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stoutfellowOne of the things about actually starting to write a paper is the process of realizing how unclear your mental work has been. I said a while back that the Varignon transformation acts in half a dozen different ways, and it's predictable which way it will act in any given situation. That turns out not to be quite true; or, rather, the ways it acts are a bit less well-defined than I thought.
My first training in advanced math was in abstract algebra, and that's still the area where I feel most comfortable. One of my advisors, though, was of the opinion that abstract algebra was - let's say "navel-gazing" rather than the metaphor he actually used, unless it was brought to bear on something more concrete. In my work, that means geometry. I find it frustrating when the algebra suggests that I should focus on some particular aspect but I can't see any geometric justification for doing so.
What happened today is this: I found that if I do focus on a particular algebraically-appealing aspect, the sharpness I saw earlier reappears. This is the second time something like that has happened in my current research, and it kind of restores my faith in algebra. (Euler once said that he sometimes felt that his pencil was smarter than he was - that purely symbolic manipulations kept coming out right, with beautiful results. On a much lower level, this has a similar effect on me; but I find it a source of joy rather than dismay.)
My first training in advanced math was in abstract algebra, and that's still the area where I feel most comfortable. One of my advisors, though, was of the opinion that abstract algebra was - let's say "navel-gazing" rather than the metaphor he actually used, unless it was brought to bear on something more concrete. In my work, that means geometry. I find it frustrating when the algebra suggests that I should focus on some particular aspect but I can't see any geometric justification for doing so.
What happened today is this: I found that if I do focus on a particular algebraically-appealing aspect, the sharpness I saw earlier reappears. This is the second time something like that has happened in my current research, and it kind of restores my faith in algebra. (Euler once said that he sometimes felt that his pencil was smarter than he was - that purely symbolic manipulations kept coming out right, with beautiful results. On a much lower level, this has a similar effect on me; but I find it a source of joy rather than dismay.)
