Terence TaoOne of the quirks of professional mathematics is that researchers are discouraged from speculating too far beyond the range of what they can actually prove. To quote Minhyong Kim (from https://mathoverflow.net/a/38694/766): "it's almost as though definite mathematical results are money in the bank. After you've built up some savings, you can afford to spend a bit by philosophizing. But then, you can't let the balance get too low because people will start looking at you in funny, suspicious ways."
It can take some conscious effort for junior mathematicians, once they actually have earned enough "theorem credits" to afford to speculate, to actually venture opinions and make broader conclusions. Easier to play it safe and only stick to what their theorems and results can objectively verify. While this does cut down on a lot of nonsense, I sometimes wonder if we should have more spaces to encourage mathematical speculation.
We do have the mechanisms of posing open problems and formulating conjectures, which work well enough, but we don't really have a culture of throwing out less precise, but still informed, speculation on where a subject might be headed. (Except perhaps during conference teas and dinners. Which is one good reason to attend such conferences... .)
fresh@tao Yes, but as soon as someone has left the circus, the ideas are gone, as well. I have a representation for Lie algebras that is trivial for semisimple ones, and non-trivial for solvable ones. It is not really complicated
A(L):={f | [f(X),Y]+[X,f(Y)]=0} and X.f:=[ad X,f].
This is again a Lie algebra and therefore opens the doors to really many cute toys. That little thing looks like a nice key to treat the solvable world better than it is now, but I cannot find the damn door lock.
A(L):={f | [f(X),Y]+[X,f(Y)]=0} and X.f:=[ad X,f].
This is again a Lie algebra and therefore opens the doors to really many cute toys. That little thing looks like a nice key to treat the solvable world better than it is now, but I cannot find the damn door lock.