Biological NetworksNetwork models describe connections among entities in a system. Such models are used across many scientific disciplines, including economy, sociology, biology, medicine, and physics. We develop general frameworks for network models and investigate their properties in applications. We primarily focus on networks in biology and medicine. For example, we infer brain connectivity networks from fMRI data. Our models show how different parts of the brain are connected with each other, and how the connectivity patterns differ between healthy subjects and subjects with Alzheimer’s or other brain related diseases. We also determine gene regulation networks in the genome. Our studies uncover genegene interactions in bacteria, mice, or humans, and give insights into smallscale biological processes. 

Highdimensional TheoryThe number of observed parameters in contemporary data sets is often much larger than the number of samples. To obtain good estimates in such settings, all information about the data needs to be incorporated, and the methods need to be carefully calibrated. We develop corresponding statistical tools and equip them with mathematical theory. For example, we establish calibration schemes that satisfy rigorous mathematical bounds. 

Machine Learning and Big DataThe advent of highthroughput technologies allows one to collect data at unprecedented frequencies. This leads to enormous data sets that are too large for traditional analysis techniques. We develop algorithms and software that can address these new challenges providing fast analyses even of data with many millions of samples. 

Empirical ProcessesEmpirical processes are a pivotal concept in mathematical statistics. We are especially interested in concentration bounds for empirical processes. Classical examples are Bernstein’s and Höffding’s inequalities. In our research, we develop new bounds that relax the assumptions on the underlying model. These bounds serve as a theoretical basis for evaluating statistical methods, both in classical as well as highdimensional settings. 