Secret tips for successful research

[MCKAY john]

Take an interest in all areas of mathematics - pure and applied.

The wider the explanatory power, the simpler the idea.

Minor knowledge in one field may have major impact in another.

Exceptional situations are fertile research areas.

Ask: When this work is complete, what wider impact will it have?

What is the general context for this work?

Be familiar (not necessarily when you start) with the literature.

Know how to find reference material.

Put semantics into your notation.

Recognize and study patterns.

And personal favorites:

You cannot know too much about:
0. The Platonic solids

1. Reals -> Complex -> Quaternions -> Octonions
(The purpose of the octonions remains unrevealed!)

2. E8

3. The Mathieu groups

--
But leave the wise to wrangle, and with me
the quarrel of the universe let be;
and, in some corner of the hubbub couched,
make game of that which makes as much of thee.