Sticking with the theme of the Riemann Hypothesis, the AMS has recently posted some articles to appear in an upcoming issue of the AMS Bulletin, one of which contains a long interview with Atle Selberg, who died last summer at the age of 90. Selberg had been a professor at the IAS and an expert in analytic number theory, responsible for some of the most important developments in the subject during the 20th century. A large part of the interview concerns in one way or another the Riemann Hypothesis, which is a central concern of Selberg’s mathematical research, with his work on it beginning during the German occupation of Norway when he was still a student, Some thoughts from Selberg on the subject:
“If anything at all in our universe is correct, it has to be the Riemann Hypothesis, if for no other reasons, so for purely esthetical reasons.” He always emphasized the importance of simplicity in mathematics and that “the simple ideas are the ones that will survive.”
About whether there is a spectral problem that gives the zeros of the zeta-function, useful for proving the RH:
That is certainly a thought that several people have had. In fact, there have been some people that have been able to construct such a space, if they assume that the Riemann hypothesis is correct, and where they can define an operator that is relevant. Well and good, but it gives us basically nothing, of course. It does not help much if one has to postulate the results beforehand—there is not much worth in that.
About his own attempts to find a proof:
Once I had an idea that I thought perhaps could lead to a proof….
[gives some details]
After a while I became more and more convinced that it would not work as I had thought initially. It just seemed unlikely to me. However, I have now and then seen that people have attacked a problem in a way that seemed “hare-brained”, to use an English term, but then it turned out that they could make it work. They have proven something that would not be easy to prove in another way. On the other hand, I have seen people have ideas that seemed absolutely brilliant, but the only problem is that if one follows these to the end one is not able to get anything out of it after all. So it works both ways: sometimes a good idea does not work, and what seems like a bad, even idiotic idea, may actually work.
About Connes’s work on the RH:
Yes, that is a new way to arrive at the explicit formulas—a new access, so to say—but it basically does not give more than what one already had. Connes undoubtedly believed to begin with that what he was doing should lead towards a proof, but it turned out that it does not lead further than other attempts. When I last talked with him he had realized this. This often happens with types of work that are rather formal. There was, for example, a Japanese mathematician, Matsumoto, who gave several lectures that made quite a few people believe that he had the proof.
and finally:
I think it is a good possibility that it will take a long time before it is decided. From time to time people have been optimistic. Hilbert, when he presented his problems in 1900, thought that the Riemann hypothesis was one of the problems that one would see the solution of before too long a time had elapsed. Today it is a little more than one hundred years since he gave his famous lecture on these problems. So one must say that his opinion was wrong. Many of the problems that he considered to be more difficult turned out to be considerably simpler to solve.
What was surprising to me in that interview is his negative view of collaboration. These days if somebody gives a non-trivial contribution, they get their name in the paper. I found Selberg’s attitude (re Erdos) bordering on the unethical, but those were different times and maybe different standards.
It appears that the IHES is currently working on editing and publishing Grothendieck’s “Récoltes et semailles”. It is mentioned that it would be out before the end of this summer. Something sure to please at least Alain Connes 😉
ninguem:
Clearly Selberg did not like to collaborate but I think the way Selberg and Erdos resolved this issue was quite ethical: they published separate papers in which they clearly described who contributed what. BTW, in MathSciNet the papers are reviewed in a single 4-page-article (by Ingham).
I found it refreshing to hear a mathematician of evident quality express a distaste for collaboration. It was also pleasant where he said prizes don’t advance science.
The non-collaborating mathematician still exists. Taubes is a good example of this beast that pursues its own interests in privacy and seriousness.
Perhaps I know too many folks too busy inflating their publication counts through incessant collaborations in which their principal contribution appears to be their name.
Another similar annoying thing is how it’s now expected that young mathematicians collaborate with senior mathematicians, whether they want to or not. You’re at such a disadvantage in the job market if you do it alone, because even if you’re successful, your peers will have the advantages of working with senior well-known mathematicians, not just in terms of the quality of the papers but also the top journals are more hesitant to accept papers from unknowns. I’ve seen utterly incompetent people get tenure-track positions through these “joint efforts.” One time at a talk I attempted to get the speaker to define the basic objects of his talk. He couldn’t do it, guessed incorrectly, then emailed me the next day with the actual definition. To preserve anonymity I won’t say what the terms in question were, but it would be like if an algebraic topologist couldn’t define cohomology. This person had multiple joint papers with multiple leaders, all in good journals. I was amazed that this could happen when the incompetence was so blatant. I have seen two talks from this guy, so it was not a fluke.
geometer:
One of the things that I meant was the fact that he tried to throw Erdos offtrack and he seemed almost proud of how clever he was.
George:
What you describe is a bit surprising. Why would the senior mathematicians bother to write papers with this guy? Maybe he has something to offer.
ninguem,
The answer to your question to George is simple: this guy physically writes the papers the senior mathematicians do not feel like taking the time to physically write. This is the sort of thing he `has to offer’.
Pingback: The shoulders of giants « Since it is not …