Workshop — Non-Linear Critical Point Theory in Analysis and Geometry

Minimal surfaces have inspired scientists and mathematicians for centuries and remain a central area of study with a vast number of pure and applied research outputs relating directly to their study. From art and architecture, to fluid dynamics and general relativity, they have an impressive list of applications. The main reason that they remain so popular amongst mathematicians is the fact that they are mysterious, beautiful, and lead to the resolution of challenging and abstract problems, not necessarily in obviously related fields. They can be created in the home by dunking a metal hoop into soapy water - the resulting soap film is a minimal surface (no matter how you bend the hoop). Equally they are integral to, for example, certain proofs of the positive mass theorem from general relativity. A surface is minimal if it is locally stretched tight, so that if you were to focus on a small region of the surface, any perturbation of that small region will increase surface area. In abstract terms, minimal surfaces are critical "points'' of the energy - in this case the energy of a surface is given by its area.

A large quantity of physical phenomena admit such a description - they are critical with respect to some energy, or in other words ``nature always tries to optimise energy''. For example the shape of cell membranes, event horizons of black holes and the behaviour of fundamental particles all fit within this regime. This workshop brings together young and experienced researchers in the mathematical/theoretical study of these problems. It mainly utilises the theory of differential geometry and partial differential equations, or broadly Geometric Analysis. This field has enjoyed popular notoriety in recent years with one of the modern progenitors of the subject, Karen Uhlenbeck, being awarded the Abel Prize. Similarly Perelman's proof of the Poincare conjecture, a question that took 100 years (and many contributors) to answer fully, for which Perelman famously turned down the $1,000,000 award.