Nonlinear critical point theory near singular solutions

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A large proportion of phenomena that appear in geometry and theoretical physics can be phrased in terms of an energy (or action) function. The critical points correspond to states of equilibrium and are described by systems of non-linear partial differential equations (PDE), often solved on a curved background space. For example soap films/bubbles, fundamental particles in quantum field theory, nematic liquid crystals, the shape of red blood cells, or event horizons of black holes all admit theoretical descriptions of this type. Remarkably, in their simplest form, the above examples (and many more) correspond to a handful of archetypal mathematical problems. The setting of this proposal is the study of these archetypal problems. It involves a rich interplay between analysis and geometry, chiefly in the combination of the rigorous study of non-linear PDE and differential geometry: an area that has had tremendous impact in recent years with (for instance) Perelman's resolution of the Poincaré and Geometrisation Conjectures, Schoen-Yau's proof of the Positive Mass Theorem from mathematical relativity and Marques-Neves' proof of the Willmore conjecture in differential geometry. A naturally occurring feature of the above problems (and non-linear PDE in the large) is the formation of singularities, which correspond to regions where solutions blow up along a subset of the domain. Due to their geometric nature, there is also scope for the domain itself to degenerate or change topology. For example a thin neck may form between two parts of a surface, which disappears over time and disconnects the two parts - one might think of this as a "wormhole" type singularity. The main aim of this proposal is to introduce tools in PDE theory and differential geometry in order to model and analyse such singularities (where a change of topology takes place). In this setting, there have been tremendous advances in analysing and classifying potential singularity formation, but often relatively little is understood about whether certain singularity types exist, or not. We will initiate a systematic and novel study of the "simplest" types of singularity formation and find conditions which determine whether they exist, and can be constructed, or whether there is a barrier to their existence.