Slow down, @Paradox.

@Daminark I think Schlag assigns a lot of problems in summer but he doesn't expect you to do all of them. I didn't do most of the probability theory cause all i wanted to think about was geometry really

Though I'll say I at least somewhat appreciate Soug's problem spam. It was stressful at the time when we were burning 35 hours each week on the pset and juggling it with 3 other classes

But I think it helped with absorption somewhat

Wait did you do books simultaneously?

The original number (before we take square root, and ignoring the 4) is $2(-1-i\sqrt3)$. So this has magnitude $4$. If you draw pictures, you can see that this is an angle of $-2\pi/3$.

@Paradox. We agree on this?

I honestly think that they should take it easy on the problems for the first quarter. Chicago already has a hostile environment for studying mathematics, assigning like 60 problems to freshman and sophomores just getting acquainted to higher math is kind of cruel.

two at a time Daminark

@Eric: I mentioned to Demonark before you showed up in chat that I knew of a few students who got chased out of math at UC quite quickly.

@TedShifrin Yeah but I want to speak in terms of the isometry group to make my proofs easy :P

@TedShifrin Sorry, and yes I get this part

as a urm in math I have heard a lot of such things and it makes me so sad @Ted. I really wish there were a serious change in the department culture.

Yeah, also somehow I see Soug as being better suited to teaching the second quarter. Like, when we were doing chapter 3 of HK, someone asked Soug about dual spaces since he didn't understand them well, and the guy just went on about weak and weak* convergence

I'm sad because the Multivariable Math class at UGA seems to be shrinking down to nothing after I left. I didn't realize that I was such a draw for students ... despite being so challenging.

@Paradox: OK, so when we take square root, we take square root of the magnitude (getting 2) and half the angle (getting $-\pi/3$). Done.

@Danu: Stop asking for advice if you don't want it. :D

@Daminark I asked him him about coordinate charts for manifolds when I was a first year in his class and he started talking about weak convergence. I think he just does that when you ask him any question whatsoever.

@Eric Ugh.

Honestly if I were not super committed to math before coming to college that first quarter would've chased me away. It was not good.

@TedShifrin Yes. If we take it from the positive side though, it comes out to be $4\pi/3$ and then after taking the square root, $2\pi/3$. In theory, shouldn't this work too?

Yes, @Paradox. That's perfect. Remember that we have $\pm$ for square roots. $-\pi/3$ and $2\pi/3$ give you a matching pair!

@Eric: I chased my share of people out of math, I guess, teaching both Spivak and my course. But I also lured in an awful lot. I think students who think math should just be rote and symbol pushing like it was in high school aren't wanting a serious math course. But, to be honest, one can do a math degree at schools like UGA (and far worse) without being challenged very much.

@Eric: In math, there's a lot of big boy macho attitude. "If I could survive the torture, then my students should be able to too."

Perhaps this explains why there aren't so many women in math? :D

I think it's inevitable that people are going to be chased away just cause the subject really isn't for everyone. I just think we have to be better at recognizing when our own attitudes about pedagogy are problematic/chasing people away from certain groups (minorities, women, etc.) at a much higher rate systematically and know how to correct those attitudes for the betterment of the community.

@TedShifrin Oh ok, so wolfram was just giving one answer and the complete answer would include the other value as well. Thanks a lot!

Yeah, I think the main killer in our class was that we were alternating between Rudin and Sally, which meant that we were given 20 problems from each, and Soug didn't want to reach linear algebra in class at all, so that was also relegated away as well

People don't want to "water down" the subject. But one can still be interesting and demanding and still help students to develop and mature. But that requires caring, personality, and TIME. Research academia doesn't reward faculty for putting TIME into undergraduate students (barely into graduate students).

@Paradox: You're most welcome.

@TedShifrin What I despise is the following syllogism: "The reason to go to grad school is because you love research; if you love something, you should be willing to make any sacrifice for it. Therefore, if you're not willing to do anything for grad school then you're doing something wrong."

@TedShifrin I did :-) Talking out loud may have helped me in my thought process ;-)

Well, @Danu, I'm glad you're finally done with that talking out loud!

Yeah I really hate the macho attitude @Ted... My girlfriend had a really bad experience with the math department here even though she genuinely expressed a lot of interest and talent for higher math, so we swore off of math here, and I don't blame her one bit

@Semiclassical I've heard faculty here express that idea almost verbatim.. Seems like a really unhealthy attitude to have

@TedShifrin lol

I'll leave you to your complaining about Chicago ^^

@Danu So I take it it's a good idea not to want to go to UChicago

I have no idea. I am in Europe anyways.

And with the whole stream of current developments, maybe Europe can become more prominent, academically, in the coming decades :D

@Danu For spring vacation, or in general?

In general.

I live and study in Europe, and always have.

I don't think Europe is doing so wonderfully, either, but let's hope Brexit/Trump don't drag the whole world down the drain.

@Danu Oh! Sorry! It's Daminark who's at UChicago!

The Netherlands did OK-ish in the elections.

I'm still looking forward to my month visit. Not looking forward to the "How could you ...?" conversations.

@Akiva You'll definitely want to be careful about such a decision. I'll say that for me personally, it's been mixed but a net positive. On one hand, there are some fears I've developed (not wanting to ask questions for fear of looking like an idiot/slowing things down) that I probably shouldn't have

I'll spare you that one @Ted

Yeah, @Danu, I suppose. But the recent gay-bashing in The Netherlands wasn't too cool. :(

Demonark: Don't you dare have that attitude.

@Ted it was a single isolated incident AFAIK, wasn't it?

I think. I should double-check.

And lots of positive reactions thereto.

@Akiva I think that the school has its faults but so do a lot of schools. I got into a lot of other schools that have math departments that are equally prominent and I'd still choose to go here.

On the other hand, though, my background when coming here was IB Math HL, which was terrible, I placed only into the second quarter of normal calc originally. Doing the 160s and then honors analysis has put me in a great position, and it's not an opportunity I think I could've gotten elsewhere

@Daminark in Marianna's class you should ask a lot of stupid questions, odds are almost no one follows.

The Spivak course has pretty much disappeared at all the big universities except UC (and maybe Ohio State), Demonark. Berkeley, Rice, UGA ... all closed down shop years ago.

@Ted I wish I didn't, but somehow, in a general class setting, I almost feel like I'm rudely interrupting just because I didn't catch something I should've caught the first time.

I think IB is a giant rip-off. At UGA I determined we wouldn't give much credit for it unless students took an in-house test (basically a final exam from calculus). Those that did under my watch never did well.

Demonark: Adopt this attitude. "If I'm lost (or didn't get that), probably at least 1/3 of the class didn't get it either. So I'm doing a service to ask."

That being said, there were some people in some of my classes that kept asking really lame questions.

I'm often relieved that most people in our class are significantly less hesitant to ask

@Ted why did people stop teaching out of Spivak?? I have very fond memories of reading his calculus book

Questions to clarify what's been going on, as opposed to go off on a tangent and show off, are almost always appreciated by other students.

@Eric: Because of the prevalence of BC credit on the AP.

Most of which is garbage.

Yeah, questions of the form "Is this related to [completely different thing]" are probably bad

@Daminark one of the big differences I've noticed taking graduate courses here is that the grad students ask way more "stupid questions"

When I started the Spivak course up at UGA in 1984, we had 3 students a year with BC credit. Now it's zillions. How many of them actually know what they should? Very few.

though I suppose they're probably good to ask after class

Oh that sucks, I don't remember the BC test well because I took like 7 years ago or something but having a solid understand of spivak's book is a whole different world from getting a 5 on AP BC calc.

That said, @Eric, the smart kids who start in my Multivariable Math as freshmen can get a year farther in their studies without too much stress. But there are still a good number of students in that course who major in things other than math.

@Eric It sure is. But students think they know it all and want to move on to "cool stuff."

yeah that's a common sentiment, people always want to know fancier things than they're ready for. Personally I never understood, I always like to keep things concrete to make sure I actually know how to do things.

Honestly I'm somewhat iffy of the idea of throwing calc at high school students so early. If they're gonna do proofs, surely there's a better avenue at introducing it, and I think that being able to mechanically perform integration by parts 7 times in a single problem is not the best test of understanding, which seems to be a significant part of what AP and IB do in calc

@Eric: Your philosophy fits my teaching pretty well. But there are lots of students who don't like the more concrete approach.

@Daminark I've always thought there should maybe be an AP linear algebra course or something that integrates some calculus from the get go instead of something like BC, I haven't thought too much about whether or not this pedagogically sound though.

@Eric I also think part of it is a self-worth question. At least in my personal experience and that of some friends, that's something of a motivator

My biggest problem is that most high school teachers aren't qualified to teach what they're teaching. We need to raise the standards on our teachers, and that's not happening.

And most students who get pushed into AP don't even belong there.

@Ted judging from your notes on differential geometry I get the impression that taking your classes would've made me a very happy student

Thanks, Eric. :)

I think that maybe we need to respect teaching as a profession more so that people qualified to teach actually see it as an attractive career.

Hard to change our society. And in this day and age, even less respect for knowledge and science.

The decision to take and stay in honors was at least partially inspired by wanting to feel like I could do something right, after how I flopped in physics and compsci. There were other factors for sure, but I'd be lying if I tried to discount that.

Where's Zach, I wanna see if I can teach him stuff

LOL

Honestly I think that if there was less of the "Pl3b for not doing honors" sentiment, the environment here would improve significantly

What're you gonna teach him? He's falling behind on Ted-cercises :P

yeah it's really very sad. I feel like the US is heading down a dark path, but honestly I guess it's always been kind of heading down a dark path.

f u n c t i o n a l a n a l y s i s

jk

@Daminark Maybe you can teach me some of that

Speaking of which, DogAteMy, I'm waiting for you to get to later stuff on manifolds and integrals of forms.

Oh, right

CHEMISTRY @Dodsy

LOL

I need to do a bunch of Ted-cercises

@Akiva Lol I'm also pretty unqualified to teach functional for now, I've had a grand total of 4 weeks of it. The only thing I think I could teach at all is linear algebra, and even that's unlikely

Did I send you homework problems on manifold stuff, DogAteMy? There are a few cool problems not in the book.

I don't think so

Let me see what chapter I'm on

Last I heard, you were working on Kantorovich's Theorem in 6.1.

I haven't progressed really far

I have to do the exercises in 6.2

OK, I'll email you some extra stuff "for fun."

Sure

Sent.

@Ted do you know of a good source for understanding the conformal gauss map

Hmm, I don't know that term.

Pray tell me.

I'm unclear on what it is exactly, I was told by Neves to think about it to better understand the conformal geometry things happening around willmore surfaces

@TedShifrin Whoa, that's a lot

He said something like, the gauss map is kind of associating to each point its best approximation in the form of the tangent plane, but that there was a conformal gauss map which instead associated the best approximating sphere or plane.

DogAteMy: Find the ones that apply to where you are. Although you might find a few challenge problems from early sets that are fun for you.

Pairs of primes make finite groups:

I've never heard of that, @Eric. So the conformal structure has to do with an equivalence class of metrics. The usual Gauss map is metric-independent. So these are all surfaces in $\Bbb R^3$ or are they abstract surfaces?

all are in $\mathbf{R}^{3}$

But these are not minimal surfaces (in which case you can make the Gauss mapping naturally a holomorphic or anti-holomorphic map to $\Bbb CP^1$)?

The usual Gauss map is conformal precisely when $H=0$, by the way. Do you know why that is?

So I'm wondering if Neves is talking about a conformal Gauss map or a Gauss map that happens to be conformal.

After all of this, i finally like Ted. Thank you for the training :D

Damn, @Alucard. Why do you have to say that!?

Sorry, i couldn't hold it back.

I give you primes and a group!

:D

Please see link above

Twin primes is in there because 1 is allowed in the solution set

but not sure if this phenom is limited to primes

4 upvotes so far

woot

I'm on a roll

Thanks to whoever upvoted

I'd like to thank my parents, and my university

And my dog Luna

:P

@Ted well doing a direct computation we know that picking a parametrization which diagonalizes the differential of the gauss map, the $H = 0$ condition tells us that the principal curvatures are negatives of each other, direct computation tells us that $\langle dN_{p}(v), dN_{p}(v) \rangle = \langle dN_{p}^{2}(v), v\rangle = k^{2}\langle v, v \rangle$, where $k^{2}$ is the shared value of the principal curvatures.

Hmm.. The surfaces we were working with were Willmore so not minimal. I seem to recall that he specifically indicated that this was some kind of suped up Gauss map rather than the typical one.

I think the idea was that rather than mapping to the space of all planes through the origin like the usual Gauss map is doing, it was mapping to the space of all planes and spheres through the origin. I think he also mentioned something about the isometry group of this space of planes and spheres being the collection of conformal maps on $S^{3}$ or something but I might be recalling incorrectly.

Interesting, @Eric. Does this appear in Bryant's papers? I'll try to look. I guess the osculating sphere is interesting—this comes from four points coming together (the tangent plane takes 3).

Hi

he seems to mention "the conformal Gauss map" but not sure if it's the same one we're trying to pin down. The paper is quite hard and I am working through it slowly.

@Ted We were learning about circles and so I was thinking about the equation and I realized this cool thing

yes, Zach?

hi @TedShifrin I have some idea of understanding the tensor product geometrically.

I was wondering would you like to discuss it ?

not right now, Karim

@Eric: You looking at the 84 paper?

Say I know that for $X,Y\in \mathfrak g$ and $g\in G$, $\operatorname{Ad}(g) X=Y$. Then $\exp(t (\operatorname{Ad}(g)X-Y))=e$ for all $t$ by uniqueness stuff, hence $g\exp(tX) g^{-1}=\exp(t\operatorname{Ad}(g)X)=\exp (tY)$. Is this sound?

So he actually defines the conformal Gauss map and discusses its geometric meaning on p. 33 of that, @Eric.

Ah ok I will stare at this intensely

I guess that's OK, @Danu. At first I was bothered, but I think it's ok.

Hmm. It seems too.. strong?

But an argument something like this has to work

This just seemed a little too easy, somehow

Well, in general exp is not one-to-one, so it bothered me, @Danu.

@Eric: If you want to talk at me about it, I'll try to think about it a little.

I was also worried about failure of injectivity

But $exp(t0) = e$ is correct, @Danu. :)