@TedShifrin for my hyperbolic geometry lecture, I planned to do a bit of Kathryn Mann's "DIY hyperbolic geometry" (with a paper model and drawing geodesics), a little on projections (Poincare disk and halfplane), and geodesics. Then for the tutorial they will be working through some stuff about deriving a distance function for H and the Three Inversions Theorem for the hyperbolic plane.
Since I didn't have trig, I stayed away from the hyperbolic trig stuff.
I am excited for this lecture but it's the second one, and I haven't even done the first one! I am just planning them all in advance so that in case I get stuck with other duties in TA work and stuff I do not have to work on it for a whole week.
I did a freshman seminar 12 years ago on Escher and math. We discussed Escher's lectures the first half and I taught some elementary projective and hyperbolic geometry the second.
It's got some great ideas on it! I think it was a draft sheet for instructors, though, since it is kind of rough and some things are only sketched out.
Does anyone know what the difference (i.e. regarding the implications/meaning; I am aware how the different definitions look like) is between Christoffel symbols of the first kind and of the second kind? My metric tensor does not have an inverse... so the Christoffel symbols (of the 2nd kind) diverge.... I figured the Christoffel symbols of the first kind would work... can I just switch to the Christoffel symbols of the 1st kind and use them instead of the ones of the 2nd kind? it seems too easy...
@JoeShmo I mean yes i see that it is a differential operator, however it seems that we just expose dy/dx through some manipulation of dividing the dx , which i know is through chain rule but when can we do this manipulating
@Faust There is no question... but "measured things form a diffrent point of view" is the kind of explanation I am looking for.... or more precisely: I am wondering if the Christoffel symboled of the first kind define something equivalent to the Levi Civita connection
So, aside from my book, there's a lovely new book by Jeanne Clelland which does basic surface theory (and a bit more) using differential forms. It has great, great exercises, including some Lie group stuff.
@robjohn i wasn't super healthy to begin with had some stomach and medication problems before i got it. Though i ended up in the hospital cause i couldn't breath had a resting heart rate of 150 and that was after 6 days of the highest fever i ever had
@TedShifrin eh, i like basically all math except for pde's i never met one i liked.
@robjohn luckily a friend busted me out or i woulda been stuck there alot longer. They wouldn't let me out without someone at home so he had to drive like 8 hours to come get me.
Anyhow, @Faust, if you wanna review some of the multivariable analysis/differential forms stuff, see my videos and email me if you need part of the text.
@TedShifrin we are having alot of trouble keeping covid under control ourselves, opening the border would be a disaster for us its running rampant down there.
yeah i figured i wasted a month proving something for the graceful labeling problem cause i thought it would get me the rest of the way there. Finally proved it turned out someone published the result i wanted to prove in a paper already.
In my thesis (complex integral geometry), I proved a local integrated theorem (that holds for any piece of complex submanifold of $\Bbb P^n$). But constants appear, and to get the constants I had to do the algebraic geometry computations for the compact case.
@Faust: There are beautiful theorems in there (transversality and applications) that everyone should know. I also have exercise sets from teaching that course I can share with you.
@Karim: In papers with folks at UGA, we worked with the Fermat cubic surface to get explicit examples. And in one of my crappy papers, the Whitney umbrella got worked out in excruciating detail.
Anyhow, I think examples can be hard but very illuminating.
@Faust: Here's a concrete example for you. Tell what what surface you get when you intersect the unit sphere in $\Bbb R^4$ with the cone $x_1x_4=x_2x_3$. Then prove it.
I see where a lot of your intuition comes from I mean your videos + Guillemin's book + John Milnor + your book would be a great resources for differential topology.
@TedShifrin i mean i am not entirely afraid of differential topology but i can honestly say i have no idea what they are for or why we should learn about them probally the smae goes for manifold in higher dimensions
i mean i know what it is, but at the same time i have no idea what they are for.
well I'll probably put more emphasis on proofs for the lcft course because it's a lecture course so those kinds of things will come up in exams, but for seminars I tend to be a little more informal
but I know that the proofs for both of those things are "infamous" for being ridiculously difficult so idk
but like 4 of the books on the reading list don'T mention anything about cohomology (specifically étale and algebraic de Rham), which is good because i haven't the slightest idea what either of those are
Yes, that's the third time since the pandemic started, I'm travelling between countries (Germany and Italy) between which there are currently no restrictions
Since the situation keeps changing I might need to get tested on arrival when I return to Germany, but that's not clear
Cases are rising in both countries (as they are pretty much anywhere in Europe), but nobody knows how bad it'll get or what measures will need to be implemented to contain it
@Astyx it's unclear in Italy whether another hard lockdown will be needed, but I think they'll keep that as a last resource because of public outrage and whatnot
Germany never really went into a lockdown so I guess they'll only keep restricting opening hours of places and numbers of people that can meet in the more affected regions as they have been doing so far
