Nonlinear critical point theory near singular solutions

奇异解附近的非线性临界点理论

• 批准号: EP/W026597/1
• 负责人: Benjamin Sharp
• 金额: $ 47.04万
• 依托单位: University of Leeds
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2023
• 资助国家: 英国
• 起止时间: 2023 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FW026597%2F1
• 关键词: Nonlinear; critical; point; theory; near

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.

几何学和理论物理学中出现的大部分现象都可以用能量（或作用）函数来描述。临界点对应于平衡状态，并由非线性偏微分方程（PDE）系统描述，通常在弯曲的背景空间上求解。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，在最简单的形式中，上述例子（以及更多）对应于少数几个典型的数学问题。本文的背景就是对这些原型问题的研究。它涉及到分析和几何之间的丰富的相互作用，主要是在非线性偏微分方程和微分几何的严格研究相结合：一个领域，产生了巨大的影响，近年来（例如）佩雷尔曼的决议庞加莱和几何猜想，肖恩-丘的证明正质量定理从数学相对论和马克-内维斯'证明的Willmore猜想在微分几何。上述问题（以及大的非线性偏微分方程）的一个自然发生的特征是奇点的形成，奇点对应于解沿域的一个子集沿着爆破的区域。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，一个表面的两个部分之间可能会形成一个细颈，它会随着时间的推移而消失，并断开两个部分-人们可能会认为这是一个“虫洞”类型的奇点。该提案的主要目的是引入偏微分方程理论和微分几何工具，以便建模和分析这种奇点（拓扑结构发生变化）。在这种背景下，在分析和分类潜在的奇点形成方面取得了巨大的进步，但通常对某些奇点类型是否存在的理解相对较少。我们将开始一个系统的和新颖的研究“最简单的”类型的奇点形成，并找到条件，确定它们是否存在，可以构建，或者是否有一个障碍，他们的存在。