The notion of groupoid has been important in homotopy theory (Poincare groupoid of a space). It has been extended by Ch. Ehresmann in the theory of fiber bundles. Its importance came from the efforts to understand highly singular quotient spaces, like the space of leaves in a foliation (Haefliger, Connes). In the same way as a manifold can be described by different equivalent atlases, a stack (a new kind of spaces) is described by an equivalence class of groupoids. It is possible to do differential geometry and to define topological invariants for such generalized spaces . Finally, as was first suggested by Malgrange and Umemura, in certain Galois theories (like the Picard-Vessiot theory for differential equations), Galois groupoids are better suited than Galois groups.

Organizations
Independent University of Moscow