Learn the power of other mathematicians’ tools

I suppose it is tempting, if the only tool you have is a hammer, to treat everything as if it were a nail.

— Abraham Maslow, “Psychology of Science”

You will find, when listening to talks or reading papers, that there will be problems which interest you which were solved using an unfamiliar tool, but seem out of reach of your own personal “bag of tricks”. When this happens, you should try to see whether your own tools can in fact accomplish a similar task, but you should also try to work out what made the other tool so effective – for instance, to locate the simplest model case in which that tool does something non-trivial.

Once you have a good comparison of the strengths and weaknesses of the new tool in relation to the old, you will be prepared to recall it whenever a situation comes up in the future in which the tool would be useful; given enough practice, you will then be able to add that tool permanently to your repertoire. Thus it is worth investing some time in learning about other tools, even if they are outside your field.

One way to do this is to read survey articles in other fields aimed at a general audience (one good source for this are the quadrennial ICM proceedings).