Manifolds are nonsingular, in the sense that each point has a neighbourhood that looks like a Euclidean space of fixed dimension. The ordinary cohomology of a manifold reflects certain hidden symmetries which are encoded by Poincaré duality. This duality enables the construction of powerful invariants that play central roles in understanding the geometry of manifolds.

Many spaces occurring in mathematics and physics, however, have singular points, although they can be decomposed into manifolds of various dimensions; i.e., they can be stratified. Examples of stratified spaces include orbit spaces of smooth group actions, compactifications, PL spaces, algebraic varieties, etc. Stratified spaces play an important role as a cycle reservoir for geometric descriptions of generalized homology theories. Furthermore, stratified spaces appear in applied areas, such as topological data analysis, or the study of configuration spaces for robot motion planning.

The ordinary cohomology of a stratified space need not satisfy Poincaré duality. There are essentially two (now classical) ways to address this problem -- Goresky-MacPherson's intersection cohomology and Cheeger's L2-cohomology. Since the discovery of these theories, there has been great interest and success in extending to stratified spaces the triumphs of manifold theory, including the study of signatures, characteristic classes, surgery theories, and knot theory. Such extensions from manifold theory to stratified spaces are rarely straightforward; they often involve subtle interactions between local and global topology.

Complex algebraic varieties are key examples of stratified spaces, and provide a testing ground for topological theories. The appearance of singularities in varieties turns out to be a rich source of beautiful geometry and topology. The study of topological properties of algebraic varieties is an area of research that has led to a vigorous interchange between geometric topology, algebraic geometry and mathematical physics, and has recently witnessed a flurry of activity in many exciting directions. The study of stratified spaces frequently leads to new insights even for manifolds.