Mini-Workshop: Geometric Measure Theoretic Approaches to Potentials on Fractals and Manifolds

Abstract

The workshop brought together researchers and graduate students from different areas of mathematics, such as analysis, probability theory, geometry, and number theory. The topics of joint interest were motivated by recent problems in potential theory with impacts into these areas: • discrete approximation to energy minimising measures • potential theory on fractals and manifolds • geometric measure theory on fractals • probabilistic potential theory • spectral theory on fractals and sets with fractal boundary. The format of a mini-workshop was especially well-suited for our subject, since it allowed enough time for personal discussions besides the talks given by the participants. The concept of energy of a charge distribution on a subset of Euclidean space is one of the core subjects of potential theory. Recent generalisations of this concept to hyper-singular energy kernels and discrete N –point distributions exhibit a close connection with ideas from geometric measure theory. A recent article by two of the organisers shows that N –point configurations minimising the discrete energy in the hyper-singular case can be used to characterise the Hausdorff measure on d–dimensional d–rectifiable manifolds embedded in Euclidean space. Such minimal energy point sets can be used for the discretisation of manifolds, which has numerous applications. On the other hand discretisation by graph structures is a common means for analysis on fractal structures. Usually, a diffusion and an associated Laplace operator are defined by rescaling discrete random walks and their transition operators on the approximating graphs.