Could we please continue to discuss primes? I would like to think about a hard problem in analytic number theory, however it seems to me we should first build our thinking on an appropriate for the task proof of existence of infinitely many primes. Kummers’ is considered the cleanest – is Schorn’s combinatoric enough? ]]>

You’re right of course. The reason I didn’t was that my primary purpose in writing the notes was to make sure that a certain portion of the course was adequately written up for the sake of an audience to which I had already explained the motivation for the results. But let me briefly explain what that motivation is for the sake of anyone who doesn’t know it and would like to.

The problem of whether P=NP is very closely related to the question of whether there is any problem in NP that cannot be solved with a (family of) polynomial-sized circuit(s). A circuit is a network of AND, OR and NOT gates that process an input (a string of n 0s and 1s) and produce an output (either 0 or 1). It is fairly easy to prove that if a function from to $\{0,1\}$ can be computed in polynomial time, then there is a circuit of polynomial size that will compute that function as well. This means that a potential way to prove that P does not equal NP is to prove that some function in NP cannot be computed by a polynomial-sized circuit. And this is useful because circuits are easier to contemplate from a purely combinatorial point of view than algorithms or Turing machines.

Now one might think that if the function f is monotone, which means that the set of input strings x such that f(x)=1 is closed under changing coordinates from 0 to 1, then there would be no advantage in using NOT gates (though in fact this supposition is known to be wrong). At any rate, it is interesting to try to obtain circuit lower bounds when NOT gates are not allowed. This was for a long time as wide open as the general problem, until it was spectacularly solved by Razborov. It is quite easy to reformulate the problem as the question about unions and intersections that I put at the beginning of the document.

So the short answer to why the result is interesting is that it is a very natural weakening of a statement that would imply that P does not equal NP.

Unfortunately, the insights of Razborov and Rudich and others have told us that, impressive as the result is, it is unlikely to represent a significant step towards a proof that P does not equal NP. Nevertheless, it does hugely improve our understanding of the problem.

]]>But it was only after following the lectures that I realised that monotone circuit complexity had something to do with these problems. I would have been much more excited about the lectures had I been inducted into the significance of the result – perhaps a few comments on the blog as a reminder to amateurs like me as to what it is all about??

]]>I didn’t really mean to post twice, but my first post got caught in the spam queue I think, so I tried reposting…

]]>Thank you for posting video of your computational complexity course. I am a graduate student in the US, and while I have never had the opportunity to sit in on your courses, I have studied your notes on Combinatorial Number Theory, as well as several of the other expository notes on your website. I find your courses (and notes) to be very interesting and comprehensible. As my graduate program offers very few topics courses, I was hopeful that I could persuade you to consider posting video of future courses on the web as well. While I can understand why some faculty members wouldn’t want to do this, I would also encourage anyone else reading this with the resources to to do so to consider posting video of topics courses!

As an aside, I also need to say that the video production was particularly well done in this case. Most taped math lectures I have seen suffer from bad audio and an inability to read what the lecturer is writing on the board. These were very well done!

]]>A similar thought I had while writing the account of Razborov’s proof was that it is possible to distinguish two kinds of solution to an exposition problem, which one might call “weak” and “strong”. A weak solution is what I gave: for each step of the proof, you try to explain why it is reasonable to expect somebody to come up with it. But sometimes, and it happened here, it is reasonable to expect somebody to attempt a calculation, and the calculation works, but one is still left wanting to understand *why* it works. A strong solution would be one where for each calculation there is a better explanation than “Try it — it works”. A super-strong solution is one where you can come up with such an explanation *in advance* and not just with the benefit of hindsight. That is, for each calculation you have good theoretical reasons for being virtually certain that it will do what you want it to do, and the calculation itself is just a sort of check that your ideas are sound.

I did look at it once and I think it’s a great paper.

]]>Yes, I think there are APs of primes known with length in the early 20s, but I don’t know the details.

]]>Apologies for this off-topic question: in the problem to show there is an arithmetic progression of any length consisting of consecutive prime numbers, is there a better calculation than a 7 with 120 difference? ]]>

I think that, while getting to the bottom of X might look like a purely expository task, it is probably a very important thing to do. This is because new theorems usually come from understanding basic things very well, and not the other way around. In my case, I owe my whole research program to a few months spent trying (successfully!) to get to the bottom of something important and basic but fundamentally simple and often passed over.

The reason why I mention this here is that I think open exposition problems and things we need to get to the bottom of are often indistinguishable in the beginning. So, even if our measure of value is only progress in research, I would still put a very high value on solving open exposition problems.

]]>For this particular problem, I have always been very fond of the Berg-Ulfberg proof: pdf.

]]>I suppose one answer is that you go into a dark corner somewhere and work out what linear maps do to volumes and eventually you tidy up your answer and find that, lo and behold, you’ve got the determinant. But maybe that leads more naturally to the alternating multilinear forms view than to the permutation expansion view. But the first could be said to lead to the second perhaps. (These remarks are meant more as a plan of what to do than an actual exposition of course.)

]]>There has been discussion a while back on this very blog about how to show determinants intuitively. It’s on my back burner to write a “gentle introduction” style exposition which should make it entirely clear, but if there’s enough demand I can always bump it up on the queue.

]]>My current pet peeve in this regard is the definition of the determinant. I don’t know of a good reference that thoroughly explains how the permutation expansion of the determinant could be written down by someone who’s never seen it before. And while we’re on the subject, I also don’t know a satisfying theoretical reason why determinants should be so absurdly useful in algebraic combinatorics (Gessel-Viennot, the method of Pfaffians).

]]>It’s Pierre Menard, Author of The Quixote. I remember being struck by that and the equally famous The Library of Babel when I read them (a long time ago) — of course, they have an obvious appeal to a mathematician.

]]>Incidentally, the idea of “giving a plausible account of how a proof *could* have been discovered” reminds me of a famous story of Borges, whose title I forget, in which his character decides to take it upon himself to independently write the text of *Don Quixote*. Fortunately, the exposition problem seems nowhere near that hard. Maybe once I’ve digested the meta-problem of how to write an expository paper of this nature, I’ll give it a try myself.

In the meantime, I vaguely remember liking the proof Berg and Ulfberg gave:

http://scholar.google.com/scholar?q=berg+ulfberg+monotone&hl=en&lr=&btnG=Search

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