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mitchell porter

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Atiyah is 89. He's still enormously knowledgeable, but various mathematicians are saying that in recent years he has published a number of incorrect mathematical claims, and his faculties are therefore sadly in decline, at least relative to a point in his career where he was making genuine discoveries. Presumably some expert will eventually undertake the melancholy duty of summarizing what Atiyah has been saying mathematically and what's wrong about it. (PF's "General Math" forum already has a thread on today's claimed proof.)

But I thought I would start a thread that is specifically on the

*physical*content of Atiyah's current ideas. One reason is that in the past few years he has coauthored a number of papers with alleged physical content, and while they were clearly speculative, I had not until now imagined that they might contain significant errors, and indeed they may not.

For example, with Manton he wrote a paper in 2016, "Complex Geometry of Nuclei and Atoms", proposing "a new geometrical model of matter, in which neutral atoms are modeled by compact, complex algebraic surfaces". Now, over twenty years ago Atiyah and Manton came up with an instantonic realization of the skyrmion - an old solitonic model of the nucleon - which was subsequently rediscovered in string theory, as part of the Sakai-Sugimoto model of holographic QCD. So one could reasonably wonder whether Atiyah and Manton had after all done it again, and found elegant algebraic-geometric representations of nuclei.

I don't know yet how this thread will work out. It may be difficult to segregate Atiyah's mathematics from his physics. Nonetheless, he has given a name to his physical paradigm - "arithmetic physics" - and I suppose that is what we should try to understand here. The notion is not unique to him. In the higher reaches of mathematics, there is already a refinement of algebraic geometry called arithmetic geometry, and presumably arithmetic physics is an application of arithmetic geometry to physics - it should be that simple.