Stability in physical systems governed by curvature quantities

由曲率量控制的物理系统的稳定性

• 批准号: 2601534
• 金额: --
• 依托单位: CARDIFF UNIVERSITY
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2021
• 资助国家: 英国
• 起止时间: 2021 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2601534
• 关键词: Stability; physical; systems; governed; curvature

The studentship will develop novel tools to determine the shape of physical objects from their curvature properties, by making use of cutting-edge research in Geometric Analysis. While Geometric Analysis has been at the core of recent world-leading mathematical breakthroughs related to the Poincaré conjecture and research on optimal transport (with Fields-medals being awarded to Perelman and Figalli), its role in Shape Analysis is still underexplored. An example of interest is the shape of bio-cells in relation to the elastic energy of its cell-membrane, the so-called Canham-Helfrich energy. To give a flavour of the type of questions involved, suppose there is a given number of people, each individual owns a piece of land and their total land mass covers the earth. Everyone shall make a precise "curvature map" of their piece of land; where are flat/bumpy, inward/outward curved pieces? This is described in terms of numbers; "flat" would be labelled zero, "curved inward" by a negative and "curved outward" by a positive number. Suppose every spot of each person's property is measured this way. Afterwards all individuals collect their data. The key problem is: Can you determine the earth's shape from that information?The approach that will be pursued relies on recent developments on the bridge between Spectral Theory, Hypersurface Geometry and Partial Differential Equations (PDE), developed by the first supervisor Dr Julian Scheuer (JS) in collaboration with leading experts in Geometric Analysis. The second local supervisor Dr Federica Dragoni (FD) complements this expertise with perspectives from Stochastic Analysis.

该学生将开发新颖的工具，通过利用几何分析的尖端研究，根据物理物体的曲率特性确定其形状，而几何分析一直是与庞加莱猜想相关的最新世界领先数学突破的核心。以及最佳运输的研究（菲尔兹奖授予佩雷尔曼和菲加利），但其在形状分析中的作用仍未得到充分探索，一个令人感兴趣的例子是生物细胞的形状与弹性能的关系。为了说明所涉及的问题类型，假设有一定数量的人，每个人都拥有一块土地，并且他们的总土地面积覆盖了地球。每个人都应该为自己的一块土地制作一个精确的“曲率图”；哪里是平坦的/凹凸不平的、向内/向外弯曲的部分？假设每个人的财产的每个点都以这种方式测量，那么关键问题是：你能根据这些信息确定地球的形状吗？所追求的基础是谱理论、超曲面几何和偏微分方程 (PDE) 之间桥梁的最新发展，由第一位导师 Julian Scheuer 博士 (JS) 与几何分析领域的领先专家合作开发。 Federica Dragoni (FD) 通过随机分析的观点补充了这一专业知识。