### Key facts about the grant

In July 2023, EPSRC funded the research proposal “Geometric Flows and the Dynamics of Phase Transitions” led by Huy Nguyen (Principal Investigator) with **£413,909**. This grant will run for three years from 1 December 2023 until 30 November 2026.

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?

This project aims to develop the theoretical framework of the singularity formation of the Mean Curvature Flow and the Allen–Cahn 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 Allen–Cahn Flow in Euclidean space and in curved background spaces, developing new concentration compactness results to analyse the singularities and surgery procedures to geometrically undo the singularity formation in mean curvature flow, and finally exploring the relationship and 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.

### What will the grant pay for?

The grant will finance **research time** for the PI and **one postdoctoral researcher positions** as well as money for the **travel** of the research team and its visitors and for the organisation of an **international conference** in 2025, which will be announced soon.

### Further information

You can find out more about the project on the EPSRC website here.

## Previous Grant – Advances in Mean Curvature Flow: Theory and Applications

### 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 was 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 did 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.

### Further information

You can find out more about the project on the EPSRC website here.