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When giving a talk or writing a paper intended for non-specialist (i.e., mathematicians not specializing in the topic being discussed), the question inevitably occurs of what one can assume to be "common knowledge". Rather than trying to guess (e.g., I assume that almost all mathematicians know what a vector space over a field and what the Lebesgue measure are), it would probably be better to determine this experimentally. Have any surveys been conducted in order to answer this question?

Roughly which set of mathematical definitions and facts are known to a proportion at least $p$ of working mathematicians?

(Here, $\frac{1}{2}\leq p<1$ is some fixed but specified quantity. Of course, a survey like this would in fact measure how well-known various notions and theorems are, so would give results for a variety of different $p$. Also of course, this depends on some definition of what a "working mathematician" is, I'm assuming self-reporting as such, but I don't think the details are too important; even a survey limited to a particular country or membership to a particular mathematical society would be something.)

It seems to me that this would be interesting both mathematically (see above) and sociologically. More refined results indicating how results vary per country, per age group, or per specialty, would of course be of value, but any survey along these lines would interest me.

I tried Googling various terms to no avail, so I'm inclined to think that no such survey was ever conducted, but maybe I missed something.

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    $\begingroup$ @GerhardPaseman The thing is, I suspect mathematicians pick up a lot of knowledge, even outside their field, beyond what they are taught in University. (To give a specific example, I suspect that many mathematicians know what the Banach-Tarski paradox is, because it's so fun and surprising, but I gather very few students are actually taught about it.) $\endgroup$
    – Gro-Tsen
    Commented Jan 20, 2019 at 18:08
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    $\begingroup$ @YCor, legend has it that at least one outstanding mathematician didn't know what a Hilbert space was: mathoverflow.net/questions/53122/mathematical-urban-legends/… $\endgroup$
    – Gerry Myerson
    Commented Jan 20, 2019 at 19:58
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    $\begingroup$ @GerryMyerson interesting. I learnt that Euler didn't know either but it might be conspiracy theory. To take this into account, possibly let me mention that my comment concerns contemporary mathematicians, and that the OP's question is rather statistical/generical than focussing on exceptional individuals. $\endgroup$
    – YCor
    Commented Jan 20, 2019 at 19:59
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    $\begingroup$ Other difficulties in specifying what you're wanting to measure: the spectrum between "obliviousness", "general awareness", "recognition memory", and "recall memory". Not that it's not an interesting general idea of a question...! I've witnessed quite a few otherwise-supposedly-good mathematicians who were very narrow, and had forgotten anything they'd learned for qualifying exams that they hadn't used daily since. And didn't care. Some people do pride themselves on their ignorance. Also, some people have better memories, apart from taste or judgement. Interesting issue, but very complicated. $\endgroup$
    – paul garrett
    Commented Jan 20, 2019 at 21:58
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    $\begingroup$ These interesting notes by Barry Mazur address a question somewhat dual to yours: "What should a professional mathematician know?" Mazur focesses on broad areas and intuitions (as opposed to statements like 'Theorem A from area B'), which seems a wise decision. google.com/url?sa=t&source=web&rct=j&url=http://… $\endgroup$
    – user1073
    Commented Jan 21, 2019 at 21:57

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In looking over my answer, though it does introduce references, it seems to me to be opinion-based so I am making it a community wiki.

If giving a talk or writing a paper along these lines, you might consider your constraints regarding length or time allotted and try to give as much background as possible. Lay out your ideas and see if you are comfortable with the amount of background that is supplied and your target. For a paper, you should probably be working with an editor who can provide you with more specific advice relevant to your target journal. I will focus on general mathematical audience talks, aka colloquia. It seems the strongest opinions out there (measured by who was willing to write something down) seem to favor more valuing the "journey over the destination." See for example:

"How to give a good colloquium" John E. McCarthy, Washington University Canadian Mathematical Society NOTES, 31 no. 5, Sep 1999, pp 3–4 https://cms.math.ca/notes/v31/n5/Notesv31n5.pdf

University of Oregon Department of Mathematics Colloquium Guidelines https://pages.uoregon.edu/njp/guidelines.html

McCarthy gives a few examples:

The first 20 minutes should be completely understandable to graduate students....It means a student who just scraped by the required coursework in your area, and then went into a different field. So

  • you can assume they know what $L^p$ is, but not what a Sobolev space or pseudo-differential operator is;
  • you can assume that they know what a manifold is, but not what Poincaré duality is;
  • you can assume they know what a field extension is, but not what an induced representation is.

So this answer doesn't quite answer your question. In an attempt to do so, I would say assume only that covered by the basic grad courses if you can. You might even consider spending some time giving the introductory definitions and/or working through a representative example. This is especially true about the first 20-30 minutes of your talk (see for example the two references above).

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    $\begingroup$ Not everyone takes the same "basic grad courses". I remember being floored that some 2nd-year master students did not know what a tensor product was, and yet I never took any grad course on stochastic processes (random example) and I probably have other embarrassing lacks in my knowledge. I'll be honest, I think your answer doesn't... answer anything. It's a tautology. You cannot rebuild math from scratch during every talk, so a sentence like "give introductory definitions" does not actually mean anything concrete. $\endgroup$
    – Najib Idrissi
    Commented Jan 21, 2019 at 21:23
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    $\begingroup$ I didn't downvote... but, indeed, I think it is not a good strategy to give basic definitions whose ramifications are a huge, long story before getting to the beginning of one's own story. People do not assimilate definitions, much less their consequences, in real time. Even people who prefer "formal" treatments seem (in my observation) to do better with examples and analogies when introduced to alien ideas. $\endgroup$
    – paul garrett
    Commented Jan 21, 2019 at 23:07
  • $\begingroup$ I am still a bit surprised that most mathematics graduates don't know what a Hilbert space is. $\endgroup$
    – Hollis Williams
    Commented Aug 30, 2021 at 20:21

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