TRR 358: Integral Structures in Geometry and Representation Theory

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Integral structures arise 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 Z, very soon lead 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 require the usage of most advanced mathematical methods, and frequently lead 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 meet 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; and it also has an interpretation within representation theory: it is part of the classification of hereditary categories over the field of real numbers.As integral structures necessitate a combined approach from different mathematical sub-disciplines, we will embark on a broad research programme reaching from algebraic geometry to analysis on manifolds, from geometric group theory and algebraic combinatorics to representation theory of associative algebras. With joint forces from the participating universities, we intend to answer major questions in the algebraic and analytic theory of automorphic forms, categorical representation theory and algebraic geometry, as well as classical and p-adic harmonic analysis on symmetric spaces.