TRR 358: Integral Structures in Geometry and Representation Theory

积分结构出现在整个数学的许多地方：如欧几里得空间中的格，作为还原群和代数图式的积分模型，或者作为群和结合代数的积分表示。即使是关于积分结构最基本的例子，整数环Z的问题，也很快进入分析、代数或几何领域。同样，对整体结构的研究最成功的方法是从不同的角度观察它们，通常需要使用最先进的数学方法，并经常导致意想不到的联系。包含两个线性独立平移的平面等距的离散组。作为复杂的双周期阿拉伯式，我们在格拉纳达的中世纪阿尔汉布拉宫见到它们。这是一个经典的事实，正好有17个壁纸组。这个结果有一个几何方面，因为它提供了平坦的紧凑轨道曲面的数量;它也有一个在表示论中的解释：它是真实的数领域的遗传范畴分类的一部分。由于积分结构需要不同数学分支学科的综合方法，我们将开展一个广泛的研究计划，从代数几何到流形分析，从几何群论和代数组合学到结合代数的表示论。 与参与大学的联合力量，我们打算回答自守形式，范畴表示理论和代数几何的代数和分析理论，以及对称空间上的经典和p-adic调和分析的主要问题。

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.