# Research

There is an introduction to my research for non-mathematicians and mathematicians from other research areas, including general themes and a few words on selected publications. Other pages in this section may use some terminology and concepts of advanced mathematics.

Currently, I am mostly interested in various topics in machine learning, among them

Functions which are well approximated by deep, but not shallow neural networks.

Convolutional neural networks and the use of symmetry in machine learning.

Various optimization algorithms for the weights of a neural network and the impact of parameter initialization.

The techniques involved stem from optimal transport theory, PDEs, the calculus of variations, high-dimensional probability, statistical learning theory, and functional analysis. Other interests of mine are differential geometry, geometric measure theory, and the continuum mechanics of solids.

Many of my earlier projects were motivated by problems in materials science. In particular, I have worked on phase field models and/or Willmore's energy (elastic bending of thin structures) and on the motion of crystal dislocations (plasticity theory). While I have focused on analytical results, I like to test hypotheses and illustrate theorems by simulations.

You can find a complete list of my publications here, on Google Scholar, Orcid and ArXiv. Some publications are also available on cvgmt or DeepAI respectively.