Research Interest: Chaos, Dynamical System, PDE.
- Shallowly Broad: Quickly absorb as many branch as possible [master more 'toolbox']
- Narrowed Depth: Carefully focus on narrow interest as much deeply as possible.
It's said one should master many tools on hand to solve complicated problems, but will it decrease the depth/mastery level ?
- Purely Abstract: Go for abstractions theoretically as highly as possible[math developed from itself]
- Applied Intuitive: Go back and forth to the nature, to try to get new idea[math developed from nature] ?
We all saw in recent decades/centuries the rigorous/abstraction has developed a lot, it's reasonable. But some giant mathematicians also gave the warning for that
e.g. Felix Klein[He means maths, for its beauty, placed in billboard is admired by 'connoisseur', but it's originally sharp weapons to fight against the heavy enemy, but people gradually forget this original use.]; Arnold[On teaching mathematics]; Kolmogorov[He more think the maths as a whole organic entity, should not be mastered seperately]
In all, it seems to be a dilemma, should we say the better way is to do mathematics both broad and deep, both purely abstract and applied intuitive ??