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Topic: Bousfield Localization (3 lectures)

Luke says: I could give the second or third talk, on the Bousfield lattices of various categories - in particular the stable homotopy category, and derived categories of (Noetherian and non-Noetherian) rings. This is what my research is on, and it's quite accessible - with a lot of pretty lattice theory. I could also discuss cohomological localization, and cohomological Bousfield classes, since it's related and I'm also doing current work on that.
(this sounds like a great third talk. I have some ideas about what should happen in the first two talks, but I want to think about it a bit more. --Philip)

Topic: Triangulated Categories (3 lectures)

plan here, if you want.

Topic: Stable Homotopy Categories (4 lectures)

Suggestions for talks (by Anna Marie and Rolf):
--Edited by Marcy
--Edited by Anna Marie again
--Edited by Rolf

Lecture 1--basic intro (what general properties should the stable cat have, other suggestions?)--I think this is the "intro" to talk two.

(Some thoughts from Dylan): I think it would be nice to do a bit of history here to motivate the stable homotopy category, proper. My suggestion would be to come at it from three different angles at once- (1) generalised homology theories, (2) cobordism and Thom spaces, (3) Freudenthal's theorem. That should just be like 10 minutes; state the theorems and the history. Then talk about Brown representability, tie it all together, the usual thing. I think the thing that needs to be decided on in this talk is: how much of it should be technical and how much of it should be historical? I realize that others may want to start straight into what it means to have a "stable homtoopy category" but I feel like if we don't motivate the axiomatic viewpoint of the second lecture with the historical stuff for the original stable category. But I do agree this should be the intro to talk two, so perhaps the second half of this lecture can lead into picking up on the main parts of the stable homotopy category that generalize to the others: cofiber sequences, smash products, nilpotence theorems...

Anna Marie says: I like those ideas for motivation--I think we should avoid being too technical in the first talk, since the technicalities are often obfuscating. And that does do a reasonable job of motivating the axiomatic approach.

Rolf says: I'd be willing to give this talk, I would probably lean on the side of not covering technical issues, focusing instead on historical or conceptual motivations. I think I'd prefer to talk a little about K-theory as well, since it gives a cohomology theory more easily than MO/MU, and the represented picture in cohomology is more accessible via this Brown representability approach.

Lecture 2--axiomatic stable homotopy categories--anybody have a vision for this? --I don't know about vision, but I just skimmed through the Hovey-Palmieri-Strickland paper "Axiomatic Stable Homotopy Theory" and there are lots of possible talks there. Maybe talk one could cover a lot of the intro material in this paper, and talk two could cover some of the more interesting examples? Who's volunteering to give this talk?

Lecture 3--modern foundations (Rolf suggests that good motivation for S-modules is the key to a talk about them. There's also plenty of room for talking about general framework for diagram spectra) Marcy, do you have thoughts on this topic?--I would hope that the motivation came from the previous talks; I am happy to give this talk, and happy not to. I would personally probably define/discuss properties of symmetric spectra (more) and maybe S-modules(less); probably would not discuss the model category aspects of these definitions unless someone really wants to hear about that. Comments?--Anna Marie: I personally find model structures don't usually make for the most interesting talks, so I'd be happy to gloss over that. I think Rolf meant that we motivate modern foundations by asking to be able to do brave new algebra, but he should clarify.

Lecture 4--equivariant stuff (I can definitely give a basic talk about this--I would cover the importance of universes, representation and orbit spheres and restriction/induction, or something like that)-- I want to hear about these things, what would you need from the previous talks?---Anna Marie: the one thing I'd really to have had introduced is the notion of universes and indexing on universes, because you sort of can't escape that idea equivariantly. S-modules and orthogonal spectra generalize much more nicely to the equivariant world than symmetric spectra.

We could also sub out equivariant stuff for something else if that seems better. We could also go more in depth and give two talks on some subjects (perhaps the first two) instead of covering everything.

Topic: Moduli Spaces (2 lectures)

plan here, if you want.