... The approach of von Neumann and Connes to the use of non-commutative algebra in physics is naive, the situation is much more complicated.

I used to say: Everything is Representation Theory. Now I say: Nothing is Representation Theory.

Eugene Wigner wrote a famous essay on the unreasonable effectiveness of mathematics in natural sciences. He meant physics, of course. There is only one thing which is more unreasonable than the unreasonable effectiveness of mathematics in physics, and this is the unreasonable ineffectiveness of mathematics in biology.

They have the idea that non-commutative algebra should remind one of commutative algebra, but the former is more sophisticated. I believe that non-commutative algebra is just as simple, but it is different.

In the Middle Age, in Germany, if you wanted to learn addition and multiplication, you could go to any university. But if you wanted to learn division, you could only do it in one place, Heidelberg. This makes sense, since in my theory with Vladimir Retakh and Robert Wilson, addition and multiplications are cheap, but division is expensive.

One of the reasons we don't do as well as we should is that we are all over-taught.

The older I get, the more I believe that at the bottom of most deep mathematical problems there is a combinatorial problem.

People (or students) do not have shortcomings, only uniquenesses. The goal of a good teacher is to turn these uniquenesses into advantages.