Buildings and their applications in geometry and topology

In this paper, we briefly introduce different types of buildings such as spherical buildings, Euclidean buildings, twin buildings, R-buildings and describe some of their applications to many subjects: (1) differential geometry such as Mostow strong rigidity, rank rigidity of nonpositively curved manifolds, Margulis superrigidity, quasi-isometry rigidity, classification of isoparametric manifolds, compactifications of symmetric spaces and locally symmetric spaces, (2) topology such as cohomology and duality properties of arithmetic groups, simplicial volume of locally symmetric spaces and Novikov conjectures, (3) analysis such as harmonic maps, harmonic analysis and representation theory of p-adic Lie groups, (4) algebra such as finite simple groups, infinite simple groups, algebraic K-theory and representations of algebraic groups over finite fields. By putting together many different types of buildings and applications in seemly unrelated topics, we hope to present an overview of rich structures and applications of buildings and the underlying groups.