So here’s the thing: I haven’t really *done* anything today. What I mean is that I haven’t constructed anything new (a page, a list of definitions, a solution) that didn’t exist yesterday, and so – for all intents and purposes – I haven’t *done anything*.

But somehow, I haven’t *done nothing* either.

Some days, I make a plan to do something (“*do*” something), and I set out on that path. Sometimes, the path I reach has a bunch of hurdles that I’m not prepared to conquer, and so I set out on a side journey to obtain the skills necessary to progress down my original path. Sometimes – on days that are particularly unkind – the side paths have hurdles requiring sidepaths and the side-side-paths have hurdles requiring side-side-paths and so the whole journey gets twisted into some amalgamated blob of *non-progress* that somehow still manages to accomplish something.

That, ladies and gentlemen, was a metaphor. It’s a metaphor that fits my day rather well.

So as I mentioned earlier, algebraic topology was a bust. I decided, then, to finally take the plunge and to read something on -modules via Google. My professor had suggested this as a nice algebraic way to derive lots of the differential geometry results by way of learning really difficult algebra stuff like categories and stacks and sheaves and schemes and what not. That, of course, got me interested. I did a little digging and found an online resource from Harvard and decided to take a stab. I made it through about a page before I realized I was missing stuff on stuff.

I freshened up on stuff about Lie groups and took a gander at what Wikipedia had to say about Universal Enveloping Algebras. Of course, Universal Enveloping Algebras required me to know things about Tensor Algebras, and when I decided to look up something more foundational like “rings of differential operators”, I decided that I should probably concurrently try to parse through some literature regarding Differential Algebra as well. That chase has brought me to where I am now and has sustained me for the better part of three hours.

In that three hours, I’ve found lots of good resources (including an online .pdf of Ritt’s text *Differential Algebra*) and have done quite a bit of reading, but if I were to die today and pass the totality of today’s efforts off to someone else, their inheritance would consist of precisely zero tangible work.

So yea: Not doing anything while not doing nothing is a thing and it’s called “research mathematics”. Such is life, I suppose.

I think I’m going to end today’s part of my quest on the differential algebra / -modules front here: I’ve got some stuff to do and what not, yadda yadda yadda, etc. etc. I plan on working some more on Kobayashi and Nomizu before bed, though.

All I do is math math math no matter what….

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