occasional meanderings in physics' brave new world

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Location: New Zealand

Marni D. Sheppeard

## Friday, July 04, 2008

### Riemann Again

Sorry Carl, I can't resist. Recall Connes' remark regarding the Riemann Hypothesis that
it is a basic primitive question about the adelic line which we don't understand. It is a question about the way addition is fitting with multiplication.
In this light, the very simple use of the adeles in Li's paper comes across almost as an insult to the spirit of Connes' approach.

One gets the eerie feeling that Li has a strong subconscious sense of a promising line of attack, but that this has led him into the labyrinth of murky delusion, a frightening place of which there is no need to say more. But if nothing else, the paper has caused a lot of bright people to ponder the mystery that is the rational adeles. In M Theory, since a prime $p$ is pretty well always associated (as logos building blocks) to categorical dimension, even if only to count sets, the adeles must be an $\omega$-categorical construction. There is no problem defining rational numbers or appropriate limits in this setting, so the mystery lies in what it means to tack the real numbers on the end, at the infinite prime.

One nice property of the adeles as an Abelian group is that it is isomorphic to its Pontrjagin (Fourier) dual. Somehow this is analogous to the schizophrenic property of the group $U(1)$ in the full Stone duality setting, enriched to the $n$-category hierarchy setting. Or, as kneemo would put it, a string is secretly a necklace of pearls.