Ideas in Mathematics The idea of continuity. We have a beautiful formula
\[\lvert B^A \rvert = \lvert B \rvert ^ {\lvert A \rvert}.\]But what happens when at least one of $A$ and $B$ are empty? Then we set some conventions.
The most important theorems in mathematics are those that are EASY to prove.
Let other people do the unnatural things, and you do the natural thing, then life is much easier. Then you need to be fast.
At one point, Yau had many students. So he organised smeinars twice a week to let students teach each other. They worked very hard. Occasionally, Bott, who was an electrical engineer, would drop in and listen. He said ‘Don’t work too hard! Just do the natural thing.’
In fact, Chern S. S. and Yang C.N. were also like that. That’s why they live so long, they don’t work as hard. Of course, this doesn’t work in today’s China. You need to work 996.
Just like left actions and right cosets, there is duality in life. Simply simple things are very hard, while simply hard things are extremely simple.
Ideas in Mathematics Completion is one of the great ideas.
S. S. Chern once said: How to do research? Just pick up some papers (by great mathematician on good, interesting results), then improve the theorem, live dropping a condition and do generalisations.
Chern, Yang, are all not very original, but they are still great mathematician/scientist.
Ideas in Mathematics Two implies finitely many.
There was Jones polynomial which was something original for knots and links. After the paper was published for a month, four independent groups submitted papers to the same journal, coming up with the (I don’t really know what he said here, so let’s just say) Bob’s polynomial. The editor forced them to work together and submit one paper.
The magic of maths is that from something very simple we build something beautiful. You don’t want to just do difficult things, like hard hard hard then we get not interesting stuff.