Reading Ravenel's ["green book"](http://www.math.rochester.edu/u/faculty/doug/mu.html), I wonder about his question on p.15 "that the spectrum MU may be constructed somehow using formal group law theory without using complex manifolds or vector bundles. Perhaps the corresponding infinite loop space is the classifying space for some category defined in terms of formal group laws. Infinite loop space theorists, where are you?". What is the state of things on that now? 



complex cobordism from formal group laws?
As far as I know, there is still no such interpretation. The closest I've heard is some rumored (but unpublished) work in derived algebraic geometry interpreting MU as some kind of representing object.


Such a construction of MU in terms of formal group data be very welcome (probably even more now than when Ravenel wrote the green book).


EDIT: Some elaboration.


We do know a lot about MU. We know that it has an orientation (Chern classes for vector bundles), and in it's universal for this property. It's not then extremely suprising that we get a formal group law from the tensor product for line bundles, but the fact that MU carries a universal formal group law, and that MU ^ MU carries a universal pair of isomorphic formal group laws, is surprising. At this point it's something we *observe* algebraically. Even Lurie's definition of derived formal group laws, assuming I understand correctly, is geared to construct formal group laws objects in derived algebraic geometry carrying a connection to the formal group law data that we already know is there on the spectrum level, and hence ties it to the story we already knew for MU implicitly.


Some reasons these days we might want to know how to construct MU from formal group law data:


* Selfish, ordinary homotopy-theoretic reasons. It's very useful to be able to construct other spectra with specific connections to formal group law data (like K-theory, TMF, etc) and constructing them is generally very difficult. Things like the Landweber exact functor theorem, the Hopkins-Miller theorem, and Lurie's recent work give us a lot of progress in this direction, but they only apply to restricted circumstances. None of these general methods will construct ordinary integral cohomology, corresponding to the additive formal group law (only rational cohomology). If we understood how to build MU, we might understand how to generalize.
* Equivariant homotopy theory. I would tentatively say that we don't have nearly as good computational and "qualitative" pictures of the equivariant stable categories, because we don't have something like the startling MU-picture that relates it all to some stack like the moduli stack of 1-dimensional formal group laws. If we found MU by \_accident\_ then we don't really know how the analogue should play out in other, more general, stable categories.
* Motivic homotopy theory. Hopkins and Morel found that there is some data to formal group laws appearing in motivic stable homotopy theory via the motivic bordism spectrum MGL. I'm not up with the state of the art here but a better understanding of this connection would be very important too - for understanding MGL itself, but also hopefully for understanding the analogues of chromatic data in these categories related to algebraic geometry.
* (space reserved for connections to other subjects that I've forgotten)


