In this talk, I will first introduce the subject geometric measure theory which has had a strong impact on a variety of fields. For instance, in geometric analysis, the proof of the positive mass theorem (by Schoen and Yau) and the Willmore conjecture (by Marques and Neves) are two major breakthroughs which are driven by powerful tools from geometric measure theory.

The second part of this talk concerns minimal surfaces. Whereas there are 250 years of successful history for two-dimensional minimal surfaces, in the higher dimensional case, the development is blocked by the long- standing question of possible almost everywhere regularity. I will present three major achievements of mine, that give meaning to both the extrin- sic and intrinsic curvature of generalised minimal surfaces and lay the foundation to study PDEs on them.

Finally, I will indicate my research programme towards this long-stand- ing question. The approach is based on my afore-mentioned achievements, and is carefully designed to ensure significant impact on non-linear partial differential equations, the calculus of variations, mathematical models in the natural sciences, geometric analysis, and, if the question is solved in the affirmative, differential geometry.