Try to learn something about everything and everything about something. (Thomas Huxley)

Maths phobia is a pervasive problem in the wider community. Unfortunately, it sometimes also exists among professional mathematicians (together with its distant cousin, maths snobbery).

If it turns out that in order to make progress on your problem, you have to learn some external piece of mathematics, this is a good thing – your own mathematical range will increase, you will have acquired some new tools, and your work will become more interesting, both to people in your field and also to people in the external field.

If an area of mathematics has a lot of activity in it, it is usually worth learning why it is so interesting, what kind of problems people try to work on there, and what are the “cool” or surprising insights, phenomena, results that that field has generated. (See also my discussion on what good mathematics is.) That way if you encounter a similar problem, obstruction, or phenomenon in your own work, you know where to turn for the resolution.

One good way to learn things outside your field is by attending talks and conferences outside your field.

See also “Learn and relearn your field“.