Integral Structures in Geometry and Representation Theory

Integral structures are found in many places throughout mathematics: as lattices in Euclidean space, as integral models of reductive groups and algebraic schemes, or as integral representations of groups and associative algebras. Even questions about the most basic example of an integral structure, the ring of integers $\mathbb{Z}$, quickly lead us into the fields of analysis, algebra, or geometry. In the same vein, investigations of integral structures are most successfully treated by viewing them from different perspectives, often requiring the usage of the most advanced mathematical methods and frequently leading to unexpected connections.

This point is illustrated by the classification of wallpaper groups, i.e., discrete groups of isometries of the plane that contain two linearly independent translations. As intricate double-periodic arabesques, we encounter them in the medieval Alhambra palace in Granada. It is a classical fact that there are precisely 17 wallpaper groups. This result has a geometric aspect, as it provides the number of flat compact orbifold surfaces. It also has an interpretation within representation theory: it is part of the classification of hereditary categories over the field of real numbers. The analogous classification of space groups in three-dimensional space is of importance in physics and chemistry.

As integral structures necessitate a combined approach from different mathematical sub-disciplines, we will embark on a broad research programme that underlines the unity of mathematics. Our endeavour ranges from algebraic geometry to analysis on manifolds, from geometric group theory and algebraic combinatorics to representation theory of associative algebras.

This Collaborative Research Center is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) — SFB-TRR 358/1 2023 — 491392403.