Timeline for What is precisely still missing in Connes' approach to RH?
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| Jul 23, 2017 at 15:04 | answer | added | Rieendstac | timeline score: 4 | |
| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Jan 12, 2017 at 1:04 | history | edited | Abdelmalek Abdesselam | CC BY-SA 3.0 |
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| Jan 12, 2017 at 1:02 | answer | added | Abdelmalek Abdesselam | timeline score: 24 | |
| Jan 11, 2017 at 6:59 | answer | added | Will Sawin | timeline score: 72 | |
| Jan 11, 2017 at 5:39 | comment | added | nfdc23 | The reason I think this is futile (unlike the Standard Conjectures), or at the very least extremely premature, is that Connes' approach is indeed far less advanced than Grothendieck's was in 1968: in the Connes approach there is no cohomological interpretation proved for the standard $\zeta$-functions (not even the Riemann $\zeta$-function) since there is no proof of a trace formula: see the end of section 4 of arxiv.org/pdf/1502.05580.pdf. Or am I unaware of a dramatic development giving such a cohomological interpretation with a robust cohomology theory having many good properties? | |
| Jan 11, 2017 at 3:52 | comment | added | Joël | Brief, I do not think the question is futile. It may be unanswerable, which would mean that Connes' approach to prove RH is much less advanced that Grothendieck's approach to prove RH over finite fields was in the late 60's. And Grothendieck's approach, with the standard conjectures, even if unsuccessful in a sense until now, was certainly not futile: it was one of the most fecund ideas in mathematics. | |
| Jan 11, 2017 at 3:51 | comment | added | Joël | @nfdc23: but prior to Deligne's breakthrough, there was already a roadmap of a path from the etale cohomology to the actual proof of the RH over finite fields: the so-called standard conjectures of Grothendieck. I believe Santker is asking for the analog of the Standard Conjectures in Connes's approach to RH. True, Deligne found an other idea avoiding the standard conjectures and using instead some positivity arguments, and these conjectures are still open 50 years after their formulation. But the conjecture did exist, and in Connes' case, the OP is asking for conjectures, not proofs. | |
| Jan 11, 2017 at 3:41 | history | edited | GH from MO |
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| Jan 11, 2017 at 3:00 | comment | added | nfdc23 | Here is another way to convey the futility of such a speculative question. Etale cohomology was invented as an analogy to the cohomology known from algebraic topology, and Serre had proved a version of RH for Kahler manifolds, but the gulf is vast from creating some analogue to proving the hard theorems about it to make it powerful enough to establish the RH aspect of the Weil conjectures. Prior to Deligne's breakthrough one could have likewise asked "what are the issues left open" to adapt Serre's Kahlerian analogue. In view of what finally happened, perhaps the best answer is: a good idea! | |
| Jan 10, 2017 at 23:25 | comment | added | santker heboln | For an analogy to break down (whatever is precisely meant by "analogy" in mathematics, which may vary in context) you first have to try to make it. In any case it seems a worthwhile goal to try. But let's not get too far off topic by discussing opinions about how worthwhile this program is. I don't have an opinion because I don't understand it well enough. | |
| Jan 10, 2017 at 22:50 | comment | added | Yemon Choi | This is not "arguing about philosophy". This is about saying: an analogy is all very well and good, but (as @nfdc23 says) actual substance is needed. Analogies break down all the time, it's just that people don't write books about these cases and rhapsodise about them | |
| Jan 10, 2017 at 22:44 | comment | added | santker heboln | The positivity condition would follow from a Riemann Roch type formula for the corresponding "surface". @YemonChoi: That may be so (in many cases), but let's not argue about philosophy. ;-) | |
| Jan 10, 2017 at 22:13 | comment | added | nfdc23 | Serious progress on hard analytic problems cannot be achieved just by defining a topos (which is basically a piece of algebra). Positivity properties have to come from somewhere. | |
| Jan 10, 2017 at 19:24 | comment | added | Yemon Choi | analogy != proof | |
| Jan 10, 2017 at 19:21 | history | edited | santker heboln | CC BY-SA 3.0 |
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| Jan 10, 2017 at 19:08 | history | asked | santker heboln | CC BY-SA 3.0 |