance of national and international research networks. This will be achieved by a sustained programme of national and international research visits with researchers from the UK as well as Canada, USA, Italy, France, and Germany. The development of this network will have wideranging impact through reciprocal knowledge exchange, student recruitment and research training as well as maintaining the UK's excellence at the forefront of research.

Of particular interest is the financial sector which accounts for a third of the UK's gross domestic product. Examples of PDEs are the Black-Scholes equation, the Hamilton-Jacobi-Bellman equation and the forward-backward Kolmogorov equation. These equations are used to price derivatives, for calibration, pricing and hedging in incomplete markets, portfolio optimisation and optimal decision making. Well trained mathematicians with a background in PDEs are in constant demand from the financial industry and support for this area is essential for the continuing prosperity of the nation. In particular, by strengthening the Analysis and Geometry Group at Queen Mary University, we will be able to train highly skilled researchers and to provide them with the knowledge and technical abilities necessary to work in challenging positions in the financial sector.

A large proportion of the proposed research has the potential for industrial applications. Geometric flows are currently being used for image processing devices, in particular for de-noising, enhancement and super-resolution of low-quality images. There are several active approaches to solving these questions, including in particular a discrete version of the mean curvature flow. This technology finds applications ranging from facial recognition to medical scanners. As of today, the UK is not at the forefront of this development, which is partially due to the fact that it does not yet have a deeply ingrained tradition in the area of geometric analysis. The proposed research project will help to improve the current situation and establish an environment where such development can occur.

Finally, geometric flow methods have produced some of the most impressive intellectual achievements of the last hundred years, Perelman's proof of the Poincaré and Geometrisation conjecture being the most prominent examples. This has considerably enhanced public awareness of mathematics over the past years which helps to secure a steady flow of students into this subject and science in general. This area is highly geometric and visual which makes this subject amenable to outreach activities for a broader audience through public lectures, summer schools for high school students and presentations to teachers.