Stratifications first appeared as a way of dealing with singularity. For example, in algebraic geometry, Whitney showed that any singular variety could be decomposed into smooth varieties, satisfying gluing conditions. More generally, the notion of pseudo-manifolds allowed for the extensions of numerous invariants of manifolds to objects with singularities. Those extensions are only invariant under stratum-preserving homotopy equivalences however, which motivates the introduction of a homotopy theory for stratified spaces.
In this talk, I will present such a stratified homotopy theory, and explain how it can be characterized in terms of strata and links - those objects that encode the gluing conditions between strata. On the other hand, some structures in topology - such as embeddings - can be encoded as stratifications on a space. In those cases, the links and strata can usually be interpreted in terms of the structure of interest. I will explain how the homotopy theory of stratified spaces can be used to extract information about those objects.