Homotopy theory of stratified spaces

Homotopy theory, the study of paths in a space, is the source of some of our deepest topological insights. Stratified spaces, spaces with a given decomposition into ‘nice pieces’ which fit together in a reasonable way, are a natural generalisation of spaces which arise in many contexts. They have been extensively studied, but their homotopy theory has been relatively neglected. In this talk I will give some examples and outline a general framework for studying the homotopy theory of stratified spaces, and some of its consequences.