Advice on how to read a mathematical paper

Question

I am a 2nd year graduate student in Number Theory. My advisor has sent me papers to read and I have trouble getting through even a paper. Be warned that my problems may be noobish and borderline childish.

Problems: (how do I get through these problems?)

1) every sentence or two, I will encounter a definition I do not understand. Learning that definition will lead to learning more definitions and so on. Do I have to learn all the definitions?

2) there will be techniques, or theorems that the author will quote from other papers. Does it mean I have to look up the paper? Do I have to read through that paper too?

3) my advisor said I should try and get the “big picture”. I can see that the results of the paper are amazing and interesting. For example, the paper wants to find the asymptotics for number of $D_4$ fields ordered by conductor. I can’t understand what is so special about $D_4$ and why choose conductor when discriminant is the usual invariant to go with. If I understood the paper in depth, I think I will know why. But getting a big picture without understanding seems like an impossible task for me.

4) how much time in general should I spend on a paper I want to understand? After 2 hours of definition chasing, I would simply get frustrated and call it quits and be discouraged. Is this normal or do I just have bad work ethic and should perservere more?

I would appreciate any advice from someone who has been doing research for years.

Thanks!

Answer 1

Of course the answers to these questions are highly personal. People tend to read papers in very different ways one from each other. I would say that probably the most important thing to understand is the structure of the arguments, and why they work. This may be a difficult task, because certain authors don't bother too much in making this clear to the reader, and they rather just pile up technicalities on the top of each other without explaining the reason why they're doing so. In a well-written and informative introduction, you should be able to loosely follow the logic behind the proofs, and you should get a very general idea of what is going on in the paper. This is of course not enough if you want to understand everything carefully. Personally, I tend to read the same sections several times, maybe on different days. Thinking about them in the meantime helps "digesting" the math, and usually at some point things become clear.

With regards to your specific questions:

1) Essentially, yes. Of course it is also your supervisor's responsibility to give you a paper (at the beginning of your PhD) that you don't need billions of new definitions to understand. On your side, you should definitely have a solid grasp of the basic concepts. Then it's part of your work to bridge the gap. With time you'll learn that certain definitions are crucial, and other mostly cosmetics.

2) Usually no. Again, if a paper is well-written then the arguments explained in it, plus the statements of the quoted theorems from other papers, should be enough to understand everything. Certain people hate to use other papers as "black boxes" though, and go through, even if quickly, the other quoted papers.

3) This is a delicate point. I think what your supervisor means is that there are certain big motivating questions behind math, and so also behind number theory. One of them is certainly to understand number fields as deeply as possible, and knowing how many number fields are there with a given Galois group is definitely at least a good start. That doesn't necessarily mean that $D_4$ is a particularly interesting case. But very often in math in order to tackle a big, open problem one cuts it down in very little and doable pieces hoping to recover then the big picture. Certainly ordering number fields by discriminant is a natural thing to do, but in the $D_4$ case one also have Artin representations attached to them, so also ordering by conductor is a reasonable thing to do. Nobody says that one way is better than the other, they are just two different instances of the same problem. There is no systematic way of "understanding the big picture". My very personal recommendation is to try to read different things, and go to many seminars, even if they seem a bit far from your research topic. Listening to the same thing under different points of view sometimes yields a better understanding of why mathematicians work on certain problems rather than on other ones.

4) I think this question has no answer. Just spend as much time you think it is reasonable to do!

Answer 2

First, let me reassure you that your problem is a very common one. Reading a mathematics paper, particularly if you are just starting out in the area, can feel like chasing an endless stream of new definitions and references.

As has been suggested in the comments, you could speak to your supervisor about this. Perhaps also your university offers courses in this kind of thing? I know that the university where I did my PhD ran courses on these types of skills such as reading and writing academic papers.

As has also been suggested in the comments, there is no one size fits all approach. However, seeing as you have asked, I will tell you what I do, and maybe some of it will be useful to you. I hope it is.

Summary of what I am about to say: Read through the paper several times each time in increasing detail.

I read the paper with a pen in my hand, ready to be able to annotate the paper. I used to use a physical print out of the paper with a pen and highlighter. Now I use an electronic version on a tablet and annotate it on there. Either way I think it is very helpful to be able to make marks on the paper.

First I try to read through the paper to get the “big picture”. Often you can gain a good idea of the big picture just from the abstract and introduction. On a separate page you can try to write down the main ideas/contributions of the paper. During this stage I try not to get bogged down in the details, this would just slow me down and sap my enthusiasm. At this stage I might not even look over all the proofs. It is very important for me at this stage to not fall down the rabbit hole! Usually if the author claims a result follows by another paper, or that a claim is immediate, I just accept it.

Once I have an outline of the paper then I make a decision about whether I want to invest more time in understanding this paper (If your advisor has told you to read the paper you don’t really have this luxury!). Usually there will be a technique in the paper I want to learn/imitate for my own research. In this case I move on to the next stage.

Now I read the paper again, but I go into more detail. I try to follow along with the proofs, and if there are steps that confuse me I write notes for myself on the paper. If something is really confusing for me I might end up writing the whole proof out on a separate page and fill in all the details. I repeat this process usually many times until I am satisfied I have got what I can from the paper, each time I repeat the process I allow myself to spend more time worrying about the details and “going down the rabbit hole”. I am surprised by how effective I have found “iterating” over a paper like this to be. Sometimes reading on in the paper will clear something up that puzzled me earlier, and then I cam glad I allowed myself to move on, rather than spend a long time on a single point.

This last stage of deciding how much detail to go into relies to an extent on your own discretion. If there is some lemma in the paper which you do not understand all the details of, maybe it is not so important for your purposes. On the other hand if this lemma is a crucial part of the proof of a theorem which you wish to generalise, then maybe it is a good idea to really understand what is going on here!

Some stray remarks:

• For a recent paper I was reading, I drew an actual chart/map of how all the Lemmas and Theorems fitted together.

• If there are definitions/objects in the paper I do not understand, I try to learn (or at least write on a separate page) the definition and find some examples of it first. To keep track of definitions and other material I am learning I use electronic flash cards (Anki). However I do not know anyone else in person who uses them for this.

• Also, you might be able to find a review of the paper you are reading on MathSciNet. This might give you a different overview of the paper.

• This will all get easier with practise.

Further references on this

A list of resources from UIC

An article on reading maths articles... very meta...