A global perspective on symmetries 

Stefan Schwede is a professor at the University of Bonn in Germany. Thanks to a grant from the Knut and Alice Wallenberg Foundation, he will be a visiting professor at the Department of Mathematics, Stockholm University. 

Symmetries and regular patterns are everywhere around us – from the symmetry of a human face to the repeated cycle of the seasons, the intricate structure of crystals and the deep symmetry of the fundamental laws of physics. Symmetry also plays a central role in mathematics. One of mathematics’ main tasks is to discover symmetry and extract information from it.  
Mathematics encodes the concept of symmetries in the notion of a group. Roughly speaking, a group in mathematics means the collection of symmetries that an object may possess. For example, the hexagonal symmetry of a snowflake will form a group with six elements. 

This project advances a global approach to the study of symmetries. The philosophy here is that one does not just focus on the symmetries of one object at a time, or on one group at a time, but considers simultaneously all possible groups, and all possible types of symmetries, as parts of one big picture. The intent is to extract information about the symmetries of an individual object by studying its interactions with many other objects.

The global approach entails a new way of looking at symmetrical objects and has already led to dramatic progress in our understanding of the relationships between geometric shapes, which is also the subject of this research.