Learn and relearn your field

Even fairly good students, when they have obtained the solution of the problem and written down neatly the argument, shut their books and look for something else. Doing so, they miss an important and instructive phase of the work. … A good teacher should understand and impress on his students the view that no problem whatever is completely exhausted.

One of the first and foremost duties of the teacher is not to give his students the impression that mathematical problems have little connection with each other, and no connection at all with anything else. We have a natural opportunity to investigate the connections of a problem when looking back at its solution.

George Pólya, “How to Solve It”

Learning never really stops in this business, even in your chosen specialty; for instance I am still learning surprising things about basic harmonic analysis, more than ten years after writing my thesis in the topic.

Just because you know a statement and proof of Fundamental Lemma X, you shouldn’t take that lemma for granted; instead, you should dig deeper until you really understand what the lemma is all about:

• Can you find alternate proofs?
• If you know two proofs of the lemma, do you know to what extent the proofs are equivalent? Do they generalise in different ways? What themes do the proofs have in common? What are the other relative strengths and weaknesses of the two proofs?
• Do you know why each of the hypotheses are necessary?
• What kind of generalizations are known/conjectured/heuristic?
• Are there weaker and simpler versions which can suffice for some applications?
• What are some model examples demonstrating that lemma in action?
• When is it a good idea to use the lemma, and when isn’t it?
• What kind of problems can it solve, and what kind of problems are beyond its ability to assist with?
• Are there analogues of that lemma in other areas of mathematics?
• Does the lemma fit into a wider paradigm or program?

It is particularly useful to lecture on your field, or write lecture notes or other expository material, even if it is just for your own personal use. You will eventually be able to internalise even very difficult results using efficient mental shorthand; this not only allows you to use these results effortlessly, and improve your own ability in the field, but also frees up mental space to learn even more material.

Another useful way to learn more about one’s field is to take a key paper in that field, and perform a citation search on that paper (i.e. search for other papers that cite the key paper). There are many tools for citation searches nowadays; for instance, MathSciNet offers this functionality, and even a general-purpose web search engine can often give useful “hits” that one might not have previously been aware of.

See also “ask yourself dumb questions”.