and is compact

Hmm, so I don't know any geometry (prepares to duck a Ted smack), though Mike has mentioned this before

Hello @KennethDang welcome to this chat.

Hi. Thanks.

@Daminark I know some geometry, namely, that the angles in a triangle sum to 180.

I feel like you would have to appeal to the classification of surfaces for a topological proof

My solution was using covering spaces, you know by Poincare duality + Hurewicz that $\Sigma_2$ has contractible universal cover, so if $\mathbb{Z}^2$ were a subgroup of its fundamental group you'd have a cover that's a $K(\mathbb{Z}^2,1)$ and which is a surface

But the torus doesn't cover it by Euler char

@JasperLoy I know a little bit. Azerbaijan and Angola have exclaves, whereas South Africa has an enclave.

This sounds like college professors...

@Daminark You can actually find that in section 3.2 of my notes.

I guess it is good enough to know that the torus is the only surface with that fundamental group, but I don't know off the top my head how easy that is to prove

@Kenneth: Mostly not. Only me. :P What brings you by?

I'm a professor.

I guess you might not need that it's the torus?

I was looking for some high school students/teachers who want to help/practice for the American Mathematics Competition this Feb.

You know its homology groups because you know it's a $K(\mathbb{Z}^2,1)$

And as a result you know the Euler characteristic

@KennethDang Ah, from what I know of this chat, hard to find them here.

Ah okay

Hmm, we have a room for people in India preparing for math exams, but I don't know about an AMC room, @Kenneth. @CaptainAmerica: You gonna take that test?

Yes, and hopefully make it to AIME this year.

@KennethDang Do you have access to past papers? Can you work on them yourself and then post questions if you have problems?

@TedShifrin I contemplated it, but I just don't think I'd do very well.

Think of it like this: math is fun and you get to be out and about.

It's still good experience for you, @CaptainAmerica.

I study them on the Art of Problem Solving. I have a couple of questions posted on mathematics exchange. Are you looking for one?

I personally lament not doing more of that kind of thing in high school, @CaptainAmerica, and not because I think I would have won or anything. I just like solving cool problems.

Yeah, Kenneth, this is more down the line of the AoPS kids. I'm actually teaching an AoPS course that isn't competition stuff.

Oh, nice.

@KennethDang No, I am not looking for the papers. I just mentioned because I thought that might help you.

@CaptainAmerica: You're going to want to compete in the Putnam when you're in college and it's infinitely harder, so get some experience.

I think you can only compete in Putnam four times.

@TedShifrin Feb is not very long to prepare.

That's correct, @Kenneth.

I suck at comp problems

Just do some problems and take the test, @CaptainAmerica.

and indeed at regular problems

When you compete in Putnam, just Put down your Name on the paper.

lol.

smacks @Jasper

I remember the 2 I got quite fondly.

Putnam was always too close to exam week for me. I never actually looked at the problems to see if they were the types that I'd enjoy or not

You would have found a number of them interesting, Demonark.

Quite a few people here (me included) seem to not be very fond of contest math

Yeah, it was always right in the middle of finals at UGA, but still students did it.

I don't know if I like the idea of math-as-competition, but I love the problems.

I'm really bad at competition problems

I'm not particularly fond of contest math, @Alessandro, but problem-solving is problem-solving and experience is experience. One gets better with experience.

I took Cal I, II, and III, and DE in my highschool career. But it was from a community college. Probably doesn't help that much with Putnam.

I can basically literally never do them, but, hey.

Unfortunate that you didn't save that stuff for a real math department at a real university, @Kenneth.

@KennethDang That's a nice school you have that allows you to do that.

Its a early college highschool.

*an

I don't think it's necessarily great, @Jasper. There are a lot of horrid "college" courses out there, not just horrid high school courses.

Well, currently, the A level exams they take in the UK do some calculus and some differential equations. Everybody takes some, to prepare for university math.

Perhaps, but these professors came from Rice University. I could easily do AP Calculus AB/BC problems, which is accepted at MIT. It was decent.

Is LATEX enabled here?

@TedShifrin I'll think about it, but I already have insomnia. I'll probably stress about this too.

@KennethDang see math.ucla.edu/~robjohn/math/mathjax.html

@CaptainAmerica16 You will learn that stress is a tensor.

Is that from physics? lol.

I think first year college calculus is basically the equivalent of BC so at many places it's available. High schools having multivariable calculus and ODEs (not just the week of how to divide by $dx$) strike me as more rare

I agree. But this was a HS on a college campus.

The BC course and exam are not in fact comparable to a good college Calc II course.

@KennethDang If you're at a computer in a browser, there's a link at the right of the page that will give you instructions.

And a 5 on the BC can be anywhere from a 60 to 100 on the exam. You can literally get a 5 without knowing the series stuff at all.

I plan on restudying calculus after I complete BC.

Okay, thanks.

@TedShifrin Oh. That might explain a lot.

@TedShifrin Thanks for the heads-up. I can take the rest of the year off then.

I had students who had excellent teachers, got 5's, and were very strong. I had others who were trained for the test, got 5's, and ended up not doing so well in Calc II.

You're supposed to be getting through more of Spivak than chapter 1, CaptainAmerica.

Yeah...

Oh, I should send you the worksheet I just gave my AoPS kids on the Fund Thm of Calc, @CaptainAmerica.

I just did that too

I'm going to be doing Spivak over the break.

@TedShifrin Then it might explain less. I think I had a singularly good teacher. >_>

Hmm, question: is going through the proof of classification of surfaces worthwhile or nah?

@TedShifrin Send the worksheet

please

Alright, I'm off. Bye everyone.

bye, Fargle

@TedShifrin, did you ever compete in AIME/USAMO?

See you Fargle

Bye.

@Daminark What we did in my topology course was to assume existence of triangulations and then show it from there. It's probably a huge pain to do the whole thing

@Kenneth: I remember doing AMC a few times. I don't remember if AIME was around in the 60s.

Bye Fargle

Did you do well? Like training and stuff?

Ah yeah I think Andy Putman has a writeup that's basically a page long once you know triangulations. I guess there's the question then of how bad it is to show that surfaces can be triangulated/are the ideas interesting enough to warrant the suffering

Demonark: You got his name right :P

Oh. I see.

I am more careful now that I know, I legitimately thought Putnam was correct when I first saw it. Dyslexia or smth idk

Andy hates it when you get it wrong

I encountered him through FB posts on a mutual math friend's posts, Demonark. I've never met him.

:'(

Compared to my class anyway

@Paul might've submitted my Notre Dame SoP when I thought that it was Putnam

I'll give Andy the right to butcher my last name at least once. To be fair asking people to get it right is cruel

I'm going to work on this over the break too :D

Danimook

did I get it right?

Close enough

@Ted let me show you the kind of stuff I get, give me a second.

Oh, I know what is in standard courses, CaptainAmerica.

I'm trying to give my best kids something halfway to Spivak (but we're not doing that much in the way of proofs).

I feel like I'm not learning everything as well as I could, and I never have much time to expand on things either.

@Daminark well hopefully you got it right or he exaggerated his dislike

regrets giving Demonark nightmares over misspelling

I committed that entire post to memory. Your secrets are never safe on the net.

I am working on answering a post on main, so I totally missed it.

It was a beautiful work of literature.

You'll regret this forever.

Oh well.

Any suggestions for finding the zeros of this expression for $A$? It looks very complicated.

Hi @Ted.

Hi, @MikeM.

I feel really sad right now. I don't know what a partition is.

There are many proofs of the classification @Daminark. You always need some hard input somewhere. The triangulation statement is essentially Schoenflies, but the proof is still not totally obvious given that.

I should have just googled it before I typed that.

Now I know.

I prefer assuming smooth structures exist and are unique and thinking about handle decompositions, since the combinatorics is easier (and more clean to me). Or I have a proof written up somewhere using homology that still requires the triangulation requirement but rather minimal combinatorics.

@CaptainAmerica16 I can't disagree.

Oh, sorry, @CaptainAmerica. I gave them the actual definition of upper and lower sums. A partition is just a subdivision of $[a,b]$ into finitely many subintervals.

I should have just looked at the context clues: "Partition of the interval..."

Oh, never mind.

Wait, I'll send you that page of notes.

cool, thanks

@CaptainAmerica16 you have stumbled upon wisdom that I think a lot of MSE askers could benefit from

XD

sorry, @CaptainAmerica: Sent a blank message by mistake. Correct one right after.

You should probably understand some geometric / combinatorial content to the classification, one way or another.

@TedShifrin Got it now.

I got to go Mr(s). Cya.

Bye Kenneth

@Mike gotcha, that makes sense

I'm going to work on some of these problems tonight. I love stuff like this.

Spivak guilt returns

@CaptainAmerica16 A partition just partitions a set into subsets. These subsets are nonempty, disjoint, and together make up the whole set. So you see the name is a very suitable one, because a partition partitions a set. Sorry, that is for another kind of partition!

@JasperLoy wrong partition, this was meant as a partition of an interval for use in integration

@Ted Can you give me the answer to the convexity problem?

As Fargle (or perhaps Demonark) suggested: Think of averaging the piecewise-linear function with the quadratic.

;-;

What does that face mean? He gave you the answer...

What do you think it means?

Ask Facebook what it means. Facebook = Book of Faces

Okay, let me rephrase. Stop whining. :)

Do you find me annoying or something?

I never joined Facebook. I am glad it didn't get my data.

I'm irritated that I didn't figure out the problem mainly.

@Jasper: A number of my friends who've been longtime users are leaving. I'm too slothful to do so, because I have so many connections there.

As a differential geometer ought

One can use the website messenger.com for the messaging functionality without the main page. They can't monetize you quite as effectively.

@TedShifrin Oh, I didn't know you have Facebook too. I thought you were too old for that. =)

True enough, Demonark. And sometimes connections with torsion.

LOL, @Jasper. Many years ago, my students formed a group called "Tortured by Teddy" and made me join FB so I could join the group.

Odd kids.

It was a rambunctious group, not the best Multivariable Math class I ever had, but a good one.

I think the latest generation of kids are not as into social media as the second latest generation.

I think I disagree, Jasper.

@Ted does that problem have anything to do with linear approximation?

Which problem?

convexity, the averaging thing made me think of that.

:P

Not that I can see.

Howdy, I am implementing a boundary value problem solver in C++ called the initial value adjusting method. Although I have followed the algorithm correctly, the solution does not converge. I'd appreciate if someone can help me. I can share the files with you. The paper is: sciencedirect.com/science/article/pii/0022247X79901495. Asking a question on the forum may not be a good idea as it involves considerable exchange of ideas

I honestly don't know if I'll do well in a math degree

Yes, that is very hard to say.

An interest doesn't seem like enough to carry you in academia.

Yes, and some people think that if they cannot be the best at what they do they might as well not do it at all.

I don't think I'll ever be anywhere near the best, but I don't want to absolutely suck either.

But you can do it as long as you think you are good enough to do some things.