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I have a question about the history of two important results from the eighties in symplectic geometry. In both cases it seems that important results were developed (almost) simultaneously by different people, which I find quiet remarkable and unusual. I would be interested in knowing if the corresponding researchers were aware of each other, working on the same projects, and if so at what stage of their project they did become aware of the other one. (Highly hypothetical: do you think that these days people accept more easily to publish a joint paper in such a situation?)

There is the convexity result by Atiyah, Michael F., Convexity and commuting Hamiltonians, Bull. Lond. Math. Soc. 14, 1-15 (1982). ZBL0482.58013. and by Guillemin, V.; Sternberg, S., Convexity properties of the moment mapping, Invent. Math. 67, 491-513 (1982). ZBL0503.58017. Both articles have been published in the same year... of course, it takes some time to get a paper published and there was also nothing like the arxiv to maybe have a better idea of the developments.

I'm also surprised because the localization result Duistermaat, J. J.; Heckman, G. J., On the variation in the cohomology of the symplectic form of the reduced phase space, Invent. Math. 69, 259-268 (1982). ZBL0503.58015. was published the same year, followed by Berline, Nicole; Vergne, Michele, Zeros d’un champ de vecteurs et classes characteristiques équivariantes, Duke Math. J. 50, 539-549 (1983). ZBL0515.58007., which seems to describe a similar thing.

I admit that I do not have a very precise question, but if anybody could give some insight on how so many interesting things developed in such a short time by different authors, I would be extremely interested.

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  • $\begingroup$ We often have migrations suggested to [HSM](hsm.stackexchange.com) that are resisted because the askers want to get MO's opinion, but this sort of highly opinion-based question seems like it really belongs there much more than here. $\endgroup$ – LSpice May 3 '20 at 17:25
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    $\begingroup$ The work by Atiyah and Guillemin-Sternberg were indeed done independently. I believe Kostant may have also proved the same theorem independently. I have often told the story about how Atiyah, Guillemin, and Sternberg discovered they had proved the same theorem. Atiyah was giving a seminar talk about the theorem at Harvard. As the seminar progressed, the front row became more and more agitated. Finally, Raoul Bott spoke up, saying something like "Michael! You should know that these two gentlemen proved the same theorem two weeks ago!" and pointed to Sternberg sitting next to him and Guillemin. $\endgroup$ – Deane Yang May 3 '20 at 17:29
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    $\begingroup$ @DeaneYang Thank you very much for this story. This is very interesting. I would have expected that somebody had come up with the conjecture, and that different people would have worked in parallel to solve the claim, but not that they also came up independently with the statement. I can imagine what a shocking moment this must be to discover somebody else is just giving a talk about "your" result. $\endgroup$ – Klaus Niederkrüger May 4 '20 at 13:06
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    $\begingroup$ @LSpice I did not know about HSM. Maybe you are right, but I think migrating my question to HSM is just the same as deleting it. $\endgroup$ – Klaus Niederkrüger May 4 '20 at 13:13
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    $\begingroup$ Obligatory mention of the Wikipedia article on multiple discovery. These examples could perhaps be added to Wikipedia's list. $\endgroup$ – Timothy Chow May 4 '20 at 14:44

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