I was browsing through different category listings on the arXiv today and noting the changes in numbers of papers over the years. As you might expect, there are more and more papers being posted to the arXiv every year. However, one category defies this trends: QA (quantum algebra).

There are actually less papers being posted to QA in the past three years (2010 317, 2009 308, 2008 323), than there were in the late 90s (1998 364, 1997 434, 1996 395). By contrast, there are about 4 times as many AG papers in the past few years compared to the late 90s, about 10 times as many RT papers, and about 5 times as many GT papers.

What do you make of this? Does it represent a trend in the kind of math that people are doing? Or are people just classifying their work differently?

It would be interesting to see if one can use this arXiv category to get a sense of which fields are becoming more and less popular over time.

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math.QA is somewhat special in that it was the first arXiv category. One might be able to attribute some of that decline to the arrival of new competing categories.

I never realized that. How did it come to be the first arXiv category?

Would Noah expand on his comment? I’m not sure what ‘first arXiv category’ means. As I heard it, the physics arXiv existed first and then there was something started by David Morrison for mathematics (actually, I thought for algebraic geometry). Eventually these were merged. When this first happened, was all of mathematics in the same list? Neither arXiv nor Wikipedia

seem to have much information about the history of arXiv.

Hopefully at some point Greg or Terry will show up with the right info, but what I meant was that the first *math* arXiv category was the ancestor of QA (it was then called q-alg). Prior to that there was a lot of physics. I’m not sure when exactly which subjects showed up, but I did find the following helpful info from the arXiv’s 27 Dec 1997 update

“The math archives will be reorganized starting 1 January, 1998. The archives alg-geom, dg-ga, funct-an, and q-alg will be subsumed as subject classes in the new “math” archive (started 12/97 — see math info for more information).”

So those are the first four in some order.

I arXived a couple of papers in q-alg in the mid 1990s. One of those papers was about knot theory and I’m not sure that had we arXived the paper today its primary classification would have been math.QA. So maybe Noah is right; although that would not explain the slow decline after 1998.

This year I thought I had noticed a small decline in hep-th submissions, but the data does not seem to quite reflect that.

On the other hand, it would interesting to see the growth of math.MP (mathematical physics) matched against math.!MP and against hep-th/similar. There seem to me to be more and more cross posting into MP than even last year.

Actually looking at the back issues, the first month there was a category theory paper was in March 1995 – one on orbifolds and a cross-listing from q-alg and I could guess what it was before I read it: Baez-Dolan’s ‘Higher dimensional algebra and TQFT’ (i.e. HDA 0). I conjecture that a lot of what would have gone into q-alg in the mid 90s now goes into math.CT.

The first math category, then a separate top-level archive, was alg-geom in 1992. It was a direct offshoot from the very first archive of all, hep-th in 1991. Soon after alg-geom came funct-an. Then in 1994 four other math archives were tried, and two of these succeeded: dg-ga and quant-alg. The other two were too narrowly defined to catch on. In December 1997, math.GT and math.CO were started. Then in 1998, the whole list was unveiled, only slightly different from now. The four pre-existing categories became math.AG, math.DG, math.FA, and math.QA.

The decrease in usage of math.QA is explained by two things: (1) that it was somewhat narrower in conception, but was started at a time when the topic was popular. (2) That it has long had to compete with the more traditional math.RT, math.GT, and math.AG. (And a little in math.CT.) So there are both scholarly and bureaucratic reasons for the trend. But it still gets enough use, I think, that it’s not so bad.

I think it’s also less clear to the mathematics community as a whole what “quantum algebra” really means. In fact, if a student came to mean and asked me what it was (a not unreasonable question) I would have a hard time articulating it. I’ve never used it as a primary class (I have for AG, RT, SG, GR, GT and RA) though I’ve certainly done plenty of quantum algebra in my life, and I don’t really anticipate that changing.

Oh, and since no one else seems to have pointed this out, you missed a potential explanation, which I think is a big part of the issue: the set of people using the arXiv has changed. Of course, this is obvious, but it may have changed different amounts in different subjects. Given that most of the quantum algebra community was already using the arXiv in the late ’90s, it doesn’t take very much bleed-over into other subjects to get the stagnation in number of papers.

I was having exactly the “what is QA anyways” discussion with a grad student here recently, and indeed it is somewhat hard to articulate. I gave as a criterion “Does it have anything at all to do with the Jones polynomial? If so then it’s quantum algebra.” But this isn’t really right, as there are probably a few GT papers that fit that description but shouldn’t be cross-posted in QA, and there are QA papers that involve pretty distant subjects (like Gromov-Witten theory).

Ben, in your case I would have argued that Part 1 of your recent paper should definitely have had QA as its primary, while Part 2 is a closer call (either QA or GT as primary is a reasonable decision). I also think you could have swapped QA and AG on the Markov Traces paper if you’d wanted to. In sum, this is all your fault.

Well, I was a physicist working in QA in the mid ’90s and now most (somewhat math minded) physicists have moved into different areas, such as category theory, condensed matter, QG etc.

Noah- The reason that happened is that part 1 was originally the whole paper, which I felt went under GT (though I agree that QA would also have worked).

In short, I confess.

Comments others have made are mostly to the point, but to get a real historical perspective you have to go way back to the era before arXiv when “philosophy” encompassed all sorts of things now divided into separate disciplines: natural philosophy (physics, astronomy, etc.), mathematics, and so on. Sometimes the specialties are too arcane or at the other extreme try to take in too much. The labels people put on their research activity may or may not stick forever. Probably NT will endure, but QA and even my beloved RT are broad and also hard to define for non-specialists including most students (and most faculty colleagues I’ve dealt with). Specialization won’t go away, but there has to be at the same time an awareness of the hoped-for “unity of mathematics”. The only certainty is that the future will look different from today.

Joel Kamnitzer wrote:

Ah, kids these days. Once upon a time, shortly after the invention of horseless carriages, a scientist at CERN invented the world-wide web. And in 1991, before the web even really caught on, Paul Ginsparg started the arXiv. It operated mainly using email at first, and it was dominated by string theorists and particle physicists.

So, it’s not surprising that the biggest arXiv group was hep-th, dominated by string theory. And it’s not surprising that mathematicians who were into closely related subjects, like topological quantum field theory and quantum groups, were the first to start reading papers on the arXiv.

Then we started posting papers there, too. Our papers weren’t all precisely “physics”: we just wanted to make our stuff available! So, we put papers on hep-th that didn’t strictly belong there.

To solve this problem, eventually someone decided to set up q-alg. And then, when more mathematicians got serious about the arXiv, a more rational classification system was developed, and q-alg became math.QA.

By now, mathematicians of all kinds have embraced the arXiv, so math.QA doesn’t quite make as much sense as it once did. But it was very important in its day!

What is the official description of what belongs in the QA category? Is there one?

Do any of the arXiv categories have actual definitions? Submissions will be reclassified if you put them in the “wrong” category (this has happened to me when I made a mistake, but I recall hearing of people who didn’t agree with the reclassification), but my guess is that this is just a judgement call on the part of the “czar” of each category.

Here are the descriptions of the math arXiv categories. I don’t know if this information appears anywhere on arxiv.org, or if any comparable descriptions exist for the physics categories.

By the way, Lieven Le Bruyn has more convincing numerical data on the decline of quantum algebra.

Thanks Mark for that link!