Stability in physical systems governed by curvature quantities

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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.