I got a grant to do independent research on the AI alignment problem and was able to quit my job.

Yeah Thinkpads are obviously a popular suggestion here. I'd check out System76 too (their cheapest stuff may or may not fit your definition of "cheap").

I know exactly one party trick based on mathematical group theory, which I have actually used to impress non-mathematicians at a party.

There’s a concept called the “center” of a “group”, which is the set of operations that commute with every other operation in the group. The center always contains the identity operation of doing nothing. The group of scramblings of a Rubik’s cube happens to contain exactly two elements in its center: the identity, and a move called the “superflip” which takes a little bit of effort to memorize how to do, but it’s not so hard. Much easier than actually solving a scrambled Rubik’s cube. It’s like you do a simple move repeated 4x, and then you do that whole 4x set 3x with some rotations in between. Not terribly complicated. Importantly, once you memorize it it’s not difficult to do just by feel, since it’s a fixed sequence of mechanical motions.

So, the party trick goes like this:

You have a Rubik’s cube that is exactly a superflip away from the solved state. You hand it to an unsuspecting party guest and say “go ahead and make one or two turns” (it’s important to say something like “one or two” because if they do 3 the trick becomes challenging, and if they do 4 or more it might become impossibly difficult unless you’re actually good at solving Rubik’s cubes, which I am not). They take this obviously unsolved cube and make a couple more moves so now it appears even more scrambled.

You take the cube back and do the superflip behind your back, without looking at the cube.

Then you move the cube out from behind your back, and at the same time (trying to be slick about it) you undo the one or two moves remaining before it is solved. Everyone gasps and say “omg he solved it behind his back” (when really you did no such thing).

This works because if S is the superflip and X is the simple moves they did to it, S X S is equal to just X because S commutes with everything. (S is also its own inverse, so that S S = 1.)

This may be true but I hate the practice of referring to “plastic” as if it’s a single substance. It’s a bunch of different materials that don’t really have that much in common with each other, especially from a health/toxicity standpoint.

For example, people treat it as common sense that “you shouldn’t burn plastic” because the smoke is “toxic”. For PVC this is totally true, it makes very nasty stuff like dioxin that will poison you. But on the other hand you can burn polyethylene (think milk jug) and it’s no more toxic than burning a candle. Definitely way healthier to breath than wood campfire smoke, for example.

There’s also such a silly pattern where people learn some chemical might have some effect on the body and suddenly everyone is up in arms about it. For example Bisphenol A in many applications was replaced by the very similar Bisphenol S just so things could be labeled “BPA Free”. BPS probably has similar estrogenic effects to BPA.

I’d say the moral of the story is be wary of received wisdom about chemical toxicity from people who aren’t chemists.

Pro tip: if you call zoning out “meditating” (because that’s what it is) it makes you feel more put-together and successful. Congrats on your practice of hydrotherapy meditation.