Mathematical models of the world around us have proven to be astonishingly successful. Much of our technological progress is based on quantitative descriptions of natural phenomena. Mathematics is often said to be “unreasonably effective” in this regard. Nowadays, mathematical models and statistics pervade all natural and economic sciences, and are becoming increasingly important in medical and social sciences. Mathematical models are used to describe and understand quantitative relations between events or measurements. In many applications there is special emphasis on statistics, as a mathematical method to extract meaningful information from large amounts of data.
A large part of the department’s research concerns mathematical modeling of real life phenomena in various areas like physics, life sciences, forensics and finance. For example, much is known about how patterns such as the red spots typical for measles, our finger prints, or the convection rolls in the atmosphere and oceans that shape our climate, develop from a homogeneous state. However, we are still unable to predict or analyze the features of fully developed patterns. These nonlinear problems require the integration of computations with topology, with the final goal to create topologically validated computational machinery for finding the paths along which dynamical systems change from one state into another.
Phenomena and situations in which chance (“randomness”) plays a role are typically described by probability models. Typical questions are: how likely is it that a suspect committed a crime, which genes play a role in the recovery from a spinal injury, what is the optimal strategy for reducing the waiting times in a hospital? The randomness may be intrinsic to the phenomenon (e.g. in genetics, the outbreak of an epidemic, or financial time series), arise as unavoidable noise or sampling error in a biological experiment, or be the result of conscious randomization in clinical trials. Statistics is the art of drawing conclusions about random phenomena. Our statistical research is directed at developing new probability models, and at inventing and investigating statistical methods to apply these models to empirical data.