Key facts about the grant
In September 2018, EPSRC funded the research proposal “Advances in Mean Curvature Flow: Theory and Applications” led by Reto Buzano (Principal Investigator) and Huy Nguyen (Co-Investigator) with £613,223. This grant will run for three years from 1 January 2019 until 31 December 2021.
In addition, the Faculty of Science and Engineering at Queen Mary University of London (QMUL) is supporting the project with a fully funded PhD studentship (via their Research Support Fund) and a further Research Enabling Fund. In total, the contribution from QMUL exceeds £80,000.
What is this research project about?
The project aims to develop the theoretical framework of the singularity formation of the Mean Curvature Flow, 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.
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), or image processing and material science (Martensite, nonlinear plate models).
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. A variety of related problems will also be studied.
What does the grant pay for?
The grant will finance research time for the PI and Co-I and two postdoctoral researcher positions (Mario Schulz and Shengwen Wang, September 2019 – August 2021) as well as money for the travel of the research team and its visitors and for the organisation of an international conference in 2020, which will be announced soon.
You can find out more about the project on the EPSRC website here.