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I learn and produce mathematics. In that process, I had to read quite a number of research articles.

Question :

What are some efficient ways to keep a note of results when reading a research article in mathematics?

I keep a note of definitions (in detail) and results (with out proofs) for each paper I read.

Are there any other efficient methods to keep track of what results does a paper contain?

Reading introduction or abstract of a paper does not count as they are written in cryptic manner.

I asked the same question in Academia but it was not sufficient.

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    $\begingroup$ I find your question quite confusing. What is "keep a note of results of a research article"? The article precisely keeps a note of your research. If you mean keeping note of side results, auxiliary proofs, which are not included in the article, please be more explicit. I have no idea what's your point on introductions and abstracts and why you find them "cryptic", if of any relevance. $\endgroup$
    – YCor
    Commented Apr 11, 2020 at 15:58
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    $\begingroup$ @YCor "your research".. Not my research.. This question is asking about reading some one else's research article... I have specified in the edit .. Is it less confusing now? $\endgroup$
    – Praphulla Koushik
    Commented Apr 11, 2020 at 16:06
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    $\begingroup$ Introductions really ought not to be written in a cryptic manner - for many articles, finding the introduction impenetrable is a good sign that one is not yet ready to read the article (although there are also plenty of papers with horribly written introductions). $\endgroup$
    – Noah Schweber
    Commented Apr 11, 2020 at 16:13
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    $\begingroup$ I know a colleague who has boxes filled with small, A6 sized, cards on which he writes the generic information about the article (e.g. title, author, year, etc) and then a few notes about the results which he found useful in there. He keeps them sorted in there in various ways -- I really like this offline approach myself, and have tried setting one up myself. The key point about it is that the info he puts in is what he finds interesting or relevant to his own research, not just a summary of the articles. $\endgroup$
    – Carl-Fredrik Nyberg Brodda
    Commented Apr 11, 2020 at 16:16
  • $\begingroup$ @NoahSchweber I am not sure if I could have written more clearly.. This feeling of introduction and abstract being "cryptic" comes after I read the full article.. I feel may be the author could have said one/two lines extra about some thing in the introduction itself... But I see this is the procedure followed in most of the papers :D $\endgroup$
    – Praphulla Koushik
    Commented Apr 11, 2020 at 16:17

1 Answer 1

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In my own notes, my general approach for summarizing a paper is the following:

  • I copy the abstract verbatim, aside from correcting blatant typoes. (Every so often there's an exception here for papers with ridiculously long abstracts - which sort of defeats the point of an abstract, after all!)

  • I generally don't copy definitions exactly; instead I just record the term(s) introduced, the type(s) of object(s), and the intended role. (That said, particularly snappy/amazing definitions do also get stated in full.)

  • I copy the main results verbatim, again aside from correcting blatant typoes. I use my own judgment as to what constitutes a "main theorem" here, and that doesn't always coincide with the authors' although they usually do.

  • I never copy proofs in their entirety, but I don't fully omit them either: for each theorem I mention I include a summary in 1-10 sentences of its proof. This summary may only be limited to a description of the major obstacles; I err on the side of brevity. This is often quite hard, but is a super useful exercise.

(In each case above I also record the page number(s) of course.)

  • Finally, I also record some information about the background of the article. I write a summary of the bibliography (or at least part of it), recording how the various sources are relevant. I'll also record the general gist of each paper, even if it goes beyond how it's used at the moment: if Paper 1 just needs one lemma from Paper 2, I'll still give a very brief summary of Paper 2 itself as well as mentioning the lemma. This summary is again on the order of a few sentences, usually taken from the abstract or introduction (I don't need to read Paper 2 itself for this). Finally, I'll make a list of a small number of very relevant sources from the bibliography. This list could change over time as I understand the topic better, and often is pretty meaningless initially.

This is quite a bit of work in general, but note that the only step which requires deeply engaging with the article is the proof-summary bit; the rest can be done fairly mechanically or instinctively. The result is usually a 1-3 page document. A huge amount of information is generally lost by doing this, of course, but I find the whole process extremely useful and the result is still quite helpful in recalling the paper much later.

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  • $\begingroup$ Thanks... This is definitely a good approach... Do you mind to share an example? I can send an email if you do not want to put it here,,, $\endgroup$
    – Praphulla Koushik
    Commented Apr 11, 2020 at 16:41
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    $\begingroup$ @PraphullaKoushik I prefer not to - they're fairly messy, and I don't like sharing unpolished stuff in general. (And I don't have the time at the moment to polish one up.) $\endgroup$
    – Noah Schweber
    Commented Apr 11, 2020 at 16:42
  • $\begingroup$ That is understandable.. Never mind :) :) Thanks for the answer.. $\endgroup$
    – Praphulla Koushik
    Commented Apr 11, 2020 at 16:43

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