They also talk near the end about when you let balls of specific radius roll along the entire sphere. Somehow that's related to the exceptional Lie group G2...
"You'd never guess it, but the really amazing stuff happens when you roll a ball on another ball that's exactly 3 times as big. In that case, the geometry of what's going on turns out to be related to special relativity in a weird universe with 3 time dimensions and 4 space dimensions! Even more amazingly, it's related to a strange number system called the split octonions."
@TedShifrin I kept answering prof question in class he told me he likes my peformance. He told me he will introduce local rings, which will allow us to kinda of consider nbhds. I thought that was really cool.
I am reading Eisenbudd and michael atiyah. Eisenbudd introduced the notion of local rings quite early and discussed the geometry behind stuff in algebra.
I was hoping that 'something' about the complex structure would be stable w/r/t changes in a,b but not to c.
But as far as I can tell there's nothing like that.
Plus, just computing the Igusa invariants of my curve y^2=f(x) is a pain. If c=0, the roots of f(x) are of the form 0,1,-1,a,b,-a-b. If $c\neq 0$, then the first three roots are c,1,-1 and the others are (complicated) functions of c.
But "I can easily compute the Igusa invariants when c=0 but not otherwise" is not a terribly helpful characterization.
About the only 'obvious' thing that $c\neq 0$ does is move the root at the origin.
And, well---so what?
About the only other thing that I could think of playing a difference when it comes to period integrals on said curve is whether the differential form I'm using would have a residue at infinity.
Alas, I checked: while it does have a residue at infinity, said residue doesn't depend on $c$.
So that's something that -doesn't- change with $c$ but does with $a,b$---precisely the reverse of what I want. aggh
@TedShifrin Ah, I see. I multiply by the conjugate, to get that $\sqrt{\dotsb}-\sqrt{\dotsb}=2cx/a$. I then add that to the original thing and divide by $2$ to get $\sqrt{(x+c)^2+y^2}=cx/a+a$. And then I manipulate that.
Aside from teaching it myself a few times, I've mentored grad students teaching this numerous times and have criticized/monitored their exams, so I have a bit of experience.
On the other hand, one of the reasons I retired is that I had always prided myself on motivating students to work their butts off and pass, and even in upper-division classes, I was no longer succeeding with some of 'em.
I suppose I failed back 30 years ago, too, but it seems more decisive now.
Modern assembly line precalc and calc tests competence, speed and accuracy in middle school arithmetic and high school algebra and perhaps a slight effort on skimming through calculus material.
And that is not a sexist remark. A lot of math professors need to be taught basics of teaching and exam-writing. I feel very sorry for the students. They should complain to the Head for sure.
