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School of Mathematical Sciences

Regularity theory and singularity analysis of geometric PDEs via phase field approximations

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.

The primary supervisor for this Phd project is Dr. Shengwen Wang and the secondary supervisor is Dr. Huy The Nguyen. They both conduct research in geometric analysis and geometric partial differential equations. They are members of an active research group in geometry, analysis and gravity based at Queen Mary University of London and have very strong links with other London universities with active research in Analysis and Geometry in close proximity, Queen Mary University of London has become one of the most interesting places to pursue a PhD in Geometric Analysis.

We encourage applicants with strong backgrounds in either partial differential equations or differential geometry to apply. If interested, please contact Shengwen Wang or Huy Nguyen by e-mail for further information and advice on the application procedure. For information about our group please visit 

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