Timeline for Dictionary of arithmetic symmetries and Langlands
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 10, 2021 at 1:27 | comment | added | P.H. | Yes, that's precisely the thought that had occurred to me yesterday. Rather than treating as a mystery I should perhaps think of it as its resolution! A mere shift in the angle of vision and ... until one day we'll come to see it as part of a larger whole which makes it seem all too natural. | |
| Feb 9, 2021 at 22:08 | comment | added | Will Sawin | @P.H. This might not be so helpful, but isn't the Langlands program itself an answer to that question? | |
| Feb 9, 2021 at 2:25 | comment | added | P.H. | Finally posting the question after living with it for a while, and then this exchange, will hopefully make me think more deeply and precisely along these lines, and in the end either reject it as fallacious or tautological or accept it as satisfactory as an insight. Thanks for helping along the process! | |
| Feb 9, 2021 at 2:21 | comment | added | P.H. | I promise to think more about what you have written. It's my fault that I was not able to convey in my question my central concern: is there some point to thinking that the collection of all algebraic groups acting "arithmetically on themselves" (i.e., automorphically) is the totality of a certain class of algebro-arithmetic symmetries, that seems to be strangely and intimately related to Galois actions. How to characterize that class? | |
| Feb 9, 2021 at 2:18 | comment | added | Will Sawin | @P.H. You might want to edit your question to post more detail about what, exactly, you are looking for. | |
| Feb 9, 2021 at 2:16 | comment | added | P.H. | O, certainly. It spurred me to go through Langlands' writing once again - quite hurriedly but I'll do so again when I have some time. And I'Sorry if it comes across as churlish; these messages lose all nuance. | |
| Feb 9, 2021 at 2:13 | comment | added | Will Sawin | @P.H. I am sorry my answer was not to your liking. I hope the small bits of positive information I included (e.g. on unipotent groups, and on the connections to physics) were helpful. | |
| Feb 9, 2021 at 2:08 | comment | added | P.H. | I just want to understand for myself why the two should be related at all in intuitive terms because class field theory is a flimsy rationale for it, and makes for one only in its re-interpretation in light of Langlands’ extrapolation. I do wish historians of contemporary mathematics would weigh in on it with concrete evidence, but that is at one remove from the spirit of my question. | |
| Feb 9, 2021 at 2:08 | comment | added | P.H. | I did not say that that’s how Langlands came up with it; it was indeed in part extrapolation - extraordinarily bold extrapolation - from known cases, but also more than a dollop of imagination and intuition. I wish I could know what his thoughts and insights were when he formulated his conjecture, but his 1967 letter to Weil throws no light on the matter, nor do his subsequent writings or talks as far as I am aware. | |
| Feb 9, 2021 at 2:07 | comment | added | P.H. | Your answer starts with a gross mis-interpretation of my question. No wonder the rest of its sounds to me like a screed based on misguided premises. I am indeed looking for a high-level understanding of why “automorphic” should have anything to do with “Galois”. More detailed the better, but one begins somewhere. Indeed it is not a substitute for getting into the weeds and doing the dirty work - exceptional mathematicians have been doing for 50 odd years - but it is not contradictory to it either; rather it complements it. There is the forest to admire as well as the trees. | |
| Feb 8, 2021 at 14:48 | history | answered | Will Sawin | CC BY-SA 4.0 |