Michael Atiyah

Sir Michael Atiyah at RSE

Sir Michael Atiyah died yesterday.

I attended part of a course of lectures he gave on algebraic geometry in the 1970s (until term got too busy and I was forced to drop the course). They were excellent lectures, the sort that make you feel you understand everything. (The down side is that, half an hour later, you can’t remember anything, since you haven’t had to work at it.)

The highlight of the lectures was a surprising theorem. Given a “generic” polynomial f of degree 6 over the complex numbers, in how many ways can you write f = g2+h3, where g has degree 3 and h degree 2? Clearly multiplying h by a cube root of unity or g by −1 doesn’t change anything, so ignore this.

The answer is 40. Atiyah proved this by writing down a curve with a horrendous singularity at one point; after dealing with that point, the rest was well-behaved, and he could come up with the answer.

There is a Galois group associated with this situation, of course, which (if I recall correctly) is the finite simple group PSp(4,3). In discussing this, it seemed to me that Atiyah was less sure of his ground. As is well known, he was no great friend to algebra. The Wikipedia article has a quote in which he describes it as the invention of the Devil, which you must sell your soul in order to become proficient in.

I disagreed with him on that, of course.

The picture at the top of this piece is of his portrait in the rooms of the Royal Society of Edinburgh, of which he was president at one time. Indeed, there was a period when he was at the same time President of the Royal Society of London, Master of Trinity College Cambridge, and director of the Isaac Newton Institute. Nigel Hitchen told me that, at this time, if he wanted to have a mathematical conversation with Michael Atiyah, it was necessary for him to stay overnight at the Royal Society and try to catch him at breakfast.

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About Peter Cameron

I count all the things that need to be counted.
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5 Responses to Michael Atiyah

  1. Maxis Jaisi says:

    Should it be g^3 + h^2 instead of the other way round?

    • cubic squared and quadratic cubed give polynomials of degree 6

      • Maxis Jaisi says:

        Whoops, silly me!
        I hope this isn’t too much to ask for, but what does “generic” mean here? I’m an undergrad, find the assertion very fascinating, but clearly don’t have enough background to hunt for resources about this.

      • To be honest, I am not quite sure what “generic” means here. The lectures were more than 30 years ago, and I am sure that Michael Atiyah explained the concept, but I am less sure that he gave a precise definition. Certainly one would want to assume that there are no repeated roots; there may have been more to it. I have never seen the result in print.

  2. Pingback: A fascinating problem in Peter Cameron's tribute to Atiyah - Nevin Manimala's Blog

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