Mathematics underpins many of our scientific and technological advances. It allows us to model and analyze phenomena within the physical universe and, more recently, aids us in the creation and understanding of the digital world. At the heart of mathematics are four key pillars: algebra, analysis, geometry, and stochastics. Together with their many fascinating interrelations, they provide the basic foundations and tools for a large landscape of important subbranches of mathematics. Generally speaking we can say that
- Algebra studies discrete structures and their relations. It includes for example the study of groups, rings, and graphs. These abstract concepts are key in modelling concrete problems in a conceptual framework.
- Analysis deals with the continuous, and models this by means of the tools of calculus, such as limits and derivatives, as well as more abstract theories such as topology or measure theory. This field is perfectly suited for studying the behavior of natural signals and systems, such as vision or audio and dynamics governed by (partial) differential equations.
- Geometry is, in its simplest form, the study of shape and position of structures such as points, lines and surfaces defined on finite or infinite sets. More recent breakthroughs in mathematics have completely transformed our concept of space, leading to many novel insights and forms of geometry with which physical theories can then be built to understand the universe around us.
- Stochastics studies randomness and uncertainty, and models these using probability theory and statistics. It involves the analysis of random processes, probability distributions, and statistical inference. This branch of mathematics is indispensable for understanding and predicting phenomena in which variability and chance play a role, such as in financial markets, quantum mechanics, and artificial intelligence.
To solve problems in the mathematical, physical or digital world, we often create toy models and examples, or collect data to gain insights into the underlying patterns and relationships. Due to recent advances in the field of hardware, we can now combine mathematical modeling with data science, using techniques such as statistics and deep learning, to create powerful models and predictions. Think of applications such as Computer Vision, Large Language Models, Automated Theorem Provers, and Quantum Computing.
More information about our member's research areas.
Head of the Large Research Group: Prof. Dr. Ann Dooms