Hall algebras and Donaldson-Thomas invariants I

I would like to tell you about recent work of Dominic Joyce and others (Bridgeland, Kontsevich-Soibelman, Behrend, Pandaripande-Thomas, etc) on Hall algebras and Donaldson-Thomas invariants.  I don’t completely understand this work, but it seems very exciting to me. This post will largely be based on talks by Bridgeland and Joyce that I heard last month at MSRI.

In this post, I will concentrate on different versions of Hall algebras. Let us start with the most elementary one. Suppose I have an abelian category \mathcal{A} which has the following strong finiteness properties: namely Hom(A,B) and Ext^1(A,B) are finite for any objects A, B . Then one can define an algebra, called the Hall algebra of \mathcal{A}, which has a basis given by isomorphism classes of objects of \mathcal{A} and whose structure constants c_{[M], [N]}^{[P]} are the number of subobjects of P which are isomorphic to N and whose quotient is isomorphic to M .

The main source of interest of Hall algebras for me is the Ringel-Green theorem which states that if you start with a quiver Q, then the Hall algebra of the category of representation of Q over a finite field \mathbb{F}_q is isomorphic to the upper half of the quantum group corresponding to Q at the parameter q^{1/2}.

The obvious question concerning Hall algebras is to come up with a framework for understanding them when the Hom and Ext sets are not finite. This is what Joyce has done and he has applied it where A is the category of coherent sheaves on a Calabi-Yau 3-fold.

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Why isn’t the arXiv more like Craigslist?

While some people might be suspicious of this analogy, I think the arXiv and Craigslist have a lot in common;  both took a service previously only available through an expensive print intermediary, and instead made it freely available on the web.

But there’s one point where the analogy breaks down: the aggressiveness with which they have expanded.  Craigslist started out only in San Francisco, but has since opened a bewildering number of local sites.  Of course, at the time, most them were very sparsely used at first, since people in those cities had never heard of Craigslist.  But relatively quickly, the ones all over the US took off (for a sense of which are getting used, I think best of craigslist is an excellent way to get a (highly NSFW) cross-section).  I think some of the UK ones are also getting some use, though there are also dozens of mostly empty Craigslist sites for random foreign cities like Seoul or Buenos Aires.  However, when the demand gets in place for such sites, Craigslist will be there, and who’s to say where and when it will happen first.  After all, it was very little trouble for craigslist to set up the foreign sites, having already created the US ones.

Similarly, the arXiv has a model which scales brilliantly, and in the time that Craigslist has gone from an SF-only site to covering 500 cities around the world, the arXiv has…added sections in Quantitative Biology and Quantitative Finance.  I don’t want to be too hard on the arXiv.  I mean, they do an incredible service to the mathematics and physics communities, but I don’t understand why they’ve taken such a restrictive view of their possibilities.  Maybe there’s some serious obstruction I’m not seeing, but I’d like to know what it is.

I mean, why not an arXiv for economics?  For literature?  For history?  My understanding is that most other disciplines don’t have a centralized preprint server (this experience is most based on my dating of grad students in other disciplines, so it’s rather, ahem, anecdotal), so why isn’t the arXiv at least providing the opportunity?  Maybe they wouldn’t be used much at first, but what’s the harm in trying?

L’affaire El Naschie

So I know I’m a little late to the party on this, but I couldn’t resist commenting on the strange case of M. el Naschie (I assume that this is just the German transliteration of the name English speakers would be more likely to spell al Nashi). Zoran Škoda brought him up in the comments to a post at the n-Category Cafe, and John Baez did an excellent job exposing the level of intellectual bankruptcy at the journal Chaos, Solitons and Fractals. The details are better recounted elsewhere, (unfortunately the posts above have been removed. Those interested in following the case can try Richard Poynder’s blog Open and Shut) but in a nutshell, El Naschie published dozens of papers in his own journal (he’s the editor-in-chief) which appear to be of no scientific or mathematical merit (this is my judgment based on excerpts and titles, and also seems to be the consensus of commenters at nCC), which make rather grandiose claims based on rather incoherent numerology. John Baez characterized him as “worse than the Bogdanov brothers,” which is pretty high up in the food chain of physics hoaxes.

But my intent here is not to beat up on El Naschie. He’s already set to retire in shame. The people who really have egg on their face here are those who enabled the man who is for all intents and purposes a crank to run a superficially prestigious-seeming journal. Continue reading

Bleg: testing algebraic integrality by computer.

Update 2: we’ve found a nice answer to our question. Maybe it will appear in the comments soon. –Scott M

Scott, Emily, and I have an ongoing project optimistically called “The Atlas of subfactors.” In the long run we’re hoping to have a site like Dror Bar-Natan and Scott’s Knot atlas with information about subfactors of small index and small fusion categories. In the short run we’re trying to automate known tests for eliminating possible fusion graphs for subfactors.

Right now we’re running into a computational bottleneck: given a number that is a ratio of two algebraic integers how can you quickly test whether it is an algebraic integer? Mathematica’s function AlgebraicIntegerQ is horribly slow, and we’re not sure if that’s because it’s poorly implemented or whether the problem is difficult. So, anyone have a good suggestion? After the jump I’ll explain what this question has to do with tensor categories (and hence subfactors which correspond to bi-oidal categories as I’ve discussed before).

To whet your appetite, here’s an example. Is a/b, where

a=-293 \lambda^{11}+4624 \lambda^9-23668 \lambda^7+50302 \lambda^5-44616\lambda^3+14017 \lambda

b=131\lambda^{10} - 2033 \lambda^8 + 9974\lambda^6-18951\lambda^4+12233 \lambda^2-1475

and where \lambda is the largest real root of

1 - 58 x^2 + 175 x^4 - 186 x^6 + 84 x^8 - 16 x^{10} + x^{12},

an algebraic integer? Mathematica running on Scott’s computer (using the builtin function AlgebraicIntegerQ) takes more than 5 minutes to decide that it is.

Update: Thanks to David Savitt for pointing out that both this example and an earlier one are answered instantly by MAGMA. Blegging is already working. But what’s the trick? Is it something we can teach Mathematica quickly? –Scott M

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Theta functions of one variable

I’m closing in on the source of my confusion. In this post, I’m going to explain as much as I can in the case of Jacobians of genus one curves, that is to say, the case of elliptic curves. Of course there are about a zillion books on the classical topic of theta functions, and other elliptic functions, in one variable. I’m going to do a few things that I haven’t seen elsewhere though. I’m going to work entirely in the analytic world. You’ll never see a complex conjugate, a Hermitian matrix or, with one exception that I’ll discuss when I get to it, a real or imaginary part. As a result, my constructions will be analytic so, if I get down to compact spaces, I will be able to apply GAGA and conclude that they are algebraic. Also, I’m going to try as hard as possible not to make any arbitrary choices. Finally, of course, I have been thinking about the higher genus case, and I am trying to choose notation that will generalize well. There will probably be a followup post shortly, discussing what changes in the higher genus case. So far, it looks like the things that are different are basically orthogonal to the things that interest me.

For those who haven’t seen \theta-functions before, let me give this advertisement — just as every polynomial is a product of linear factors, every function on an elliptic curve is a product of \Theta-functions. If you care about elliptic curves, it should be pretty obvious why you care about \Theta-functions.

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Out-of-print books

If you ever need an example of how unhelpful and badly designed our current publishing system is, the existence (or rather, lack of existence) of out-of-print books is ready-made.

Now there was a time when not publishing a book could make serious economic sense.  Publishers couldn’t afford to publish runs of books below a certain number, and the demand for some books can become so small that there was no way to profitably print them.  It’s a shame but an understandable economic reality.

This is simply no longer the case.  Print-on-demand services (for example, lulu.com) can now print books as people order them for a cost considerably lower than the list price of any math textbook.  All a publisher needs to do is put PDFs of their books on such a website, put a $30 markup on them (or more, considering how much math books cost), and let the money roll in.  If they don’t have PDFs, I bet Google Books would make them for free.  In short, publishers are leaving money they could be making on their back catalogue on the table, and hurting the mathematical community at the same time.  Thanks, guys.

This rant was engendered by a post of Timothy Chow’s at What’s New (a.k.a. Terry Tao) about a new website, where one can express one’s desire for a old math books to be brought back into print.  The website’s a good idea but ultimately getting specific books that are particularly popular back into print is a short-term fix.  The real problem is that publishers’ mindset still hasn’t caught up to the advances in technology. When are they going to enter the 21st century?

[Ed. – last paragraph edited a bit in response to comments]

The revenge of the return of the son of talk blogging: Denis Auroux on mirror symmetry

As you may have heard, we’re having a conference here at IAS on derived categories and algebraic geometry. Last night, at the banquet for said conference, some time after the discussion of how the derived algebraic geometers should start a blog called”The (\infty,n)-Category Cafe,” I got some complaints from someone who will remain unnamed (though for convenience, let’s call him “Navid Zadler”) about the lack of talk-blogging thus far (and particularly that he was concerned his own talk would not be blogged).

So, to put his mind at ease, I thought would give a little update. Yesterday was sort of symplectic day (well, except Olivier Schiffman, but we can discuss him later) and featured a very nice talk by Denis Auroux, on homological mirror symmetry.

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Stable marriages

So, one of my odd mathematical fixations (totally unrelated to my research) is the stable marriage algorithm. Don’t ask me why, I just find it a remarkably appealing piece of math.

The stable marriage algorithm is, unfortunately, not a fool-proof way of keeping one’s marriage stable, but rather a way of taking two sets of parties (the obvious example being a group of men and a group of women, though the real life applications tend to involve much more boring things like matching medical students to residency programs), who would like to be paired up with each other. But rather than decide who marries who in the normal way, they want to give a mathematician a preference list, and have him sort out things in a sensible way.

Being a mathematician, this chap first has to come up with definition of a rigorous “sensible.” Certainly things won’t work very well if after the matching, there are any couples who would prefer to give the mathematician the middle finger and elope with each other to Vegas rather than stick with their assigned mates. A matching with no such couples is called “stable.” Now, this doesn’t uniquely specify the matching, and could leave many people still unhappy (if one gender all has similar preference lists, somebody still has to take the duds), but anybody they prefer to their spouse will not want to leave their own spouse for our unlucky lover. But at least no couple can complain the mathematician that he should have matched them instead. Continue reading

The Cheewhye Diagram

About five years ago, Cheewhye Chin gave a great year-long seminar on Langlands correspondence for GL_r over function fields, building up to a description of Lafforgue’s proof of the theorem. In the beginning, he drew a diagram that captured the general architecture of the proof, and I liked it so much that I stole it for a talk I gave at Talbot in 2005. It seemed to get a good reception, and Mark Behrens pointed out that the Eichler-Shimura correspondence also fits into the picture with minimal alteration.

If we remove all of the explanatory text, the diagram looks like this:

. \qquad \quad \text{Geometric}
. \qquad \nearrow \qquad \qquad \searrow
\text{Spectral} \qquad \qquad \text{Algebraic}
. \qquad \nwarrow \qquad \qquad \swarrow
. \qquad \quad \text{Analytic}

I was a bit hesitant to draw this, because my advisor once told me, “If you ever find yourself drawing one of those meaningless diagrams with arrows connecting different areas of mathematics, it’s a good sign that you’re going senile.” Anyway, I’ll explain roughly how it works.

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