I'm in Las Vegas for FOCS 2010. It's been a while since I've attended FOCS, and the short distance from Salt Lake was definitely a factor.
Kudos to the organizing committee for the tutorial sessions today. I've long whined about the lack of non-paper-presenting events at theory conferences, and the topics for today were great ones. Unfortunately, a spectacular debacle on the part of the hotel restaurant meant that I (and 11 other poor souls) missed most of Mihai's tutorial on data structures.
But the first tutorial by Ketan Mulmuley was fantastic. Let's be clear. It's impossible to convey any depth of material on GCT in 1.5 hours, but he hit (to me) just the right level of high level exposition and low-level technical conjectures. I also learnt something brand new from the presentation, which I'll try to summarize.
In Godel's incompleteness theorem, we all know that the key step is phrasing a statement of the form "I am not true". The self-referentiality of this statement is what drives the paradox within the logical system that allows you to frame it.
The part that's important to remember is that the logical system has to be powerful enough in the first place, so that it's possible to even do self-referencing. A weak logical system can't do that, which is why Godel's theorem only kicks in once the system is strong enough (a very Greek tragedy concept this, that it is your own power that sows the seeds to destroy you).
It turns out that a similar mechanism comes into play with P vs NP. Mulmuley starts with a "trivial proof of hardness" for a function f. Write down all circuits of small complexity, and for each circuit write down some input for which C(x) != f(x). This table is of course of exponential size, but never mind that.
He now introduces the flip operation. This is the idea that instead of trying to manipulate this large table, we actually store a smaller table of certificates, with the property that from one such certificate, you can reconstruct efficiently a set of circuit and witness pairs from the table, and the set of such certificates cover the entire table. You also need to able to decode these certificates efficiently and quickly verify that an input produced in this manner actually does demonstrate a counter example.
There's a problem with the flip operation though. We're using this flip structure to show that P != NP (nonuniformly for now). On the other hand, the success of the flip operation requires us to construct a mapping from the certificates to the table that can also be verified in polynomial time. But since f is an NP-complete problem, what we are essentially saying is that we can efficiently construct short witnesses of membership or non-membership, which means that we've collapsed P and NP !
You could of course retort by saying, "well I don't want to use the flip operation then !". It turns out that you can't ! He states a result saying that ANY proof separating P and NP will essentially look like a flip-based proof. So no matter what you do, you have to deal with the self-contradicting structure of the flip. The geometric construction (GCT) is his program for constructing the flip so that this cycle of referencing can be broken.
But that's not where the analogy to Godel's theorem comes from. When we start digging into separating the permanent from the determinant, we can translate the problem into an embedding problem over algebraic varieties. This is because both the permanent and determinant have unique characterizations as the only polynomials satisfying certain properties over matrices.
Now the separation problem can be framed as the problem of finding an efficient obstruction that demonstrates the separation between these two varieties. What's even more bizarre is that ANY proof that separates the permanent and the determinant will end up producing such an algebraic-geometric obstruction, even if the proof itself doesn't use algebraic geometry.
So let's review. Any proof of P vs NP will induce a "flip" construction that generates small certificates that explicitly show hardness of a problem. The permanent is one such hard problem, and lends itself to an invariant representation that yields a key obstacle that we need to construct in polynomial time in order to make the flip work.
In other words, we need to be able to construct an object in polynomial time (that to date we don't know how to do in less than triply exponential time) in order to show that these very same poly-time computations CANNOT solve the permanent.
And this is the core paradox at the heart of the program. Our inability to prove that P != NP is because we can't show that P is strong enough to express within itself the certificate that it is NOT strong enough to capture NP.
Mulmuley makes another very interesting point at the end of the tutorial. Many people have wondered why the GCT framework can't be used to prove "easier" bounds for other separations. While he mentions some examples of problems (matrix multiplication) where people are attempting to use the GCT framework to show poly lower bounds, he also points out that it is only because of the beautiful invariant representations of the permanent and determinant that we can bring the machinery of algebraic geometry to bear on the problem. As he says, we got lucky.