Timeline for Examples of major theorems with very hard proofs that have not dramatically improved over time
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Jul 21 at 5:18 | comment | added | Yizhen Chen | Yakov Eliashberg lectured a proof of the uniformization theorem in his undergraduate complex analysis course. It does not depend on Riemann Roch, of which he did not have time to finish the proof. | |
| Feb 10, 2020 at 13:23 | comment | added | Andrea Ferretti | @AlexandreEremenko of course it depends, but Riemann Roch is a pretty standard result proved in many courses. Once that is known, obtaining the uniformization theorem for all surfaces is pretty easy, and I tried to give a hint of how to do it in my comment above. I mean, of course the comment does not have many details, but the fact that the outline of the proof + reference can fit in a single Mathoverflow comment makes me classify this as pretty simple. | |
| Feb 10, 2020 at 13:18 | comment | added | Alexandre Eremenko | @Andrea Ferretti: a) It depends on your students background, b) How much time is required for Riemann-Roch? c) Riemann-Roch is for compact surfaces, and the Uniformisation theorem is for all surfaces. | |
| Feb 10, 2020 at 13:14 | comment | added | Andrea Ferretti | It seems to me that this proof could be covered in full detail in a couple of hours max, once Riemann-Roch for Riemann surfaces is done | |
| Feb 10, 2020 at 13:11 | comment | added | Andrea Ferretti | A simple proof as Theorem VIII.11.12 is in Demailly's book. First do the case of domains of the plane, which is elementary. Then compact surfaces, using for instance Riemann-Roch. For a noncompact surface $U$, take an exhausting sequence of relatively compact connected open sets with smooth boundary $\Omega_k$. The connected sum $\Omega_k \cup \overline{\Omega_k}$ has a complex structure by reflection, and by the previous case you embed $\Omega_k \subset\mathbb{C}$. Up to subsequences, these converge to an embedding of $U$ | |
| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Sep 24, 2015 at 17:50 | comment | added | Alexandre Eremenko | This result has somewhat long history. It was first stated by Klein who published it, and since then the long story of the proof began:-) See the paper of Abikoff MR0628026 or the book of Saint-Gervais. | |
| Sep 24, 2015 at 15:58 | comment | added | Lasse Rempe | I believe this is a (very early) 20-th century result, proved by Koebe and Poincaré, rather than a 19-th century result? | |
| Dec 29, 2013 at 23:22 | history | edited | Amit Kumar Gupta | CC BY-SA 3.0 |
Use superior Hindu numeral system instead of Roman
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| Dec 21, 2013 at 15:42 | history | edited | Gil Kalai | CC BY-SA 3.0 |
added 110 characters in body
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| Dec 20, 2013 at 18:33 | comment | added | Lubin | A few years before I retired, teaching a fairly advanced graduate course in complex analysis, I foolishly promised the students that I would get Uniformization proved by the end of the semester. Hah! | |
| S Dec 20, 2013 at 17:15 | history | answered | Alexandre Eremenko | CC BY-SA 3.0 | |
| S Dec 20, 2013 at 17:15 | history | made wiki | Post Made Community Wiki by Alexandre Eremenko |