Two more comments on Mark Hovey's problem list.........DMD ______________________________ Date: Fri, 5 Mar 1999 13:36:01 GMT From: Subject: Re: Hovey response It seems to me that there ought to be a lot of problems in geometry and "physics" where algebraic topologists could say something interesting, but as far as I know there isn't much work going on in that direction. There are a lot of moduli spaces that people like to think about (in Yang-Mills theory, Seiberg-Witten theory, solitons, etc) that people like to think about, but I've never heard of anyone looking at invariants more sophisticated than rational homology. The Morava K-theory of these spaces must surely be interesting, and any insights leading from such calculations would be a good advert for the subject. I'd love to explore in this direction if I had more time. Neil Strickland _______________________________________________- Date: Fri, 05 Mar 1999 09:52:17 -0500 (EST) From: Jack Morava Subject: re: Mark Hovey's problem list I think Mark Hovey deserves a lot of thanks for his problem page. I hope the list grows, like a crystal or a snowball, by additions and commentary from the mathematical community. Everybody is going to have something to suggest. There is a lot of activity in equivariant homotopy theory in general [and equivariant elliptic cohomology in particular] which deserves some discussion; and there is a classical cycle of questions about relations between geometric and homotopy-theoretic equivariant bordism, and equivariant transversality [with a large, distinguished, and nowadays almost forgotten literature]. There is a large and growing European school of `brave new algebra', which proposes (more or less) that anything you can do in an additive category is probably a specialization of a more natural construction in categories enriched over spectra... In Frank Adams's heyday the gap between algebraic and geometric topology was just beginning to yawn. It's pretty clear that Mark's list is primarily concerned with {\bf algebraic} topology, but I don't think it's not connected to other parts of mathematics; it's just closer to algebra. Classification of differentiable structures on manifolds led to a deep reinterpretation of the classical theory of quadratic forms, to algebraic K-theory, and to the J-homomorphism. Modern stable homotopy theory is heavily involved with modular forms and their generalizations, the p-adic symmetric spaces of Drinfel'd, generalized triangulated categories, and many other very active subjects. I was told (not by Mike Hopkins) that, during the course of Mike Hopkins' talk at Barry Mazur's 60th birthday conference, Mazur and Tate finally understood that the Adams-Novikov E_2 term is the sheaf cohomology of the moduli space of one-dimensional formal groups. I would like to have seen that.