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Geometry Mathematics

Sir Michael Atiyah

A picture with Sir Michael on the right and me on the left, in Edinburgh. I think it was taken on July 16, 2018.

When CAMS (Center for Advanced Mathematical Sciences) was first founded circa 1999, I was an Electrical Engineering student at the American University of Beirut (AUB) where CAMS is physically located. I was curious and wanted to attend a talk by Sir Michael Atiyah, whom I will refer to as Sir Michael for the rest of the post. It took place in Hotel Al-Bustan actually. I think it may have been at some point in December 1999, or perhaps in January 2000. The talk was about an “elementary” problem in geometry, which is the problem on configurations of points. This was also the first time I had see Sir Michael in person, and also the first time I had learned about stereographic projection (remember, I was an Electrical Engineering student back then, not a Mathematics student).

I was really interested in the problem, and of course in Mathematics in general, particularly geometry (broadly understood). I decided to email Sir Michael, and asked him about careers in Mathematics. He answered me that there were many career opportunities in Mathematics nowadays. That email was key in making me decide to study Mathematics later on.

I emailed the poor man several times with ideas and sometimes wrong proofs. He used to reply to me with short emails, explaining to me why it does not work etc. Sometimes he would give me some advice too: try proving this, or that. Once also, Sir Michael told me to “gry my hands” at the planar case of the problem.

Let us move forward in time to October 29, 2016. I emailed Sir Michael about an idea for how to generalize his problem to a Lie-theoretic setting. On November 1, 2016, he replied to me, and the following is an excerpt from his email (which was also sent to Paul Sutcliffe and Roger Bielawski):

“I like your idea, which ties up closely with mine but is more algebraic.  Please send me a fuller version (which need not be perfect).  Do not rush to publish as yet.  I will advise and help publication and see how it fits with my own work.  It may be that that we can write a joint paper.”

I was really happy of course. My favorite Mathematician, who is also one of the best Mathematicians in the World, liked one of my ideas. This was a turning point.

I then visited him twice in Edinburgh, once in February 2017 and once in July 2018. Both times, we discussed many mathematical subjects (including of course the problem on configurations of points), with mostly him doing the talking. I must say it was really impressive to listen to his thoughts.

As many people may know, Sir Michael came a lot under attack in the past couple of years of his life, especially after uploading a series of preprints on various subjects: an argument for the lack of complex structures on $S^6$, a fixed point idea for simplifying the proof of the Feit-Thompson theorem, an explanation for the value of the fine structure constant and an argument for the Riemann hypothesis.

While flaws and/or inaccuracies were found in these preprints, I would like to say a few remarks.

Sir Michael’s idea for the Feit-Thompson theorem is, in my humble opinion, quite plausible. I find it possible that someone may manage to apply a fixed point argument to show the statement that any finite group of odd order is solvable. Perhaps the details in that preprint were incorrect, but the general idea itself is in my opinion quite plausible, and certainly worthy of further study. Alain Connes took up this idea and pursued it a bit further, his way, but so far the approach has not been successful. Anyway, I do think it is worthy of further study.

Regarding his preprint on the fine structure constant, I think that there are some interesting ideas in that preprint, regardless of whether or not they can applied to the fine structure constant. They may possibly have other applications. I did not understand all the Math behind this article, because, as of now, I am not familiar with Von Neumann algebras.

I have to say that, even when he was facing all that opposition to his ideas in the last couple of years of his life, this did not stop him from expressing himself. Some people thought this was sad. I thought it was remarkable. Moreover, his brilliant intuition never suffered a single bit. I think he was less concerned with details so much, but wanted to gain a deeper understanding. As he used to say, there are those who like to think, and those who like to understand. Sir Michael was definitely among those who wanted to understand.

By Malkoun

A Mathematician who is also interested in Mathematical Physics and Programming.

4 replies on “Sir Michael Atiyah”

I liked your views on Atiyah. He wanted to understand and that requires sometimes putting details aside.

Yes Dennis. He also did not like formulas much. They may prove something, but they do not explain it. I agree with him (though sometimes, this is the best that one can do, at a given point in time, before a better, more explanatory proof comes along).

Joe, thank you for sharing this. It’s wonderful to hear how this great mathematician played a pivotal role in inspiring your career. I can’t say I’m too familiar with Sir Michael Atiyah’s work, besides a basic grasp of the famous Atiyah-Singer Index Theorem. From my point of view, I think it’s great he contributed these ideas you mention (regarding the lack of complex structures on S^6, etc.) They add more approaches to our toolkit for looking at these and similar problems, whether or not they provide a direct proof.

Hey Paul. I particularly like the two preprints I have mentioned: the Feit-Thompson preprint, which has a very nice idea, which could possibly be made to work ultimately (in my humble opinion) and the article on the fine structure constant which made me reflect on mathematical constants and contained an interesting construction, some kind of renormalization process, inspired from physics, yet applied to mathematical constants. Real numbers form a continuum. What makes special constants such as $e$, $\pi$, $\gamma$ (or $i$, if one includes complex constants too) and so on, so special to us? Of course, one may give an answer for each one of these special constants but, as far as I know, there is no uniform answer.

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