Tuesday, April 29, 2008

Should we care about geometric complexity theory ?

There's been an interesting debate over on the complexity blog with Lance's first diss of Ketan Mulmuley's program, and Ketan's response. I think Lance essentially trolled himself with the line,
Also while knowing the resolution of P v. NP is very important, knowing the details of the proof, especially if it requires deep and complex mathematics, is not nearly as important. I was excited to see Fermat's Last Theorem resolved in my lifetime but I have no desire to actually understand the proof.
Although I have a hard time imagining how someone could say this and mean it (after all, P vs NP is not Fermat's theorem in either impact, breadth or importance), and even more so how someone steeped in complexity theory could say it, I think the larger point he makes is essentially reasonable. Namely, (and I paraphrase),
if you want to lead people to the top of the mountain, you have to show them the way, or give them some hope that they can get there.
What would make the whole GCT program more plausible (and I think that people would like to believe in it) are some intermediate results. Maybe some weaker lower bounds using more primitive techniques that suggest that more powerful statements are lurking underneath. I don't think more surveys will help, frankly. I've read the Regan survey, and even the initial parts of GCTflip, and it's really heavy going. In fact, I was talking with a professor who spent an entire semester, together with students trained in complexity theory AND representation theory, trying to work through some of the early GCT papers. It was definitely more than a month's worth of effort.

Of course, there is one example of "a weaker lower bound using more primitive techniques": KM's result separating P and (NC without bit-wise operations). This paper appeared in STOC 1994, and was subsequently published in SICOMP in 1999. I've always wanted to read this paper, for one thing because of its intriguing way of mapping computations to algebraic sets (if that isn't cool geometry, I don't know what is).

So since summer is here, and semester is out, I've decided to forcibly inject some content into this blog. I'm going to read the P vs NC paper and blog about it as I go along. I don't know how many installments it will take, and there might be digressions along the way, but at the end I hope to understand this paper, and with it maybe even get a glimmer of the general direction KM is pushing in the GCT work.

For anyone interested in following along, I'll be mainly using the SICOMP paper as a reference, and will tag all posts with the tag 'p-vs-nc'. As far as possible I'll try to do one post a week (more if time permits). Let's see how this goes.
Post a Comment

Disqus for The Geomblog