I got a bit behind on responding to comments to my post on Li’s preprint, so I thought I would just start a new thread.

Now, I don’t want to concentrate too much on the particular’s of Li’s case, since I don’t know Li or too many of the specifics.  I’ll just clarify that I don’t think Li is a crackpot, or did I say in my past post that I though that.  But that’s exactly why I feel like releasing his preprint the way he did was a mistake.  Certainly, there’s a bit of my taste for facetious exaggeration (I’m sure my readers have noticed this tendency in the past) in describing the unfortunate aspects of that as “crackpotesque,” but I think that also helps convey what about it I thought was a bad decision.

I’m more interested in expanding on some of the comments in that thread.  (more…)

There was a request containing the phrase, “theory of modular forms,” so I’ll write an introduction to that. Chris seems to be taking care of the rest of that paragraph.

Pretty much all of the material below is 50-150 years old. Don’t expect too much originality.

(more…)

Since people who don’t get alerts from WordPress probably are no longer paying attention to this post, I thought I would top-post a comment left by Christine Cheng, a researcher who thinks about stable matchings. The full-text is below the fold. The main upshot is that there are some deterministic algorithms which produce fairer matchings than Gale-Shapley’s but they’re hard to implement. (more…)

One reader was curious if we had anything to say about the recent preprint by Xian-Jin Li entitled “A proof of the Riemann hypothesis”. Unfortunately, analytic number theory seems to be a weak spot of the mathematical blogosphere, so none of us seemed inclined to go through the paper and look for mistakes. Luckily, Terry Tao did and thinks he has found a mistake (which the author may claim to have fixed…things are starting to get a little confusing). Alain Connes also seems to be unconvinced. Oops.

Which leaves the rest of us to wonder what happened. I mean, this paper looked promising precisely because it didn’t look like the work of a crackpot. Li has a Ph.D. from Purdue (in mathematics) and is a mathematics professor at Brigham Young, and analytic number theory is his research area. He has several other unsuspicious articles on the arXiv, and the style of his Riemann hypothesis article is wholly unremarkable (considering that it claims to prove probably the most celebrated open problem still at large in the mathematical world). Why would someone risk the level of embarrassment involved in putting a proof of RH which had not been really thoroughly vetted on the arXiv, apparently with no warning (whether it can be fixed or not, if Terry Tao found a problem in less than 24 hours after it was placed on the arXiv, it definitely was not vetted thoroughly enough before being released on the world. It’s also on its 4th version on the arXiv in 3 days)? What was the hurry?

I can’t really speak to Li’s situation, since I don’t know the guy. It may well be that he sent his preprint to Tao and Connes and they didn’t get around to reading it. But if he didn’t, that was a huge mistake on his part, one which definitely makes him look more crackpotty than I expect he wants. If he didn’t give any conference talks on the subject before releasing the preprint, that was a huge mistake. Honestly, I think putting it on the arXiv, where it will remain forever, taunting him, rather than his personal webpage was something of a mistake. After all, you want a chance to get comments from the people who might be able to point out any mistakes you made before you end up on Slashdot. While this goes double, or perhaps n-uple for some large n if trying to prove an important problem like RH, I think it’s a good point in general that you should tell people about your work while it is still in its formative stages. It could save you a lot of pain. Admittedly, some people worry about being scooped, but I feel like this is the sort of thing that people are naturally more paranoid about than they should be. Ultimately, it would be better if we shared all our good ideas. After all, if somebody else does something cool with an idea you had, that just makes you look smarter for having such a good idea.

[Ed. - I changed the title of this post, since the original one was a bit more inflamatory than I intended]

Reading the Economist last night, I came across the following rather strange advertisement.

It’s an ad for a watch, the “C1 Concord” apparently, not at all uncommon in the Economist (although it is rather on the ugly side…) Looking closer, though, there’s unmistakably some maths in the background.

Anyone know what the maths is about? It looks pretty plausible — there are some things looking like long exact sequences in there, as well as some fundamental groups, and power series for exponentials. Best of all there’s some text lower down in the ad - “What other watch has a formula for the ultimate construction?” I’m guessing they found a real mathematician to write some pretty looking maths for them — I wonder if they realised they were being asked to write “a formula for the ultimate construction”?

Here is a very basic question that has come up in some work I’m doing with Diane Maclagan. There is lots of algebraic geometry in our intended application, but I think that what I really need is a better understanding of the underlying linear algebra. First, let me review some even more basic ideas. Let and be two finite dimensional vector spaces over a field and let be a bilinear pairing. For a subspace of , define the subspace to be the space of those vectors such that for all . We can also define for any subspace of .

Then the “Fundamental Theorem of Bilinear Pairings” is the following: for any subspace of , we have . In particular, if and only if

OK, that was pretty simple. My situation is that, instead of having a pairing down to the ground field, we have a bilinear pairing into another finite dimensional vector space. 

What is the new fundamental theorem characterizing ? In particular, under what conditions do we have ?

Many thanks!

So, Scott’s comment about being dictator of a math department got me thinking (very idly), and I came up with a hypothetical for you guys: imagine you earn a preposterous amount of money one way or another, and for whatever reason you decide that the best use of it is a new institution dedicated to research in mathematics (I don’t actually agree with this premise, but one can always assume that one has a truly preposterous amount of money, and have already spent a bunch of it on family planning NGOs and smart growth advocacy, or whatever other causes you might think are a higher priority than mathematics). What model would you go with? (more…)

Subversion, often abbreviated as SVN, is a “version control system”. Prompted by Nathan’s request to hear about collaborative software for mathematicians, and the comments on Ben’s post on the subject, I’m going to briefly describe how you might use Subversion to collaborate on a maths paper. Even better, I’m offering to set up a subversion repository for any mathematician who’d like to try it. Jump to the bottom if you already have your subversion-fu, and just want the goodies.

(more…)

So, looking at Google Reader stats today, I discovered that our blog has a bit over 400 subscribers (for reference, that’s a bit over half of the n-category cafe’s subscriber base, and between a third and fourth of Terry Tao. I guess it pays to update regularly).

On the other hand, our comments feed has exactly two subscribers. My question is: who’s the other one?

Nathan Dunfield (a new commenter!) supplies our first request:

How about a discussion of long-distance collaboration tools and methods, beyond just using email and talking on the phone? It seems like there are a lot things that might work, e.g. pointing a cheap webcam at piece of paper, using collaborative text editors (e.g. SubEthaEdit), IM’ing (some clients have LaTeX support, I tnink), virtual whiteboards (e.g. Scriblink.com), but which might also turn out to be useless in practice for all sorts of annoying technical reasons. So it would be interesting to hear from people who have had success or failure with various methods.

Unfortunately, I have nothing insightful to say on this topic (I would be really excited to hear if any one else has exciting ideas, for reasons which will be clear below). This is a little sad, since I’m a perfect candidate for having done something interesting in this area. I’m pretty technophilic, even for a mathematician, and am currently writing two different papers with two different people in Germany, and working on another paper in a group of 4 where I don’t think more than 2 of have been in the same state simultaneously in over a year. Almost all this work has been done over email, or face-to-face, with occasional phone conversations and one video chat on Skype (but with no attempt to write anything on a board or paper, just gesticulation). In particular, the last paper I mentioned has been written entirely while we were all permanently in different locations (me in Princeton and Boston, one in Oregon, one in California, and one in Amherst), and generated an enormous number of emails, I think around 500 (thank Gbus for Gmail).

So, why haven’t I done anything more exciting? (more…)