@AkivaWeinberger Say, I found an approximation to $G(13)$, the amount of steps it takes for $13$ to reach $0$ in the Goodstein sequence. $$G(13)\approx g_{2\uparrow\uparrow\left(2\uparrow \uparrow\left[2\uparrow\uparrow\left(2^{ 2^{2^{70}+70}+2^{70}+70}\right)\right]\right]}$$ Where $$2\uparrow\uparrow n=\underbrace{2^{2^{2^{\dots}}}}_n$$ And $g_n$ is the $n$th term of Graham's sequence with $g_{64}$ being Graham's number.
See [here](https://math.stackexchange.com/a/2345650/272831) for an explanation.
I'm sure better bounds are possible. I butchered the $2\uparrow\uparrow$ part though
Once you start writing out the first few terms, you should be able to get a feel of what's occurring and then apply lots of induction lol
Basically, in base $n$, you'll want to try and see what $+k<n$ does (It just adds $k$ to the base) and then you should see that $+nk$ does something close to doubling, etc.
The end result is $n^k$ is akin to $k$ Knuth's up-arrows, which is how Graham's sequence sneaks in.
And I think $G(14)\approx g_{2\uparrow^46}$. Anyways, goodnight.
@BalarkaSen Hey, remember a while ago we were talking about the preimages of $S^n$, $n\le2$ in the Hopf map? I wonder what do we get if we do the same construction for the other two Hopf maps ($S^7\to S^4$ and $S^{15}\to S^8$ if I recall correctly)
I mean, obviously higher dimensional nonsense that can't be grasped by human minds.
But I wonder if maybe there's like an actual simple description.
Hm, we could do this for the nonzero element of $\pi_4(S^2)$ (which is $\Bbb Z_2$ I think) for something more manageable. Since it's made out of the usual Hopf map, I seem to recall, it should "look" similar to the picture from the usual Hopf thing.
It was like, you suspend the Hopf map to get something from $S^4\to S^3$ and then compose it with the Hopf map ($S^3\to S^2$)
That was a bit of a longer comment that went into the details, basically it boils down to the participant background and whether people want to go more on the topology or geometry trains
I didn't mention this there, but if you want both sides, you probably don't want to spend too much time on any given topic, so Lee would really be way too long
The thing is a lot of the people who joined the study group (based on comments) had already started Lee (including myself :p), so I was thinking of doing maybe a slimmed down version of the sort of topics that are covered in Lee
I'm hoping some more people will reply to the thread and voice their opinions though
My idea basically was to get a group together with a bunch of books and work through the material together, and people could choose from a selection of recommended texts as they saw fit
@Dami, I think Tu will definitely be one of the texts we use
Nah we thought we'd be assigned them, I posted on our FB group so let's see what happens. I may volunteer to give the dynamics lecture on Friday if I feel good about it closer to then
@EricSilva Apart from here on MSE, /r/math is the only other place I know on the interwebs that has an active math community, that would be interested in a study group
@Perturbative I think what Eric was going for was that it was a good place to gather people together, yes, but that you should find another platform once you have gathered people so as to actually do the stuff. The advantage seems to be that if someone does miss a bit, Reddit has a nicer arrangement for getting back on top of things, but it is somehow slower than a chat
I think Chris will likely be first to go, he at least read and TeX'd notes on the first chapter of Brin/Stuck so he's prepared that much at least, though nothing is decided yet
Also, I guess, what were the common pitfalls of the first lectures?
there was a moment last year where someone was trying to justify an inequality and schlag was like "ah yes, it's obvious!" proceeds to describe a sequence of very complicated computations for the next 40 minutes @Daminark, it was very illustrative of what he wanted to do
@Eric So we're having that first lecture on complex analysis, is that supposed to be a preview or are we gonna start from after when Titchmarsh covers that material?
i think she said it would be two things, one would be she would go over some of the stuff you may know that you definitely should know before starting Titchmarsh and the second would be a run through of why complex is cool (no proofs obviously)
He said it'd start from scratch, that we'd go through the defn of a holomorphic function, integrals around a triangle being 0, Cauchy's theorem, power/Laurent series, and residue theorem, but that it goes very quickly through that stuff so it'd be helpful if this isn't the first time you hear those words
Or I mean, I think I know the proof of that as well, like knowing that two curves are homotopic iff they have the same winding number, choose a curve of winding number 1, then integrating $\frac{1}{z-a}$ and dividing by $2\pi i$ gets you 1, so multiply that by $f(a)$ and tada
But yeah, the intuition is that this doesn't change if you scale $S^1$, so if you take a limit as $z\to a$ this should be integrating its derivative somehow? So that should get you $f(a)$?
But yeah I mean, explicitly I guess it's like, you let $z = a+re^{it}$ so that this is $|\frac{1}{2\pi i}\int_0^{2\pi} \frac{f(z)-f(a)}{re^{it}}(rie^{it})dt|$ (good lawd...)
(Note: I am someone who finds the weirdness of quantum mechanics interpretation interesting, including at the philosophical level. What I cannot stand, however, is consciousness crap.)
Lol, that kinda philosophical stuff and the general dankness of QM, plus the fact that the math in there seemed cool, did a lot to make me excited by physics
(Plus, before coming here I actually thought it was something that I was reasonably good at, not just that I did well because I was in a school where people mostly didn't care)
AP Chemistry: Students with a score of 5 may accept credit for CHEM 11100-11200-11300 Comprehensive General Chemistry I-II-III, or they can register for CHEM 12100-12200-12300 Honors General Chemistry I-II-III in Autumn/Winter/ Spring Quarters. Students who complete one to three quarters of Comprehensive General Chemistry or Honors General Chemistry forgo AP credit for all quarters completed at the University of Chicago.
Like, started with countability, suprema, Dedekind cuts, rational approximation, then a week on metric spaces, a couple days on continuity and limsup/liminf
The next 2 weeks were topology again (BCT, AC, BV, uniform convergence), then one day he was like "Yeah learn Riemann-Stieltjes kthx", the next he was like "Okay now do it on R^n khtx", then did change of variables, then curves stuff for the last 2 weeks
@Semi i think at the local uni i grew up near it was like partials, lagrange multipliers, line and surface integrals, green's, div, stokes' theorem and div grad curl stuff
when i was a freshman i got bored and taught myself calc and then did spivak, and then sophomore year i taught myself rudin, and then went over to take classes at the local uni and skipped all those so i never got see what it was actually like
yeah i would throw that in under surface integrals basicaly
I'm pretty sure this is of a different sort than what you were talking about, but nowadays, if you talk to someone and say you're a math major, do they either a) tell you about how much they hated it, or b) ask you to multiply some crazy numbers in your head?
but yeah anyway there were a few teachers who basically went out of their way to make my life hell even when i was doing my work and following the rules
they actually give us an exam at the beginning of each class and if we either do to well or they believe we are throwing the exam, you get in big trouble.
i mean, idk, sometimes it was justifiable, bc i have kind of a streak of not coping well with authority, but sometimes it just seemed like it was spite for not having to pay attention to do well in their classes or something? i had at least one teacher who definitely was like this
@Typhon that's just a stupid policy. It doesn't make the teacher's life harder if a student or two know what's happening as long as the student isn't making a big deal about it
