 Location
 MSRI: Simons Auditorium
 Video

 Abstract
This will be a survey talk on the close relationship between the local structure of a nite group or compact Lie group and that of its classifying space. By the plocal structure of a group G, for a prime p, is meant the structure of a Sylow psubgroup S G (a maximal ptoral subgroup if G is compact Lie), together with all Gconjugacy relations between elements and subgroups of S. By the plocal structure of the classifying space BG is meant the structure (homotopy properties) of its pcompletion BG^p . For example, by a conjecture of Martino and Priddy, now a theorem, two nite groups G and H have equivalent plocal structures if and only if BG^p ' BH^p . This was used, in joint work with Broto and Møller, to prove a general theorem about local equivalences between nite Lie groups a result for which no purely algebraic proof is known. As another example, these ideas have allowed us to extend the family of pcompleted classifying spaces of (nite or compact Lie) groups to a much larger family of spaces which have many of the same very nice homotopy theoretic properties.
 Supplements

