Via Richard Lipton, news that HP research scientist Vinay Deolalikar has put online a 102 page draft of a paper purporting to prove that P is not equal to NP.  (Follow the link to Deolalikar's page, and the paper is the first one linked under "Selected Publications".)  It is being taken seriously by serious people  (like Lipton)--- in the sense that they are trying to understand it and understand whether or not it is correct.   The length, of course, increases the likelihood of a so-far unspotted error.  Lipton thinks Deolalikar's use of finite model theory is an interesting and promising twist.  I have only just downloaded the paper, and am not a complexity theory expert, but what leaps out at me from the abstract is that it involves graphical models for sets of interacting random variables, an area that interests me.  Not that surprising, I guess, as lots of difficult problems---as well as ones that are easy but perhaps not obviously so, because of structure described by a graphical model of the dependencies between variables---are naturally described in graphical terms, and the relation between graphical models and complexity is a well-studied---though mostly relatively recently studied---topic.  I can no longer find the link to the online draft of Marc Mezard and Andrea Montanari's book on the subject (and its relations to physics and coding theory), perhaps because it's been published.  [Correction:  it's still online here.   Thanks to commenter Kristal Cantwell for the link.]  One key notion in such models is conditional independence---independence of two subsets of the random variables in question, when conditioned on a third set.  A really nice paper on a quantum notion of conditional independence, relating it to the equality conditions for the strong subadditivity inequality for quantum entropy, is by Hayden, Jozsa, Petz, and Winter.  Matt Leifer and David Poulin, and others, have described quantum graphical models.

You can bet that if Deolalikar's proof looks good, a slew of people will be applying the research strategy I like to call the "quantization functor" to it.

For those for whom P and NP are not their bread and butter, to give you an idea of how big this would be: I just goofed by telling my wife that if this pans out, it will be one of the most important results of twentieth century mathematics.  Well, it would have been if done in the twentieth century.  But I'll say more: although we're only 10 years in, I think it will be one of the most important results of twenty-first century mathematics.  I haven't said what it really means but that's what Wikipedia is for.

I should point out that Scott Aaronson is dubious, though not based on close perusal of the proof.  He has pledged to personally supplement the million dollars the Clay Mathematics Institute will pay Deolalikar if the proof is correct, with two hundred grand of his own.