I fully agree with what you’ve written, but I think this is a different issue. Perelman was treated poorly by the Yau school, but he also behaved poorly himself after he announced his proof. The difficulty is that he wrote up his results to his own satisfaction but to nobody else’s. In particular, he wrote just barely enough to convince everyone (eventually) that he had a full proof in his head, while providing minimal explanation and guidance for the rest of the community. In the process, he wasted several person-years of effort to reconstruct what he already knew and didn’t want to explain.

Part of the honor and respect given to solvers of famous problems is a recognition of their contribution to the mathematical community. When someone like Perelman contributes as little as he can get away with while still getting credit for the result, it’s no wonder some people become angry with him.

For example, I’m not sure he should have been offered a Fields medal. It’s tantamount to saying that if your work is sufficiently brilliant, then you’re excused from fully documenting or explaining it. This doesn’t seem like a good message for the IMU to be endorsing. It’s certainly true as a practical matter that you can get away with a lot more if many people are eager to decode your manuscript, but I feel that the community should resist this phenomenon rather than give in to it.

In the case of Perelman, well, what can one say? One could argue that his behavior has seriously compromised his standing in the mathematical community, but I suppose he doesn’t really care.

Well, isn’t it true that a bunch of highly-gifted mathematicians (we all know who) actually tried to publish their version of Perelman’s proof giving almost no credit to Perelman and that they soon withdrew their paper only after there was an “outcry” (for very obvious reasons) over it? To put it bluntly, it was a case of “poaching” of ideas and personally I feel that is much worse than some mathematician, such as Li, submitting a false proof and later retracting it! Anyway, it is hard to see how Perelman compromised his standing in the mathematical community given that the latter actually failed him and not the other way around.

(are there other examples I’m missing?)

If my general understanding is correct, Taubes’ solution to the (3-dimensional) Weinstein conjecture might tend to fit in Gil’s first category. Interestingly, in this case I think people’s general expectation had been that the conjecture was going to be cracked by symplectic field theory, which had a sizable number of people talking to each other a lot and writing lots of joint papers about it with partial results and so on. So in some sense there was a competition between (moderate versions of) the first and second avenues here. And the first won. Of course, you could argue that this had more to do with Taubes’ methods just being more suitable for this particular question than SFT, and it’s certainly true that SFT has led to plenty of other interesting results.

If you’re a young postdoc looking to maximize your publication count and to make sure as many people as possible know your name, then it’s clear which of these avenues is most appropriate for you. But with at least some more senior people, it is simply the case that they know more about certain topics (the analysis of the Seiberg-Witten equations in Taubes’ case) than anyone else in the world (well, perhaps except for two people whom he could meet for lunch whenever he wanted), and it sometimes pays off greatly for them to slowly mine that expertise as extensively as they can.

It seems unlikely to me, but only because the set of abelian subgroups of G generated by two elements doesn’t sound like a very natural collection of subgroups (unless you’re interested in classifying G-torsors on an elliptic curve). Your question reminded me of Theorems A and C in a paper by Hopkins, Kuhn, and Ravenel (just Google Scholar the names), but I couldn’t tell if they would give you the reconstruction you want. On the other hand, if they did, I’d be surprised if you really needed complex oriented descent to prove your case of Euler characteristics of 0-manifolds.

Ben,

I’m having trouble finding the fact you mentioned in your paper. Could you point to it?

In the case of Perelman, well, what can one say? One could argue that his behavior has seriously compromised his standing in the mathematical community, but I suppose he doesn’t really care.

I think the opposite is true. There are certain members of the mathematical community whose behavior has turned him away from mathematics. Given the way his results were treated after he had written them up, I think he was justified in not publicizing his techniques until he had finished writing them up to his own satisfaction. If you work in a field that has predatory mathematicians, you can either change fields, or employ some secrecy.

As for other successes of the lonely researcher school, I’m sure there are many, if one looks a bit carefully. I’ve been told that my colleague Demetrios Christodoulou did not really tell many people about his recent work, before putting it on arXiv, and it seems to be an extremely remarkable, 594 pages long, piece of work (if I understand right, it shows that general relativity permits the creation of black holes even in the absence of matter; unfortunately I don’t know much about the subject…)

But there are mathematicians who advance not just a program, but entire fields with mind-blowing new ideas. I think for these people collaboration is more about “team-building” than about anything else. While I completely agree with your attitude for the former mathematicians, the latter (much rarer) mathematicians should obviously do mathematics however they like. After all, they are the ones who do most of the important mathematics. Also, based on my few experiences with these latter mathematicians, I doubt they particularly care how the rest of us view their decisions about openness or secrecy.

While it may be better for most people to work with collaborators, I think a certain type of person can’t function in this traditional way.

A minor point, but historically the “traditional way” was for mathematicians to work alone, if perhaps not in secret; collaboration has increased markedly in past 30 years, perhaps as a consequence of things like email. There was a Notices article a while back that documented the increasing proportion of mulit-author papers in MathSciNet, for instance.

In Perelman’s case, his first introduction to the university level math world was being accepted under a strict anti-Semitic quota. And he had to show up and see these people every day. Maybe some people liked to justify the policy. One would imagine there was all sorts of unpleasantness unrelated to this issue as well; it’s hard to believe math departments are going to be nasty in only one way. Who knows what else he had to put up with as he progressed further along in his career.

Anyhow, I imagine a sensitive/loner type might have a lot of trouble handling the real world of math research at the top levels. Suppose Perelman discussed his early efforts on Poincare. Can you imagine the feeding frenzy that would have occurred once people realized he was on to something? It would have been intolerable for someone like him. Not to mention the issue of someone beating him to the punch and then deciding that Perelman didn’t really know how to get to the end, and besides Hamilton developed the whole program anyhow, and so…

I think one should not be critical of people with unusual psychological makeups, and just be glad that he was able to accomplish what he did, and was willing to show the world what he had done. I’m sure he was doing the best he could under very difficult personal circumstances.

I don’t have time to go into detail right now, but two G-sets are isomorphic if and and only if each *subgroup* has the same number of fixed points on X and Y. This is all in my paper on the subject, Small linearly equivalent $G$-sets and a construction of Beaulieu