Subject: Re: comments re problems From: Mark Hovey Date: 05 Mar 1999 02:24:37 -0500 ---------- I agree with both Brayton and Jim. But I don't know anything about unstable homotopy theory, except a bit about model categories, so I need some help here. I should certainly add the Dwyer-Wilkerson program of classifying all the finite loop spaces... I do actually think there are problems external to algebraic topology on the list, but the problem is that all the good differential geometry applications of algebraic topology seem to be gone. The general idea might that be that elliptic cohomology is to infinite-dimensional differential geometry what K-theory was to ordinary differential geometry, but we have not been able to do much with that yet. It seems to me that the geometry we might be able to say something about will increasingly be algebraic geometry, and that this will be less satisfying to those of us who remember algebraic topology answering comparatively elementary questions like classifying the central simple division algebras over R. But hey, if you have some ideas for more applied problems, by all means send them to me. I can think of one I missed--the Rosenberg conjecture about positive scalar curvature metrics when the fundamental group is unpleasant, and the corresponding questions about metrics of positive Ricci curvature. Mark