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Geometric measure theory – a mathematical toolbox to study surfaces

Yushan Young ScholarIssued by:National Taiwan Normal UniversityNumber of click-through:610
Year of approval:107/Year of project result:107 /Academic field:Science/Scholar name:Ulrich Menne

Introduction to the event

Generally speaking, my research aims at the understanding of the complex local structure of surfaces occurring in many models from the natural sciences. In this regard, a mathematical surface may correspond to a variety of different physical objects: for instance, soap films, horizons of black holes, membranes of cells, and boundaries between different phases of a material, or between different grey levels in a digitally reconstructed image.

It is the power of mathematical abstraction, that allows to devise a model of surfaces capable of covering all these cases at once and to derive theoretical conclusions (e.g., regularity results), that typically are applicable in all of these settings. By a regularity result, one means a mathematical theorem, that says, that a surface satisfying a given optimality condition admits a simpler (i.e., more regular) local description, than an arbitrary surface in its class.

The study of models of surfaces with a complex local structure pertains to the field of geometric measure theory. The success of this theory does not only stem from the many applications, that its results have found in other larger fields within mathematics (e.g., differential geometry, geometric analysis, and mathematical models in the natural sciences), but also from the new methods that it has contributed to the big fields of partial differential equations and the calculus of variations.

The core motivation for my research is to make progress towards a fundamental regularity question formulated by Allard in 1972. This question concerns the local structure of surfaces in a particularly versatile class of surfaces (namely, integral varifolds) under the natural optimality condition for this class. This class of surfaces is employed in models of all of the above-mentioned physical objects.