In the last two days a firestorm has erupted over a claim by HP Labs computer scientist, Vinay Deolalikar, http://wwww.hpl.hp.com/personal/Vinay_Deolalikar/Papers that he has proven one of the six remaining Clay problems for which a million dollar prize is offered. In particular, he claims to have proven that polynomial time algorithms (P) cannot in general solve exponential time problems (NP), or are not equivalent in power to them. It has long been suspected that this is the case, but has not been proven so far. This has implications for computational complexity theory and computable and computational economics.
The paper is over 100 pages long and involves ideas drawn from model theory in logic as well as statistical physics, an approach not used for anything previously, and so far in the enormous debate that has erupted, there is not yet a definite answer if Deolalikar is right or not, although some minor errors have definitely been found. An aggregator site for this debate is at http://michaelnielson.org/polymath1/indep.php?title=Deolalikar%27s_P!%3DNP_paper. This site provides links to others where more detailed debates have been occurring, with the ones starting with http://rjlipton.wordpress.com being especially insightful and providing most of the mathematical meat picked up on the aggregator site.