In their book, ''Noncommutative Geometry, Quantum Fields and Motives,'' Alain Connes and Matilde Marcolli begin their preface by saying:

The unifying theme, which the reader will encounter in different guises throughout the book, is the interplay between noncommutative geometry and number theory, the latter especially in its manifestation through the theory of motives. For us, this interwoven texture of noncommutative spaces and motives will become a tool in the exploration of two spaces, whose role is central to many developments of modern mathematics and physics: ² Space-time and ² The set of prime numbers. One may be tempted to think that, looking from the vantage point of those who sit atop the vast edifice of our accumulated knowledge of such topics as space and numbers, we ought to know a great deal about these two spaces. However, there are two fundamental problems whose difficulty is a clear reminder of our limited knowledge, and whose solution would require a more sophisticated understanding than the one currently within our immediate grasp: ² The construction of a theory of quantum gravity (QG) and ² The Riemann hypothesis (RH). The purpose of this book is to explain the relevance of noncommutative geometry (NCG) in dealing with these two problems. Quite surprisingly, in so doing we shall discover that there are deep analogies between these two problems which, if properly exploited, are likely to enhance our grasp of both of them.

Can someone explain in the simplest possible terms what really the link between the RH and QG that Connes and Marcolli were talking about ?

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    $\begingroup$ Any reason you would exclude an author from the title of your question when you are quoting a text written by two people? And later on, refer to her by her first name when you refer to him by his surname? $\endgroup$ – Najib Idrissi Jan 26 '19 at 10:31
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    $\begingroup$ @NajibIdrissi, the title was getting too long, as one can see, nothing less nothing more. Anyway, i will edit the parts that you mentioned. $\endgroup$ – CuriousTatenda Jan 26 '19 at 10:45
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    $\begingroup$ This pdf suggests it is about letting $q \to 1$ in the RH for curves over $\mathbb{F}_q$ to obtain it for "curves over $\mathbb{F}_1$". I don't understand what it means with the space of adeles classes of a global field being "non-commutative" $\endgroup$ – reuns Jan 26 '19 at 11:17
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    $\begingroup$ While I agree that there is sexism in the mathematical world and anyone can be guilty of it, I would encourage people not to rush to judgment on the basis of only a small amount of evidence. For example I know that there have been times when I knew two collaborators personally, and in informal interactions would call one by first name and the other by last name because that was what they preferred. Also there have been times when I knew that one collaborator contributed much more than the other. (cont'd). $\endgroup$ – Timothy Chow Jan 27 '19 at 20:31
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    $\begingroup$ While I would try not to allow these informal habits to slip into more formal communications, sometimes it would happen inadvertently, to my embarrassment and I would make up some excuse to make myself not look bad. I'm not saying that this is necessarily what is going on in CuriousTatenda's case, but just that just as we should take care not be sexist, we should also take care not to jump to conclusions about sexism, especially since an accusation of sexism can be highly damaging to someone in today's environment. $\endgroup$ – Timothy Chow Jan 27 '19 at 20:33

Not a direct answer, but maybe worth pointing out:

An apparently substantial insight into a relation between gravity and the zeros of the Riemann zeta-function (hence the Riemann hypothesis) was recently found via $p$-adic string theory by Shing-Tung Yau et al.:

  • An Huang, Bogdan Stoica, Shing-Tung Yau, "General relativity from $p$-adic strings" (arXiv:1901.02013)

Seems quite remarkable to me.


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