Stability in physical systems governed by curvature quantities

该学生将开发新的工具，通过利用几何分析的前沿研究，从曲率特性确定物理物体的形状。虽然几何分析一直是最近世界领先的数学突破的核心，这些突破与庞加莱猜想和最优传输研究（菲尔兹奖被授予佩雷尔曼和菲加利）有关，但它在形状分析中的作用仍未得到充分的探索。一个有趣的例子是生物细胞的形状与其细胞膜的弹性能量的关系，即所谓的canham - helrich能量。为了对所涉及的问题有一个大致的了解，假设有一定数量的人，每个人都拥有一块土地，他们的土地总面积覆盖地球。每个人都要对自己的土地做一张精确的“曲率图”；平坦/凹凸，向内/向外弯曲的碎片在哪里？这是用数字来描述的；“平的”标记为零，“向内弯曲”标记为负数，“向外弯曲”标记为正数。假设每个人的财产都是这样测量的。之后，所有人收集他们的数据。关键问题是：你能从这些信息中确定地球的形状吗？将采用的方法依赖于谱理论，超曲面几何和偏微分方程（PDE）之间的桥梁的最新发展，由第一导师Julian Scheuer博士（JS）与几何分析领域的领先专家合作开发。第二位当地导师Federica Dragoni博士（FD）用随机分析的观点补充了这一专业知识。

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.