Collaboration in mathematics

In October 2009, Nature published an account of a remarkable experiment by Tim Gowers. On his blog, he proposed that his readers should collaborate on a research project, to discover an elementary proof of the density version of the Hales–Jewett Theorem. A month and a half, and with 800 comments from 27 individuals later, the goal had been achieved.

Collaboration between mathematicians is not new. Hardy and Littlewood had a legendary collaboration in the early 20th century. The rules of their collaboration are described by Harald Bohr (reprinted in Littlewood’s Miscellany, edited by B. Bollobás):

To illustrate the strong feelings of independence which, as a part of the old traditions, are so characteristic of the English spirit, I should like to tell how Hardy and Littlewood, when they planned and began their far-reaching and intensive team work, still had some misgivings about it because they feared that it might encroach on their personal freedom, so vitally important to them. Therefore, as a safety measure, … they amused themselves by formulating some so-called ‘axioms’ for their mutual collaboration. There were in all four such axioms.

The first of them said that, when one wrote to the other, …, it was completely indifferent whether what they wrote was right or wrong …

The second axiom was to the effect that, when one received a letter from the other, he was under no obligation whatsoever to read it, let alone to answer it …

The third axiom was to the effect that, although it did not really matter if they both thought about the same detail, still, it was preferable that they should not do so.

And, finally, the fourth, and perhaps most important axiom, stated that it was quite indifferent if one of them had not contributed the least bit to the contents of a paper under their common name …

I think one may safely say that seldom — or never — was such an important and harmonious collaboration founded on such apparently negative axioms.

The story of W. E. Opencomb

I have been involved in many collaborations, with well over a hundred co-authors. Here is the story of one of the most interesting of these.

In 1982, ten mathematicians, all interested in combinatorics, gathered at the Open University in Milton Keynes for a research week. (The mathematicians were R. A. Bailey, P. J. Cameron, A. G. Chetwynd, D. E. Daykin, A. J. W. Hilton, F. C. Holroyd, J. H. Mason, R. Nelson, C. A. Rowley and D. R. Woodall.) The organisers expected that many problems would be posed, and that progress would be made on several of these by subsets of the ten of us.

What actually happened was a bit different. In the opening talk, David Daykin posed a problem; we concentrated on that problem to the exclusion of all else, though in a different way from what David probably expected.

There are many situations where a mathematical object is constructed one step at a time. Sometimes, this is straightforward: a basis in a vector space is found by choosing vectors, each independent of those previously chosen; a spanning tree in a connected graph is found by choosing edges so as to avoid creating a cycle.

In other cases, however, it is possible to get stuck. A partial Latin square is a square array of side n partially filled with entries from the set {1,…,n} so that no entry is repeated in a row or column. It is a Latin square if every cell is filled. Latin squares cannot be made simply by choosing entries arbitrarily. Here is an example where the last entry in the top row cannot be filled.

1 2  

Daykin observed that this partial Latin square can be broken into two pieces, each of which can be completed:

1 3 2
2 1 3
3 2 1
1 2 3
2 3 1
3 1 2

This had led him, with Roland Håggkvist, to the following definition. The intricacy of a sequential construction procedure is defined to be the smallest number k such that any partial solution can be broken into k pieces, each of which can be completed to a solution. The example of a non-completable partial Latin square shows that the intricacy of this procedure is at least 2. Daykin also observed that it is at most 4. For a theorem of Ryser says that a partial Latin square can be completed if all the entries lie in an r×s rectangle, where r+sn. If n is even, divide the square into four pieces of side n/2; each can be completed, by Ryser’s theorem. (For odd n, a small further trick is required; I leave this to you.) Daykin proposed that we decide whether the exact value of the intricacy for this problem is 2, 3 or 4.

We were unable to solve this in a week, and as far as I know, it is still unsolved. But we did what mathematicians love to do: extend the idea of intricacy to many other situations in finite geometry and combinatorics and prove a body of results about these variations.

The ten authors wrote a paper on the results, and published under the nom de plume W. E. Opencomb. It appeared in Discrete Mathematics 50 (1984), pp. 71–97.

The story then is not entirely different from that of Gowers’ blog, except that it was done by mathematicians meeting and talking without the benefit of modern technology.

Some other collaborations

Every collaboration is different. In 2003 I visited Tehran for a conference, and ended up writing five joint papers as a result. One was a previously written paper which I was presenting at the conference; the others involved either work on problems presented at the problem session, or work with postdocs at the Institute in the few days I stayed after the conference.

My favourite of all happened in the 1970s. At that time I visited Eindhoven University regularly, usually at the end of the academic year to spend the money left in their visitor budget. I had two long-term collaborators there, Jaap Seidel and Jack van Lint.

One year, Jaap was working with Jean-Marie Goethals in Brussels on a geometric approach to graphs with smallest eigenvalue -2 (classified by a very complicated, unpublished theorem of Alan Hoffman). When I arrived, Jaap told me that they had reduced it to classifying “star-closed” sets of lines in Euclidean space, and that he had shown that there was one infinite family of such sets and three “sporadic” exceptions.

The next day we drove to Brussels to work with Jean-Marie. He told us that he had also found that there was one infinite family and three sporadic exceptions. But on comparing notes, we discovered that the infinite families were different! Each of them had slipped and left out one case, so that there were actually two infinite families and three exceptions. Immediately the penny dropped and I said, “Root systems”.

Root systems are geometric objects which arise in the classification of simple Lie algebras over the complex numbers. If all the roots have the same length, the classical results assert that they fall into two infinite families, called An and Dn, and three exceptions called E6, E7 and E8. It took almost no time to check that this was indeed what was happening. So we had our proof, with a clear pointer to where to find the “exceptional” graphs that had given Hoffman so much trouble. This paper (Ernie Shult was added as an author as he had done similar things at about the same time) was published in Journal of Algebra 43 (1976), 305-327; this is now one of my most cited papers.

Some collaborations arose because I refereed a paper and suggested (anonymously) a strengthening of the result. In such a case, the editor may suggest that the authors and referee get in touch directly.

Writing papers with students is a somewhat vexed topic. I would much prefer that students stand on their own feet and write their own papers. Yet I realise that in this imperfect world, it is sometimes easier for a paper with the name of an established researcher to be accepted than one written by a student or very recent graduate. So if my students ask me to be a co-author, I try gently to dissuade them, but yield to pressure.

The largest number of “named” collaborators on one of my papers is 7. This happened because three teams of two, and one individual, were working independently on the same problem, and (fortunately) discovered this before the paper was written.

Mathematical collaborations on the rise

Collaboration among mathematicians increased dramatically over the course of the twentieth century. Detailed analysis of data from Mathematical Reviews is given here. A couple of pointers: in the analysis by decade, the proportion of authors involved in joint papers rose from 28% in the 1940s to 81% in the 1990s, and the average number of collaborators of an individual author from 0.49 to 2.84. This trend has continued in the present decade: the number of mathematicians with no joint papers has actually fallen slightly!

Many factors contributed to this dramatic increase. One was certainly the mathematician Paul Erdős, whose career coincides almost exactly with the Mathematical Reviews data: he began publishing in around 1930, and died in 1996, though, remarkably, his joint papers are still appearing! Many good biographies of him are available; all stress the huge number of collaborations he was able to maintain as he travelled around the world. He may not be the most prolific mathematician ever (this crown goes to Leonhard Euler), but was certainly the greatest collaborator, having written papers with over 500 mathematicians.

The development of new tools such as blogs and wikis will certainly increase the trend, though probably not in directions which the bean-counters would approve. In any collaboration, it is impossible to allocate credit numerically to the people involved (though I believe some institutions are demanding that academics should make such judgments). Indeed, the whole idea of authorship is called into question.

And a good thing too, in my view!


About Peter Cameron

I count all the things that need to be counted.
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8 Responses to Collaboration in mathematics

  1. Interesting discussion, particularly your collaboration stories were quite nice to read. It just shows how mathematics can happen in ways not thought possible. 🙂

    • Thanks for your comment. While mathematics is certainly nothing like big science in terms of collaborations, they almost all have some story attached to them. Mathematics is a human activity, after all!

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  3. Pete says:

    Something that seems to have happened over the past century is a shift from the ideal of one man, one world-changing paper, to that of one group, many papers. In some ways, this may be a bad thing – the demands on time of the group may pull the individual (now of unspecified gender…) away from the insight needed to solve some major problem. But in general I think it is a good thing: increasingly, there are stable groups of researchers in mathematics who routinely publish papers in which only a fraction of the listed authors have made any contribution of ideas.

    Why do I think this is a good thing? For two reasons. Firstly (and perhaps most importantly) because the authors who have not had ideas leading to the solution will undoubtedly insist that the ideas are explained, and that they help write up: this usually makes the paper much more intelligible, and consequently more useful to the whole community. Secondly, because there is a wider group of people to whom the people with initial ideas may freely talk. The result is typically wider connections made to interesting areas, and more new ideas leading to solutions. Typically, in these groups, if you ask the group about one paper then only some of the authors will know the details; sometimes some members may find it hard to even sketch the argument. But the members with less knowledge will be different on different papers. If, for example, I pose a problem to you, and you tell me you have no idea how to solve it, then this is a very useful contribution – I know that I should not ask people who are good at algebraic arguments. Instead I should ask some other part of my group who know (perhaps) probabilistic methods to solve the problem. But when one of these probabilists is stumped by their problem, perhaps the right attack will be an algebraic method.. when the entire group authors everything, then these questions will actually be asked, instead of the algebraist wasting months trying to learn probability and vice versa, followed by neither managing to solve problems that the other would be able to solve easily.

    • This is undoubtedly true. Perhaps it is slightly longer-term than you suggest: look at Hardy and Littlewood (whose combined talents made them an impressive team). But the examples of Andrew Wiles and Grigori Perelman show that the old model coexists with the new. Personally, I hope that this continues; both have something to offer.

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