Archive for January, 2009

Background to a Polymath project

January 30, 2009

This post contains brief descriptions of some mathematics that one would need to know in order to have a realistic chance of contributing to the collaborative research project I shall suggest in the next post. If you are familiar with the Hales-Jewett theorem, Szemerédi’s regularity lemma, the triangle-removal lemma, and the proof from the triangle-removal lemma that a dense subset of $\null [n]^2$ contains a corner, then you do not need to read it. (more…)

Is massively collaborative mathematics possible?

January 27, 2009

Of course, one might say, there are certain kinds of problems that lend themselves to huge collaborations. One has only to think of the proof of the classification of finite simple groups, or of a rather different kind of example such as a search for a new largest prime carried out during the downtime of thousands of PCs around the world. But my question is a different one. What about the solving of a problem that does not naturally split up into a vast number of subtasks? Are such problems best tackled by $n$ people for some $n$ that belongs to the set $\{1,2,3\}$? (Examples of famous papers with four authors do not count as an interesting answer to this question.)

It seems to me that, at least in theory, a different model could work: different, that is, from the usual model of people working in isolation or collaborating with one or two others. Suppose one had a forum (in the non-technical sense, but quite possibly in the technical sense as well) for the online discussion of a particular problem. The idea would be that anybody who had anything whatsoever to say about the problem could chip in. And the ethos of the forum — in whatever form it took — would be that comments would mostly be kept short. In other words, what you would not tend to do, at least if you wanted to keep within the spirit of things, is spend a month thinking hard about the problem and then come back and write ten pages about it. Rather, you would contribute ideas even if they were undeveloped and/or likely to be wrong. (more…)

A Tricki issue

January 20, 2009

This is a mathematical post rather than a Tricki post, but the title comes from the fact that the issue I want to discuss has arisen because I have made a statement in a Tricki article and can’t quite justify it to my satisfaction.

The article in question is about a useful piece of advice, which is that if one is having trouble proving a theorem, it can help a lot to try to prove rigorously that one’s approach cannot work. Sometimes one realizes as a result that it can work, but sometimes this exercise helps one to discover a feature of the problem that one has been wrongly overlooking. I have a few examples written up, but one of them is giving me second thoughts. (more…)