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This is part of an algebraic topology problem list, maintained by Mark Hovey.

1. Build MU from the moduli stack of formal groups. This has got to be doable somehow, though it is an old problem (I first heard it in Ravenel's green book). Note that we have some more tools now--the moduli stack is, I think, just a space in Voevodsky's category, so maybe it is an infinite loop space there?
   
   
2. Classify all possible Bousfield classes of E-infinity ring spectra. I know very little about this problem. Note that the Spanier-Whitehead dual of the suspension spectrum of a space X is an E-infinity ring spectrum, with multiplication dual to the diagonal map. But you are forgetting the disjoint basepoint!! When you add, as you must, a disjoint basepoint to X, you find that the E-infinity ring spectrum has the Bousfield class of the sphere. Also, I once heard somebody--Jim McClure?--say that if you start with, say, MU, and you mod out by p, you find that you must also mod out by all the higher v's too in order to get an E-infinity ring spectrum. Therefore, I conjecture that if E is an E-infinity ring spectrum that kills a nontrivial finite spectrum X, then E has the Bousfield class of E(n) for some n. (I must be p-local here). We know these Bousfield classes do occur, since Morava E-theory is an E-infinity ring spectrum. (Goerss-Hopkins).
   
   
3. This one is due to Mike Hopkins. Generalize the whole Thom spectrum business as follows. Take an A-infinity ring spectrum E. Look at the space of A-infinity self equivalences of E. I think this has a classifying space B, because of composition of self equivalences. Given a map X --> B you should be able to construct a Thom spectrum, and it should be some kind of half-smash product of E and X. Mike had a more explicit description, which I seem to have forgotten.

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