Kammern, Apartments, Gebäude : Aktuelle Trends in der modularen Darstellungstheorie
Gebäude sind mathematische Objekte, die Geometrie und Algebra auf faszinierende Weise in sich vereinen. Dieser Beitrag gibt einen Einblick in diese Theorie und erklärt ihren Einfluss auf neuere Entwicklungen in der mathematischen Grundlagenforschung.
Buildings are geometric objects of a combinatorial nature. They are built from smaller elements called chambers and apartments. The article provides an introduction to this fascinating theory and explains the use of everyday language in mathematics. It starts out from the Heawood graph in Figure 1 and describes the algebraic model needed to analyze it. The author also comments on historical developments. Starting from Felix Klein’s Erlangen programme, the interaction and the unification of algebra and geometry have become a driving force in modern mathematics. The theory of buildings was initiated by Abel Prize laureate Jacques Tits. It is part of this fascinating story. A particularly important class of buildings was introduced by Jacques Tits and François Bruhat. As an easy example, the article discusses the Bruhat-Tits tree shown in Figure 4. It also explains the connection with reflection groups via triangulated planes. At the end of the article the author briefly describes in which way buildings are relevant for his research. This concerns questions from number theory related to the Langlands programme.