Learn techniques from papers as a researcher

Question

I'm an incoming PhD student. I did some research during my undergrad and master. One of the things my advisors told me was to read papers but not too focused on "technical details". This did help me navigate literature faster by just skimming for the "main ideas" in the paper. But is this the right attitude when the purpose of reading papers is to learn techniques (e.g. how to prove certain bounds, how to analyze certain matrix operations)? I felt that understanding an overall "strategy" or "essential ideas" has not been very effective in advancing my own research. That is, I'm still not very good at doing those type of complicated analysis and my research effort remains "elementary". It's like taking a math class without doing any homework: I feel that I know what is what but tend to struggle when actually sitting for an exam and solving problems.

What is the most effective way to cultivate technical abilities as a researcher, without the aid of textbooks and problem sets?

Answer 1

As someone who struggles with this as well (and I've finished my phd in 2018 :O), I think I have some helpful things to say.

Think about some theorem you've proved all by yourself, in all the technicalities. It may be from your master's or some hard exercise. Now, think about the summarized version of a proof: something like you could describe to another mathematician during a phone call. What details of the proof would you include? Which ones would you discard? I believe this phone-call version of a proof is what you're aiming for when you try to get the overall strategy for a proof.

As an example: the main result of my phd thesis consists of, basically, complexifying a real-analytic foliated hypersurface (which is written in a bad way, too many terms in a sum), then blowing up the singularity and computing the holonomy of the leaves. I then showed that these holonomy groups are all linearizable, which allows me to use previous results to write my hypersurface in an adequate normal form, discarding higher order terms in its first scripture. As a context, this generalises to a infinitude of cases what was previously done to a few singularities classified by Arnold.

This is something another mathematician from my field (singularities, foliations...) would have a general picture. All the keywords are there.

The difference is, you now have to extract this from a proof in a paper without necessarily breaking it down to its smaller parts (since it's not something you did previously!). This is something that comes with practice and your own guts ("how deep do I have to get to understand this?"): read, read, read. Usually papers offer a brief explanation on the general technique or context of its main results: aiming to see that in the proofs may be a good way to do so.

Lastly, deciding which things to really break down or not is something that also comes from practice. This is the part I have the most difficulty. I usually try to do the first one and, if too many things seem interesting or useful, I try to perform a finer reading of the text.

Also, something that usually worries me when I'm reading stuff is: "shouldn't I have studied and mastered all these topics during my phd? Why can't I follow this proof like I can do in a textbook?". Too many topics won't be covered during your phd and get ready to face advanced new stuff out of the blue ("I wanted to study differential equations, what the **** is a scheme??", I heard someone say... :P )

Answer 2

Ravi Vakil has made some interesting comments about learning math as a PhD student.

A subtle leap is required from undergraduate thinking to active research (even if you have done undergraduate research). Think explicitly about the process, and talk about it (with me, and with others). For example, in an undergraduate class any Ph.D. student at Stanford will have tried to learn absolutely all the material flawlessly. But in order to know everything needed to tackle an important problem on the frontier of human knowledge, one would have to spend years reading many books and articles. So you'll have to learn differently. But how?

...

See if you can get one lesson from the talk (broadly interpreted). If you manage to get one lesson from each talk you go to, you'll learn a huge amount over time, although you'll only realize this after quite a while. (If you are unable to learn even one thing about mathematics from a talk, think about what the speaker could have done differently so that you could have learned something. You can learn a lot about giving good talks by thinking about what makes bad talks bad.) Try to ask one question at as many seminars as possible, either during the talk, or privately afterwards. The act of trying to formulating an interesting question (for you, not the speaker!) is a worthwhile exercise, and can focus the mind. Here's a phenomenon I was surprised to find: you'll go to talks, and hear various words, whose definitions you're not so sure about. At some point you'll be able to make a sentence using those words; you won't know what the words mean, but you'll know the sentence is correct. You'll also be able to ask a question using those words. You still won't know what the words mean, but you'll know the question is interesting, and you'll want to know the answer. Then later on, you'll learn what the words mean more precisely, and your sense of how they fit together will make that learning much easier. The reason for this phenomenon is that mathematics is so rich and infinite that it is impossible to learn it systematically, and if you wait to master one topic before moving on to the next, you'll never get anywhere. Instead, you'll have tendrils of knowledge extending far from your comfort zone. Then you can later backfill from these tendrils, and extend your comfort zone; this is much easier to do than learning "forwards". (Caution: this backfilling is necessary. There can be a temptation to learn lots of fancy words and to use them in fancy sentences without being able to say precisely what you mean. You should feel free to do that, but you should always feel a pang of guilt when you do.)