Professor Joseph Neisendorfer (University of Rochester, USA) lectures a short course on Localization Theory at Universidade do Minho, Braga. The course takes place on October 18, 20, 25 e 27, 2006, at MICA (Departamento de Matemática), from 2h30 to 4 pm. The programme is the following:

Emmanuel Dror-Farjoun's magical theory of localization

Lecture 1: The existence of geometric localization and an algebraic analogue.

From this point of view, geometric localization involves making spaces equivalent to a point and modules equivalent to zero in some sort of universal way.

Lecture 2: This localization includes completion at a prime.

The standard definition of completion at a prime is in fact only an approximation to the better homological definition involving ext.

Lecture 3: Zabrodsky' lemma and consequences of Haynes Miller' solution of the Sullivan conjecture.
                      
Miller' solution to the Sullivan conjecture amounts to saying that finite simply connected complexes are local with respect to the killing of the classifying spaces of finite groups. This allows us to reverse the process of taking connected coverings of simply connected finite complexes, provided we are willing to complete the spaces and restrict to those for which the second homotopy group is finite.

Lecture 4: Applications of localization and completion to H-spaces and loop spaces.

We shall show that the process of trying to make finite complexes into H-spaces by taking connected covers is doomed to failure. And although the iterated loop spaces of localized spheres have power maps which are null homotopic, a certain large number of loops is necessary before any power map, even on a connected cover, is null homotopic.