Title:  The mod-2 cohomology of symmetric groups
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Abstract:  The mod-p homology groups of symmetric groups, including a fairly canonical additive basis, has been known for almost exactly fifty years now.  Moreover, the linear duals of the cup product and Steenrod algebra actions have been known for almost as long.  A number of mathematician have off-and-on been searching for intrinsically cohomological descriptions of these structures ever since.  Collaborators and I took this question up recently and think we have at least tamed if not slayed this dragon, at the prime two.  The key notion is that of a Hopf ring, which can be viewed as a pleasant convenience or a deep organizational tool.
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I'll start by reviewing some history and talking about the homology in geometric terms.  Then I'll give some examples of Hopf rings from classical algebra which can be recast in this language (and thus reaffirm Jim's opinion that I'm really an algebraist, which I'm not really...).  Finally I'll talk about recent results, joint with Giusti and Salvatore.