I was working on a research problem in between watching my room full of students struggle with histograms and standard deviations, and then I suddenly saw it.
The particulars of the “it” in question are probably of no interest to most of my readers, and little enough to the rest. Hell, in the “of great mathematical significance” sense, it’s not even very interesting to me. It’s not a solution that’s going to save the world or make me rich or anything.
But the reason that we do mathematics — at least, the reason that I do, at least sometimes — is to catch these little moments when the clouds clear away and one can apprehend a Pattern in its purest form. When things, against all expectations, make sense.
The hard part, now, is to take the vision of How It Works and turn it into words and other symbols. I’m firmly in the camp that has mathematicians as explorers rather than inventors: the things that we discover are already, in some sense, there. I can see the pattern for this particular family of structures, and its construction makes it clear (to me) why it’s correct. Justifying it is going to be a bit of a pain.