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I have two questions regarding the development of mathematics:

1) Is there an example where in mathematics, a collaboration has led to the discovery of another result? I already know something like the Polymath project, or the Hilbert program and the Hamilton's program.

2) Is there an example of how individual secrecy and lone effort in mathematics has led to a breakthrough discovery without much contact with the math community? The ones I am aware of are like Andrew Wiles, and Ramanujan.

I just want some more examples cause I was quite curious with regards to certain approaches to mathematical development.

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According to the Mathscinet database (which covers most publications since approximately 1940), 37.6% of all mathematical papers have more than one author, so they are results of collaboration. This number was much less in the previous centuries. This means that examples of a) and b) are too numerous to try to make a list.

This site contains a lot of statistics on collaboration in mathematics:

http://www.oakland.edu/enp/trivia/

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  • $\begingroup$ One would hope so... But then again, any grad student's paper will automagicallly include his advisor's name in the author list. $\endgroup$ – Carl Witthoft Dec 10 '18 at 13:32
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    $\begingroup$ @Carl Witthoft: first, this is incorrect that "every graduate student will include his adviser's name" at least this is not practiced in mathematics. Second, a research of the graduate student is in most cases really joint research. $\endgroup$ – Alexandre Eremenko Dec 10 '18 at 14:39
  • $\begingroup$ Fair enough. I'm more used to looking at theses in Physics or Engineering. $\endgroup$ – Carl Witthoft Dec 10 '18 at 15:00
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Perelman's early work was impressive enough to garner several job offers from US universities. But he turned them down, returning to Russia and a research position with the Steklov Institute of Mathematics in St Petersburg. At that point, Perelman effectively disappeared — he stopped publishing papers or discussing his research with colleagues. “We would occasionally ask where he was,” says a friend. “No one seemed to know what he was doing.” Even people at the Steklov Institute didn't know what he was working on.

https://www.nature.com/articles/427388a

In the early 2000s he [Mochizuki] stopped venturing to international meetings, and colleagues say that he rarely leaves the Kyoto prefecture any more. “It requires a special kind of devotion to be able to focus over a period of many years without having collaborators,” says number theorist Brian Conrad of Stanford University in California.

https://www.nature.com/news/the-biggest-mystery-in-mathematics-shinichi-mochizuki-and-the-impenetrable-proof-1.18509

But the purported proof, which Mochizuki first posted on his webpage in August 2012, builds on more than a decade of previous work in which Mochizuki worked in virtual isolation and developed a novel and extremely abstract branch of mathematics.

https://www.nature.com/news/monumental-proof-to-torment-mathematicians-for-years-to-come-1.20342

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