Nonlinear critical point theory near singular solutions

在几何学和理论物理学中出现的很大一部分现象可以用能量(或作用)函数来表述。临界点对应于平衡状态，并由非线性偏微分方程组(PDE)描述，通常在弯曲的背景空间中求解。例如，肥皂膜/气泡、量子场论中的基本粒子、向列相液晶、红细胞的形状或黑洞的事件视界都承认这种类型的理论描述。值得注意的是，在最简单的形式下，上面的例子(以及更多)对应于少数几个典型的数学问题。这一建议的背景是对这些原型问题的研究。它涉及分析和几何之间丰富的相互作用，主要是非线性偏微分方程组和微分几何的严格研究的结合：这个领域近年来产生了巨大的影响，(例如)佩雷尔曼解决了庞加莱和几何猜想，舍恩-尤从数学相对论证明了正质量定理，马克斯-内维斯证明了微分几何中的威尔莫尔猜想。上述问题(以及大范围的非线性偏微分方程组)的一个自然发生的特征是形成奇点，该奇点对应于解沿域的子集爆炸的区域。由于它们的几何性质，域本身也有退化或改变拓扑的空间。例如，表面的两个部分之间可能会形成一个细颈，随着时间的推移，它会消失，并断开这两个部分的连接--人们可能会认为这是一种“虫洞”类型的奇点。这项提议的主要目的是引入偏微分方程理论和微分几何中的工具，以便对这种奇点(发生拓扑变化的地方)进行建模和分析。在这种背景下，在分析和分类潜在奇点形成方面取得了巨大的进步，但对某些奇点类型是否存在的了解往往相对较少。我们将开始对“最简单”类型的奇点形成进行系统和新颖的研究，并找到决定它们是否存在、是否可以构建、或者它们是否存在障碍的条件。

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.