# Event № 975

*Abstract:*

A fundamental construction in homotopy theory is p-completion. It is a functor from spaces to spaces that is characterised by turning maps that induce an isomorphism in mod-p homology into homotopy equivalences, and thus allows studying homotopy properties of spaces "one prime at a time". Of particular interest is the homotopical properties of p-completed classifying spaces of finite and compact Lie groups, and maps between such objects. The theory of p-local groups was inspired by advanced studies in modern modular representation theory of finite groups, and was developed with the aim of providing a very general framework in which the homotopy theory of p-completed classifying spaces can be studied. In this talk I will describe the basics of the theory of p-local groups, the motivation for creating it, and some of the fundamental results obtained using it.