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• CommentRowNumber1.
• CommentAuthortomr
• CommentTimeFeb 28th 2021

Oliver Lorscheid in his https://arxiv.org/abs/1801.05337 F1 for everyone states “Most of the initial goals of F1-geometry have been solved, with the exception of the most influential one, the Riemann hypothesis.”. So, what we are lacking in current F1 definitions (monoid schemes or the blueprints - the two most elaborate efforts) to use F1 math for the completion of M1 theory?

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeFeb 28th 2021
• (edited Feb 28th 2021)

I’ll reply here once. Since this question essentially duplicates the one in another thread you started, let’s stick to that if further discussion should really be needed.

So as a quick reply: You mistook a comparison by analogy (here) for statement of equivalence: Nobody claims that finding a theory of $\mathbb{F}_1$ means to find M-theory (much less $M_1$-theory, wich I’ll assume is a Freudian slip of yours). Instead, as a way of understanding how it can be that M-theory remains unknown but nevertheless thought to exist, I offer the comparison to the situation with the theory of the “field with one element”, which is or was similarly unknown in itself, is or was similarly thought to be reflected in known limiting cases, and which therefore is or was expected to exist, even if unknown.

I am not sure if it’s true that the community has agreed that the theory of $\mathbb{F}_1$ has all been sorted out. But if so, then all the better for my analogy!

• CommentRowNumber3.
• CommentAuthorRichard Williamson
• CommentTimeFeb 28th 2021
• (edited Feb 28th 2021)

The current state of algebraic geometry over $\mathbb{F}_1$ is a million miles away from the principal goal of finding a way to encompass the archimedean part of number theory in some way analogous to algebraic geometry in positive characteristic (the Riemann hypothesis is only one special case of this). I think the most likely progress in this direction will be indirect: Mochizuki’s work is strongly reminiscent of a kind of algebraic geometry over $\mathbb{F}_1$, and Scholze et al’s condensed mathematics also has this kind of flavour.