Fusion Systems and Localities

The abstract algebraic concept of a group is central in contemporary mathematics, since it allows us to handle objects of a diverse mathematical origin in a uniform way. Groups arise for example from symmetries of other mathematical or geometrical objects. On the other hand, a given group leads again to new algebraic, topological and geometric structures. In particular, to every group one associates a topological space, called the classifying space. This leads to many different ways of studying groups. However, it is a common feature of different approaches that they focus on properties related to a fixed prime p.

Saturated fusion systems provide a new algebraic framework which allows us to discover more and more connections between the different approaches related to a prime. There are however still many open questions remaining. For the prime 2, it is an important long-term goal to classify the simplest building blocks of saturated fusion systems. In this project, we will overcome some major conceptual difficulties arising on the way to such a classification result for fusion systems. This will feed into a simplified proof of the classification of finite simple groups, which is an important theorem classifying the simplest building blocks of finite groups. The new algebraic theory we develop will also allow us to study structure-preserving maps between classifying spaces of groups "at a prime p". We expect that this will open new ways for research in homotopy theory and representation theory.

Inside mathematics, our research will benefit researchers working in group theory, representation theory and homotopy theory, and it will foster interactions between these different areas. One long-term impact of our project is also a simplified proof of the classification of finite simple groups. This will be of interest to mathematicians from many branches of mathematics.

As our research is in pure mathematics, it is not possible to predict exactly how our results will impact other fields of research or contribute to societal change and economic growth. However, groups and their representations play for example an important role in theoretical physics and in chemistry, and homotopy-theoretical methods are a powerful tool in data analysis.

An immediate and predictable societal impact of our project is the training of the research associate. The research associate will be trained both in analytical and mathematical skills, and in communication skills.