An outline of the group's teaching and research
The Max Planck research group “Geometric Measure Theory”, led by Prof. Dr. Ulrich Menne, is based on a cooperation of the Max Planck Society with the University of Potsdam. Within the Albert Einstein Institute it is in close contact with the division “Geometric Analysis and Gravitation”, currently led by Prof. Dr. Hermann Nicolai on an interim basis. At the University of Potsdam it cooperates in particular with the research group “Partial Differential Equations”, led by Prof. Dr. Jan Metzger. The group "Geometric Measure Theory" may include students aiming for a Bachelor or Master degree as well as PhD students and postdocs.
In order to successfully study regularity problems in Geometric Measure Theory, a good knowledge of the more basic part of Geometric Measure Theory (rectifiability, varifolds or currents) and of, at least linear, elliptic partial differential equations (Schauder and Caldéron-Zygmund estimates) is usually indispensable. This in turn relies on more elementary parts of measure theory & real analysis (e.g. the Riesz-Radon representation theorem & covering theorems, area and coarea formulae) as well as Sobolev functions and some basic functional analysis. The teaching of the group addressing Bachelor students and Master students will reflect these needs. Moreover, for PhD students and postdocs, on important contributions, both classical and new, will be held from time to time.
The study of the higher-dimensional Plateau problem involving surfaces of dimension at least three naturally leads to the use of Geometric Measure Theory. If one intends to model soap bubbles and soap films, the same statement holds for the two-dimensional case. The reason being the existence of powerful compactness theorems in the setting of Geometric Measure Theory and the inherently singular nature of the surfaces to be modelled, respectively. The existence theory in suitable subclasses of varifolds or currents has developed into a largely satisfactory state by now. Moreover, several fundamental regularity theorems have been obtained.
Nonetheless, there still exist some very important open problems in particular in regularity theory. The main objective of the research group is to contribute to the solution of these questions, concerning especially cases in which it is already known that classical regularity cannot be expected. Here, the task decomposes into two parts. Firstly, powerful, suitably adapted notions of regularity need to be studied and, occasionally, even invented. Subsequently, it has to be proven that solutions to important variational problems in fact possess these properties.
Geometric Measure Theory has found diverse applications in mathematical theory. In Geometric Analysis, "Geometric Evolution Equations" and "Geometric Variational Problems" such as the Plateau problem should be mentioned. Further applications occur for instance in the theoretical treatment of processes such as image reconstruction or crystal growth which appear to be quite different at first sight.