The School of Mathematical Sciences of Queen Mary University of London invite applications for a PhD project commencing in September 2025.

About the Project

• Supervisor: Dr Huy Nguyen and Dr Shengwen Wang
• Funding: Competition funding through QMUL Science and Engineering Studentships, QMUL Engineering and Physical Sciences Research Council (EPSRC) studentships, or S&E Doctoral Research Studentships for Underrepresented Groups 2025/26, or QM/China Scholarship Council (CSC)
• Deadline: 29th January 2025

Project description

Geometric Analysis, and in particular Geometric Flows, led to some of the most important breakthroughs in pure mathematics in the last decades. This modern area of mathematics combines Differential Geometry (algebraic structures of geometric objects and local control of geometric properties such as curvature) with tools from Analysis (mainly partial differential equations and the calculus of variations) to obtain global differential topological results.

As the partial differential equations that are involved are generally non-linear, they often form singularities, points where the equation non longer makes sense. In many interesting problems, it is crucial to understand when and how these singularities occur and to obtain precise quantitative characterisations of them.

An illustrative example is the Mean Curvature Flow. This flow evolves an initial manifold towards an object with more symmetries, for example a potato-shaped surface is transformed to a perfectly round sphere. In more general situations however, the flow develops local singularities where the curvature becomes infinitely large while staying bounded in other regions.

One possible Phd project is to study the relationship of certain geometric PDEs for hypersurfaces (such as mean curvature flow and minimal surfaces) and a family of phase-field equations arising from physics called the Allen-Cahn equations and Waals-Cahn-Hilliard equation. The Allen-Cahn equations and Waals-Cahn-Hilliard equations are phase-field equations modelling the phase transition phenomena in physics, e.g. multi-component alloy systems. When the width parameter of these equation goes to zero, their transition layers (zero sets) are converging to hypersurfaces satisfying certain geometric PDEs such as minimal surfaces equation, mean curvature flow (MCF) equation and the Willmore equation. We will also be offering projects in high codimension mean curvature flow, mean curvature flow in curved background spaces and the singularity formation in weak mean curvature flow.

For more info on this project read here:

Center Details:

The Centre for Geometry, Analysis and Gravitation is a world-leading multidisciplinary research centre on geometric analysis, partial differential equations, topology and all aspects of gravitation. The Centre for Geometry, Analysis and Gravitation is leading the research on gravitational waves and quantum gravity at Queen Mary University of London, and it has become a reference centre in the UK and worldwide. Its members are a unique blend of pure and applied mathematicians working on different aspects of general relativity, numerical and mathematical relativity quantum gravity and gravitational aspects of string theory and holography.

The Centre has received funding from prestigious sources such as the European Research Council (ERC), Royal Society University Research Fellowships, UKRI Future Leaders Fellowship, various EPSRC standard grants, STFC Ernst Rutherford Fellowships, and STFC Consolidated grants.

Application

The application procedure is described on the School webpage: www.qmul.ac.uk/maths/postgraduate/postgraduate-research/application-process/ . For further inquiries please contact [email protected] and [email protected]

The School of Mathematical Sciences is committed to the equality of opportunities and to advancing women’s careers. As holders of a Bronze Athena SWAN award we offer family friendly benefits and support part-time study.