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.

几何和理论物理学中出现的现象很大一部分可以用能量（或动作）函数来表达。临界点对应于平衡状态，并通过非线性偏微分方程（PDE）的系统进行描述，该系统通常在弯曲的背景空间上解决。例如，肥皂膜/气泡，量子场理论中的基本颗粒，列液晶，红细胞的形状或黑洞的事件范围都接受了这种类型的理论描述。值得注意的是，以最简单的形式，上面的示例（以及更多）对应于一些原型数学问题。该提案的设定是对这些原型问题的研究。它涉及分析与几何形状之间的丰富相互作用，主要是在严格的非线性PDE和差异几何学研究的结合下：近年来在近年（例如（例如）佩雷尔曼（Perelman）解决Poincaré和Geremeriations构想的证明，Schoen-Yau的构想对数学的证明，该领域具有巨大的影响力，该领域是对数学上的证明，而Marivientive则是对数学上的构想。在差异几何形状中。上述问题的天然特征（以及大型中的非线性PDE）是奇异性的形成，这对应于沿域的一个子集爆炸的区域。由于其几何特性，域本身也具有退化或改变拓扑结构的范围。例如，在表面的两个部分之间可能形成薄颈，随着时间的流逝而消失并断开两部分的连接 - 一个人可能认为这是“虫洞”型奇异性。该提案的主要目的是在PDE理论和差异几何形状中引入工具，以模拟和分析此类奇异性（发生拓扑的变化）。在这种情况下，在分析和分类潜在的奇异性形成方面取得了巨大的进步，但对于某些奇异性类型是否存在，通常相对较少。我们将对“最简单”类型的奇异性形成类型进行系统和新颖的研究，并找到决定它们是否存在的条件，或者可以构造，或者是否存在其存在的障碍。