Regularity aspects of variational problems

该项目将研究极小子流形正则性理论及相关领域中的一个或多个未解决的问题。特别是在稳定极小超曲面的分支点奇异性分析、面积最小化(积分和模极小)电流和多值调和函数等方面的最新进展，产生了一个重要的新思想和新技术循环。这些发展为进一步的发展提供了可能，例如将各种内部正则性理论推广到边界，并将这些方法潜在地应用于各种偏微分方程组问题。博士项目最终将涉及这些问题的某些方面，在最初的一段时间内，候选人将被要求获得解决这些规律性问题所必需的实质性的地理测量理论背景。

The project will investigate one or more of the various open questions in the regularity theory of minimal submanifolds and related areas. In particular, recent developments in the analysis of branch point singularities of stable minimal hypersurfaces, area minimising (integral and mod p) currents and multi-valued harmonic functions have generated an important circle of new ideas and techniques. These developments have thus opened up possibilities for further progress, such as extending the various interior regularity theories to the boundary, and potential applications of the methods to various PDE problems. The PhD project will eventually involve some aspect of these questions, after an initial period of time in which the candidate will be required to aquire the substantial Goemetric Measure Theory background necessary to tackle these regularity questions.