Title: Spaces of Continuous Maps and the Cohomology of Commutative Rings
Speaker: Professor Paul Goerss
Speaker Info: University of Washington
Abstract: Homotopy theory might be defined as the study of certain types of topological phenomena that remain invariant under continuous deformations. Initially, the fundamental problem in the field was considered to be the computation of the homotopy classes of maps [X,Y] between two spaces X and Y. As the field matured, this problems was assumed into the larger problem of understanding the homotopy type of the entire space map(X,Y) of continuous maps from X to Y -- the set [X,Y] is in one-to-one correspondence with the path components of map(X,Y). In the last twenty years, this wider point of view has yielded remarkable successes.Date: Thursday, October 23, 1997
A fundamental tool along the way -- and a major concern of mine for some time -- has been the Andr\'e-Quillen cohomology of commutative rings. Originally developed in commutative algebra and used in deformation theory, this cohomology naturally arises in trying to understand spaces of continuous maps. Using Andr\'e-Quillen cohomology -- which I will define -- as my theme, I'd like to trace through some of the developments in homotopy theory in the last decade or so, bringing into play the work of Miller, Lannes, Dwyer, Wilkerson, Hopkins, and myself. This will give me a chance to emphasize how much the field has grown, expanded, and matured.