This is a question that has been in my mind for a long time. Now I am a M.Sc. first year student in mathematics and planning to apply for graduate school next year in Europe. But for applying to graduate schools, it is written that first I have to write an email to a supervisor along with a idea of a research project and I have to convince him that I am well prepared to work under him as a PhD student. Also, it is recommended that you should study the papers of professors to whom you are applying.

Now, I am interested in a topic. My college does not offer that topic. So, I have learned the topics on my own. Also, at this moment I do not participate in an internship program because of this pandemic situation all over the world. That is why I have decided that I will read some papers of the professor to convince him that I am well prepared to work under him.

My question is as follows: How do I demonstrate that I read and understood a professor's paper to the professor?

Please advise me. Thanking in advance.

  • Which country in Europe? Apr 21, 2021 at 10:26
  • @astronat France and Luxembourg
    – user138005
    Apr 21, 2021 at 12:22

4 Answers 4


That is why I have decided that I will read some papers of the professors to convince him that I am well prepared to work under him.

I disagree with this advice. I would rather see applicants (especially those with an MS) demonstrate their ability to do research, usually through a publication or a strong thesis. As long as their topic is somewhat related to my research, that would be a stronger demonstration of their potential to succeed in a PhD program. Being familiar with specific papers that I have written is not necessary or expected (you can always read such papers if you are accepted).

If your only option is to study the papers of professors, my advice is to implement the methods discussed in those papers numerically, i.e. repeating the results, or possibly even conducting some trivial yet new extensions (like repeating the experiments on a different dataset). That would be far more impressive than simply reading a paper (where your depth of understanding is not clear).

  • 1
    The OP does not specify which part of mathematics they are interested in, but in many topics (e.g. my own) there is no such thing as implementing methods numerically or repeating experiments on a different dataset. Apr 21, 2021 at 14:24
  • @Elodin Implement the methods discussed in the paper is a great idea. Also, I have done this type of works previously. I checked a conjecture numerically in Mathematica. But now I am going to read a paper in hyperbolic geometry. The link for the paper is here. My question: I can not implement the methods that discussed throughout the paper always. For example, please consider the above mentioned paper. So, what should I do for the paper?
    – user138005
    Apr 21, 2021 at 14:28
  • I'm not in mathematics so I can't speak to that paper specifically, but often the mathematical statements that I prove give analytical properties that could be demonstrated numerically. You could also prove some simple extensions of the results.
    – Elodin
    Apr 21, 2021 at 16:44

You probably should not handle this like an assignment "I have to read a paper and proof that I have understood it", but see the broader context.

Would you have been interested in being a coauthor for the paper? Do you have ideas for follow up papers? Would you like to work on similar projects? Do you have own ideas? Do you get the impression that you will be able to work in the same field?

Answer these questions to yourself, then talk to the professor about what you want to do and why. Being informed about their department's work will help you and having own ideas will help them to find good topics for you.


I believe I have a good answer here, from experience. Two things you should do to communicate to a professor that you understand a paper you read:

  1. Ask good questions
  2. Add on to it, even if it's trivial, like you said

Asking questions, while it would seem like the opposite thing to do, is a fantastic way to let a professor know that you put the time and effort into reading. You don't always have to understand every little detail- that's what advisors are for. Asking a good question or two about a paper, and possibly relating it to other ideas, starts a great discussion.

Saying something like "I thought it was really interesting how you did ____ instead of the more well known method of ____. What made you choose _____ ? I tried to improve some results using _____ and came up with _____." is admirable, in my opinion.


How do I demonstrate that I read and understood a professor's paper to the professor?

The most obvious requirement here is the ability to be able to summarise the essence of the paper (and its contribution to the literature) in simple and understandable terms. If you can explain the parts of the paper that you found interesting, that also helps show your understanding. Ideally, you should be able to talk intelligently about the paper at an abstract "heuristic" level, but then get down dirty and discuss the technical details of parts of it if required. Your question here is really "How can you tell if someone understands something you understand", so think about all the things that cause you to believe that other people understand things you understand.

One thing to bear in mind here is that one reason to read a prospective supervisor's papers is to see if there are any aspects of the research that you would like to try to extend in your own research work. If you can find some aspect of the paper that you are interested in, and think you could take further, then that will give you a possible avenue of research on a research question where the professor is obviously qualified to assist you. If you find something like this, take the opportunity to discuss it with the professor to get their opinion on whether your idea is fruitful. This may show your understanding of the paper, but just as importantly, it will give you feedback on a research idea.

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