Timeline for Examples of toposes for analysts
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Dec 19, 2013 at 23:43 | comment | added | Anton Fetisov | I don't really know how sheaves can help in probability theory, but my view of probability doesn't exceed a one-semester undergraduate course. I heard the word about something called "stochastic geometry" which looks like ag/prob mix, and I also assume that anyone can benefit from the study of PDE (e.g. for stochastic motion). If you also count statistical quantum physics as probability, then you should also be interested in things like quantum field theories, and these have a deep relation to sheaves etc, but that looks far too deep for any introduction. | |
| Dec 19, 2013 at 23:38 | comment | added | Anton Fetisov | Finally, see Kashiwara, Schapira "Sheaves on manifolds". It discusses a far-reaching generalization of previous themes, with applications to the study of differential equations on manifolds. It's definitely not the book to get a first acquaintance with sheaves, but it gives a nice view of (20 y.old) state of the art. | |
| Dec 19, 2013 at 23:35 | comment | added | Anton Fetisov | @TomLaGatta, since I myself am rather far from analysis, I can't be really specific here. I can only give some pointers. Firstly there is the work of Mikio Sato et al. on hyperfunctions, which are a special sort of generalized functions, but are defined and studied via sheaf theory. Secondly, most interesting analytic objects over manifolds (differential forms, tensor fields, connections etc) naturally form sheaves and that part of structure is very important. Any good algebraic complex geometry book will discuss it (e.g. de Rham's theorem). | |
| Dec 19, 2013 at 21:45 | comment | added | Tom LaGatta | Thank you for your excellent answer, @Anton. Could you expand more on the statement, "sheaves are important for analysts"? Coming from the world of probability, we don't talk about sheaves very often, and I would like to know more about why we should. | |
| S Dec 19, 2013 at 21:04 | history | answered | Anton Fetisov | CC BY-SA 3.0 | |
| S Dec 19, 2013 at 21:04 | history | made wiki | Post Made Community Wiki by Anton Fetisov |