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As it is reasonable to think the work of mathematicians will be developed/made in their offices of universities (or in eventual seminars or conferences), here are the colleagues, books and journals, connection to databases and blackboards.

My belief is that a great part of mathematicians continue, somehow, their work outside working hours of their professional environment the university. In fact I think they have enough resources in their homes for this purpose and that they communicate with collaborators or colleagues while they are in the continuation (progress/attempts) of their research in their homes. I even evoke periodic meetings of nearby collaborators to study and work in specific problems.

Question. Is it reasonable to think that the professional mathematician does research in mathematics outside the office of his/her university? Typically, under what conditions? Many thanks.

The secondary question is a general overview of this situation and scenario, in case that the work outside of their professional enviroment is remarkable and can be characterized. I don't know if there are well-known examples of proofs of theorems due to mathematicians having an origin at home, coffee shops...I say research sessions/working day outside their offices. Thus an answer for the question under what conditions? should be pedagogical and informative, so that your colleagues and the general public can to know how the research in mathematics is done outside of university and get good results (and if there are general advices to schedule research sessions, remarkable preferences or tips to research in mathematics outside your office of your university).

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    $\begingroup$ Work habits differ wildly, but I would imagine that almost all mathematicians work outside their office some, or most, of the time. $\endgroup$ – Angelo Sep 27 '19 at 11:54
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    $\begingroup$ "I don't know if there are well-known examples of proofs of theorems due to mathematicians having an origin at home, coffee shops." What does it mean? Usually people think about theorems or proofs during the whole day, what is the "origin" you are looking for? $\endgroup$ – Francesco Polizzi Sep 27 '19 at 12:22
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    $\begingroup$ Famously, a lot of mathematics was done at the Scottish Café in (what was then) Lwów in the 1930's. $\endgroup$ – Robert Israel Sep 27 '19 at 12:31
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    $\begingroup$ @vidyarthi: With a faculty job at a university, you are being paid to do research, and your job is arranged to allow a certain amount of (paid) time to work on it (at least in theory). There are other effects that make a faculty job conducive to producing research, beyond just having an office on campus: journal subscriptions and library access, the experience of teaching, advising graduate students, etc. But you can benefit from many of these things without spending all your time in your office. $\endgroup$ – Nate Eldredge Sep 27 '19 at 13:08
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    $\begingroup$ @vidyarthi: yes, that is a good point. The AMS, shamefully, no longer even allows individuals to subscribe to MathSciNet. You have to be affiliated to an institution to have access to this essential resource. $\endgroup$ – Nik Weaver Sep 27 '19 at 14:00
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Mathematics differs from most other professions in that the only "resources" which are really needed are paper and pencil. (Even these are not strictly necessary, one can use sand and stick as the ancients did. Some can do even without sand, as the examples of famous blind mathematicians show).

As a result, the working habits of mathematicians widely wary. I know one study of this variety of working habits: J. Hadamard's book An essay on the psychology of invention in the mathematical field. It contains in particular the results of a poll that he made among his friends mathematicians. (For example it cites a famous story by Poincare how he invented authomorphic functions, while boarding a bus).

Many similar examples are known from the memoirs of mathematicians, or books like Littlewood's Miscellany. The idea of the uniformization theorem came to Klein when he was recuperating from asthma in a seaside resort (described in his book History of mathematics in XIX century). He was so excited that interrupted his vacation and rushed home to write a paper. Banach had a habit of working in a cafe. Feynman (not exactly a mathematician but close to it) recalls that he used to work in a topless bar at some time. If I remember correctly the so-called "tropical geometry" was invented by a group of mathematicians in a Rio-de-Janeiro beach, perhaps this is just a legend.

There is no opposition between "an office' and "home". Many mathematicians have offices at their homes, with books, computers, etc. Some of my friends have even blackboards in their home offices. A blackboard saves paper and it is more convenient for conversations. It is just a question of habit and convenience, where one prefers to work. Some people can have nicer office at home than at the university. Some people whom I know prefer home because smoking is prohibited on most US campuses:-)

Also, in some countries many mathematicians do not have convenient isolated offices. This was the case in Soviet Union, for example. Many of them also did not have convenient offices at home. In the beginning of my career, I remember proving most of my results while walking. I had regular walks with my adviser in a park near the university (my adviser shared an "office" in the university with 6 or 7 people, so we rarely discussed mathematics in his office).

When I moved to the US and obtained a convenient office, I still remember proving several theorems while walking my dog. I even think that walking stimulates mental activity, especially walking in a nice environment, in a park or a forest.

Many people used to work in a library if a library with convenient working space was available. Nowadays computers replace books, which makes the choice of a working place even more flexible.

I also know mathematicians who come to their office at 9 and work till 5, and do not work on Saturdays and Sundays. My impression is that this is a minority, but I am not aware of any statistics.

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    $\begingroup$ Actually "tropical" in tropical geometry is named after Imre Simon who was a computer scientist (Hungarian I believe) working in Sao Paulo who used the tropical smearing to solve problems in automata theory. The French automata theorist then called it the tropical semiring in his honor. $\endgroup$ – Benjamin Steinberg Sep 27 '19 at 15:07
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    $\begingroup$ @BenjaminSteinberg: "The mid-latitudes automata theorist" :) $\endgroup$ – darij grinberg Sep 27 '19 at 16:55
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    $\begingroup$ Paper, a pencil and an eraser. The Brits call it a rubber, which people in the US use for another kind of activities. $\endgroup$ – Sylvain JULIEN Sep 27 '19 at 21:27
  • $\begingroup$ Unfortunately, most people use nowadays a ballpen and unlimited supply of paper instead of an eraser. $\endgroup$ – Alexandre Eremenko Sep 27 '19 at 23:44
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    $\begingroup$ @AlexandreEremenko : You may be conflating the origin of the term "tropical geometry" with Smale's claim that he did much excellent mathematical work on the beaches of Rio (as mentioned in another answer). $\endgroup$ – Timothy Chow Sep 28 '19 at 3:59
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Famously, A. Wiles produced his celebrated proof of Fermat's Last Theorem after working in his attic at home (in near-total secrecy) for seven years.

This is clearly a kind of extreme example; however, doing research at home is not unusual, especially when one is looking for a quieter environment than the university.

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Significant work is done by mathematicians while asleep (usually in bed). Normally, this does not happen in isolation but follows a long period of intense concentration on a problem while awake. It hardly needs to be said that this is a rare and unpredictable phenomenon.

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  • $\begingroup$ Unpredictable to be sure, but I'm not certain that it's so rare. Something about the explanatory "usually in bed" parenthesis is very funny to me. $\endgroup$ – LSpice Sep 30 '19 at 21:05
  • $\begingroup$ I think it's quite common that someone will work hard on a difficult problem, perhaps not making much progress, and find that the next day they can suddenly see how to solve it. It's hard to say how much actual work on the problem has been done during sleep, but during sleep your brain organizes the day's experiences to produce long-term memory, and this may be part of it. $\endgroup$ – Robert Israel Oct 2 '19 at 3:23
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Steve Smale won a Fields medal in 1966, at least in part for proving the Poincaré Conjecture for dimensions five and higher. He induced some controversy when, as written on Wikipedia, he said that his best work had been done "on the beaches of Rio." This led to the withholding of his grant money from the NSF.

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    $\begingroup$ Wow, that seems a very harmless statement. I can hardly imagine someone taking offense at it. Is there some subtext that I am missing? $\endgroup$ – Federico Poloni Sep 28 '19 at 9:28
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    $\begingroup$ @FedericoPoloni it is in fact clarified in the link (a note by Smale): Smale was apparently already persona non grata in Washington on account of his anti-Vietnam-war activism, and it is suggested in the link that for this reason, the science advisor to then US president Johnson accused Smale of using public funds to finance maths on Rio's beaches. Same populism, different decade $\endgroup$ – AlexArvanitakis Sep 28 '19 at 19:23
  • $\begingroup$ I have no idea how to open the linked resource. Are text files and HTML not cool enough any more? $\endgroup$ – Tomáš Zato - Reinstate Monica Sep 30 '19 at 12:16
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    $\begingroup$ @TomášZato: text files and HTML may or may not be cool, but I doubt that is the motivation of someone linking to a PostScript file! $\endgroup$ – Bort Sep 30 '19 at 14:36
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    $\begingroup$ @TomášZato: postscrip is older than html, and probably cooler :) There are various viewer, at list on Unices. And you can convert it to pdf, if this makes it easier for you, using ghostscript (the ps2pdf script). $\endgroup$ – user95393 Sep 30 '19 at 17:11
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Stimulating, interesting and prolific mathematical conversations can take place in many different environments: a famous example that I'd like to mention is the story of the celebrated Yoneda Lemma. As it is said in this email written by Yoshiki Kinoshita in occasion of Yoneda's death, it looks like the lemma was first stated during a conversation that Yoneda was having with MacLane in a café.

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I almost never do research at my place of work in a University. I teach my classes there, then go home to think. It's almost impossible for me to really think in a public setting.

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