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.
Prof Dr Chao Xia (CX), a leading expert on Geometric Analysis in China and long-term collaborator of JS at Cardiff’s partner university at Xiamen in China, completes the supervisory team.
The student will benefit from the partnership of Cardiff University with Xiamen University in China through regular virtual exchange with CX’s research group and optional personal visits hosted by CX. During such visits, access to local seminars in Fujian province will allow the student to interact directly with world-leading experts in the relevant fields.
Cardiff University provides a highly inspiring research environment for postgraduates. Its Analysis group comprises internationally leading experts in pure and applied mathematics. Local seminars on PDE will be highly beneficial to undertake this project. The student will automatically join Cardiff’s Doctoral Academy with opportunities to interact with fellow postgraduate students from other disciplines, build networks and receive further training to attain professional skills.
HOW TO APPLY Applicants should apply through the
Cardiff University online application portal, for a Doctor of Philosophy in Mathematics with an entry point of October 2021
In the research proposal section of your application, please specify the project title and supervisors of this project.
There is no requirement to submit a research proposal
In the funding section, please select "I will be applying for a scholarship / grant" and specify that you are applying for advertised funding from EPRSC DTP.
The 3.5 year studentship includes UK fees, stipend (amount for 2020/21 is £15285) and a research training grant to cover costs such as research consumables, training, conferences and travel.
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