How to Read a Math Paper

How to start – outline of steps
  • Read over the introduction, even if you don’t understand a lot of it, and check the references being mentioned throughout. 
  • Read over the introductions of a couple of those, to get a better understanding of the context of your article/problem, see what else has been done in the area. 
  • Once you feel like you know the general (not all the details yet) idea of what’s being asked and how it’s being approached, read through the article 
  • The intent of reading it the first time is not to understand everything, and in fact you may not understand anything of some of the sections; it’s to be able to fill in the gaps on re-reads, and with other resources
  • What might help during these steps:
    • Find survey articles (google “survey article [my topic]”)
    • Find PhD thesis or MSc thesis papers about your question or a related one (they might give more examples and details)
    • Take notes on at least the introduction, and maybe the first section (at least for now, start at the beginning)
    • Look for textbooks that explain these topics in more depth, especially if they have examples written out
    • Look for seminars/talks about the paper
If you’re stuck:
  • Sometimes details are deeply hidden or omitted for page constraints
    • An extended version of the paper may be on arxiv
  • Be aware that there are gonna be typos (footnote: “However, when, as a graduate student, one encounters the task of reading a technical mathematical paper for the first time, it is often the case that one loses much of one’s higher reading skills, reverting instead to a more formal and tedious line-by-line interpretation of the text. As a consequence, a single typo or undefined term in the paper can cause one’s comprehension of the paper to grind to a complete halt, in much the same way that it would to a computer.” from Terry Tao’s blog)
    • Compare with similar sections of the paper
    • Recognize what the author is trying to do, and how the step fits into the overall argument, don’t just formally read line by line
  • Try and find a simpler paper, or a paper which uses the same argument to solve an easier problem (check the references)
    • A paper will usually tell you when they’re following another paper for a section of the proof
  • Find something to try!
    • Even if you’re completely lost on how to justify a step, just try the first thing which comes to mind. It might not work, but understanding why it doesn’t is still progress
  • Find a way to turn large parts of the paper into one step
    • A paper may be going through a very complicated construction or a long computation which can be summarized as simply “getting a bound” or something similar.
    • Look at what the step you’re stuck on is trying to prove, and how it comes into play in the larger structure. 
    • In this case, there is no need to spend time understanding “unnecessary” parts of the proof, unless you’re incorporating the arguments into your own research
General advice:
  • Check your understanding of the ideas (not just the technical details)
    • Can you approximately reconstruct the proof after looking at it
    • Explain the paper to someone else (or to a rubber duck)
    • Simple outline: first step leads to second, etc, in a few sentences
  • Don’t be a passive reader!
    • Ask yourself why the definitions are set up the way they are
    • As you read statements of theorems, think of how you would approach proving it or disproving it before reading ahead in the paper
    • Try to anticipate what results follow from theorems
    • A math paper is a polished version of a longer process of trials and errors – recreate this process for yourself!
  • Prioritise which parts to read and skip
    • Don’t read everything in detail if the paper or a specific proof is very long
    • Some parts of papers are there for completeness, but might not give a lot of interesting insights, skim or skip those
    • Ask your supervisor for advice and insight into which parts are interesting general tools, which examples are important in your field, what is background reading, which parts you can skip
  • Visualize abstract/general proofs
    • Start with the proof/explanations of a special case
    • Have a leading example, which you refer to as you read through the paper and check how the general results apply in practice – this will help you develop intuition for the problem
    • (for graph theory) Draw a graph or example illustrating the important ideas or proofs of the paper
  • Once you have a good understanding, think beyond the paper
    • Check why all the assumptions are necessary – what happens when you drop one of them?
    • Can the results be generalised further? If not, what are insightful counter-examples?
    • Variations – How would the problem and results change if you slightly change the setting?
If it’s your first paper…
  • Don’t be discouraged if you need to read up on a lot of background. If you don’t understand everything on your first read –  don’t hesitate to pause and go back to fill gaps
  • It gets faster! As you will get comfortable with standard arguments
  • Papers are written with a specific target audience in mind – look for a first paper that is suitable for your background
Further links on how to read a math paper:

Now you know how to read a math paper, you’re all set to go try your own things and let us know any helpful advice we could add to this page!

Not sure how to even find which paper to read? Take a look at MathSciNet, which you can access for free through the McGill VPN or WiFi on campus

Thank you to Agnès, Bart, Maddie and Sam for organizing and giving this workshop. Thank you to Prof. Sophie Spirkl for her insightful advice.

Further links on how to read a math paper:

en_CAEnglish (Canada)