Timeline for What are the potential applications of perfectoid spaces to homotopy theory?
Current License: CC BY-SA 3.0
18 events
| when toggle format | what | by | license | comment | |
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| Mar 15, 2021 at 23:41 | answer | added | Yuri Sulyma | timeline score: 3 | |
| Mar 15, 2021 at 13:23 | answer | added | Peter Scholze | timeline score: 18 | |
| S Jul 21, 2017 at 14:24 | history | bounty ended | CommunityBot | ||
| S Jul 21, 2017 at 14:24 | history | notice removed | CommunityBot | ||
| Jul 13, 2017 at 17:35 | comment | added | Yuri Sulyma | @DenisNardin I did mean the TAF book by Behrens-Lawson, although presumably there are applications to tmf as well. | |
| Jul 13, 2017 at 13:58 | comment | added | ACL | In French, TAF is a colloquial word for "work" :-) But what does it mean in this context? | |
| S Jul 13, 2017 at 12:40 | history | bounty started | Ofra | ||
| S Jul 13, 2017 at 12:40 | history | notice added | Ofra | Draw attention | |
| Jul 4, 2017 at 3:04 | comment | added | Yuri Sulyma | @skd the TAF book is surprisingly user-friendly, and self-contained. The first 7 chapters (out of 15) tell you all the number theory you need to know, and after that it's pretty easy to construct. Saying something about TAF is, of course, much harder :) | |
| Jul 4, 2017 at 2:12 | comment | added | skd | @YuriSulyma oh, interesting, thanks! I've been too intimidated to read the construction of TAF, but this seems cool. | |
| Jul 4, 2017 at 1:43 | comment | added | Yuri Sulyma | @skd higher-dimensional formal groups are used to build $TAF$, by splitting off a 1-dimensional guy from its $p$-divisible part (if memory serves me). That's the only example I know. | |
| Jul 3, 2017 at 15:24 | comment | added | skd | Googling led me to chromotopy.org/hypothetical-abelian-varieties, which talks about the same thing as my previous comment. | |
| Jul 3, 2017 at 15:15 | comment | added | skd | (contd.) Higher-dimensional formal groups also pop up in homotopy theory via the Ravenel-Wilson computation of K(n)_* K(Q/Z, k); this is an exterior power of K(n)_* K(Q/Z, 1), and so the sum over all positive integers k of the Dieudonn'e modules of Spf K(n)_* K(Q/Z, k) should be the Dieudonn'e module of some abelian variety. What does the geometry of this abelian variety have to do with K(n)-local homotopy theory? (A more concrete problem, maybe, is understanding what happens at height 1.) | |
| Jul 3, 2017 at 15:13 | comment | added | skd | Thanks for asking this question! I'm curious to read responses to this question. This is slightly tangential, but a (more general?) question that I have is whether higher-dimensional formal groups can influence homotopy theory in some way. For instance, the Gross-Hopkins period map is a special case of the crystalline period map (section 6 of the Scholze-Weinstein paper). The former has had enormous success in chromotopy --- can something in chromotopy be deduced from the latter? | |
| Jul 2, 2017 at 10:42 | comment | added | Ronnie Brown | The paper by Vezzani arXiv:1405.4548. gives further references. I was interested to note the use there of cubical sets with connections for constructing derived functors. . There is a paper on this Patchkoria, I. ‘Cubical approach to derived functors’ Homology, Homotopy and Applications 1 (2014) 133–158. | |
| Jun 30, 2017 at 10:49 | history | made wiki | Post Made Community Wiki by Todd Trimble | ||
| Jun 30, 2017 at 8:36 | comment | added | Sean Tilson | It is also worth pointing out that Scholze has started to use more tools from homotopy theory. I would say that my interest would lie in some connections that maybe haven't been articulated (I have no clue what they are) but homotopy theorists are frequently interested when people are start using their tools. | |
| Jun 30, 2017 at 6:05 | history | asked | Yuri Sulyma | CC BY-SA 3.0 |