@BalarkaSen i do not know $\pi_n$ for $n>1$ lol. I will learn it this summer I hope

That is, X must be.

GMT is like waaaaaaay more techinical than the stuff you did in the first few quarters @Daminark, I think it might be good to gloss over things in that case, idt it's the right time to be very technical, my point is, idt that approach is conducive to doing things.

@BalarkaSen Is $\pi_n(S^n)=\Bbb Z$ always true?

@Dodsy The only compact connected 1-dimensional manifold is the circle, correct. I don't know why this is relevant to what I wrote though.

@Alessandro Yep.

X must also be this in order to be homeomorphic, is what I'm getting at.

@Alessandro the homotopy groups of $n$-spheres bigger than $n$ are fucking crazy

@BalarkaSen do you know homology ?

Is $\pi_n(S^n)=\mathbb{Z}$? I may be entirely wrong

However, I have limited knowledge.

@EricSilva I've heard there are a few open problems about them as well?

limited why ?

@Dodsy $X$ must be a circle in order for $X \times X$ to be homeomorphic to $S^2$? Why?

@Daminark fair. I still think you should not glaze too much. Epsilon delta is the workhorse of analysis

(Indeed, $S^1 \times S^1$ is the torus, not the sphere)

@Alessandro definitely, I've been told that basically entire branches of algebraic topology exist historically to study the problem of computing them

Ew

Actually, though, it's not too bad

@BalarkaSen No, $X$ must be a compact, connected one dimensional manifold in order to be homeomorphic to $S^2$ (so I thought, with my limited knowledge, I'll add again) Which means that $X x X$ may change these properties of $X$?

What's the condition for having an inflection point, for instance?

I mean for what it's worth I may not do too many more things along those lines, so I liked that Marianna's wrap up was very enjoyable

Also like, if you do grad analysis they teach you measure theory again anyway

Anyways as I said, I've only had a passing encounter with any of this and I am just going off a vague memory.

$y^{''}$ must have roots

written: again; read:*for the first time*

Right. What's y''(x) in this case?

$6ax+2b$ have roots. So III always true.

I mean, we learned a reasonable amount, and for what it's worth we had a number of important theorems that were actually careful. Fubini, for example, and Marianna was significantly more cautious about stuff like Radon-Nikodym

for I and II $y^{'}$ 's discriminant is important

More precisely, it's always got a positive root at x=-b/(3a)

@Dodsy Oh, hm, I see what you're saying. $X \times X$ is a 2-manifold, does that necessarily mean $X$ is a 1-manifold?

If it is you'd be done by what you said.

does positivity required?

I mean you say that @Daminark but I really do not believe people tend to come out of HA having learned much of anything, even if it's covered in class

You're told that a,b have opposite signs

sorry, again i wrote it wrong, $a$ and $c$ have opposite signs

Oh.

I guess that boils down to saying if $X \times X$ is homeomorphic to $\Bbb R^2$, is $X$ homeomorphic to $\Bbb R$?

Actually, is $\left(\pi_{n+k}(S^n)\right)_{n \in \mathbb{N}}$ eventually constant for all $k \in \mathbb{N}$ ? Or am I misremembering something?

Then no, strike the positive remark

so, III is a true statement

I mean, define much. By the standards of graduate school of course it's pathetic, as far as an undergraduate level is concerned it's pretty decent

@BalarkaSen I don't really know what $X x X$ means. But if you were to look at what that does to a space, perhaps you'd find that there can't exist a space $X$ that when $X x X$ has the property of being a one dimensional manifold.

@dodsy JeSuis is asking for $X$ a space, he never specified that $X$ needs to be a manifold so there might be some dogbone style weirdery?

@EricSilva That's neat, right. Have to show $k(x,y)$ is the same as $\bar{k(y,x)}$

Thanks!

@AlessandroCodenotti Hey, you could be right, for sure.

Honestly @Daminark my year was pretty good but I think a minority of people came out of the class with workable knowledge. They just knew a bunch of words.

Hi @Alessandro, @Studentmath, @EricSilva, Nate, Demonark

Right. But we haven't used the sign condition, so I doubt we're done @AbdullahUYU

Hi Ted

(I know nothing about that so probably not. Balarka is the one you should listen to :P)

Prof @Ted!

Oops, and @Semiclassic and @Balarka

Hi @Ted

Anyway, whatever the case, we're done now, my focus will soon shift toward algebra

Hey mang.

Hey @Ted!

@SteamyRoot Yeah.

Hi @ted

By Freudenthal suspension theorem.

@Balarka can $X\times Y$ be a manifold if neither space is a manifold?

it is important for the $\Delta$ of the $y^{'}$ i think

@Dodsy Yeah. Well, you raised an interesting point actually.

I don't know if your approach works.

It might not.

I won't pretend to know more than I do :)

$y^{'}=3ax+2bx+c$ and $\Delta=4b^2-12ac$

I often don't do math in here because I honestly am just not at that level that you guys are at yet.

A lot of it looks foreign to me.

@Alessandro I don't know... trying to think of an example.

@TedShifrin hi Ted

Right. Since a,c have opposite signs, what can you conclude?

hmm, i get it since $a,c$ have opposite signs

Hi @JeSuis

$\Delta>0$

@AlessandroCodenotti Oh i see now. We know that $X x X$ must be a one dimensional manifold, but that doesn't mean $X$ has to be anything more than a space.

Yup, exactly.

@Daminark I think our algebra sequence is much better than HA

I don't know if it does @dodsy I don't know anything about this topic

Right. So how many real roots does y' have?

So $y'$ have two real roots and that means we have local maximum and minimum

I and II are also true always

I mean everything depends on exactly what you're trying to get out of it

HA gives you a taste of quite a lot of things at a level that could be considered decent for undergrads

wow that is fundamental :)

Like later when you see measure theory and functional analysis, you have an idea of what's going on, and I do think there's quite a lot of value in that

Well, that's a bit quick. We should also rule out that they could both be max or both min

@TedShifrin How is your morning going? Do you eat breakfast usually?

@TedShifrin And what's the jumble today.

I haven't solved the Jumble yet. I'm working on it. :)

:)

You could also do things differently, so that you learn everything with more of the details and problem focus, but then you cover like, half as much

Nah, HA is a poorly structured class that underserves the majority of people who come into take the class. It can be done a lot better. As it is now it's kind of a problematic class.

To close the loophole, what's the average of the two real roots?

@BalarkaSen I might ask that on main, or maybe @Ted can help?

@TedShifrin Oh you only give us the final jumble, right?

Ted's stuck on the Jumble.

Yeah, Nate. I have the four words, just not the final answer.

It's one of those jumbles that has like 4 and then a final one with key letters.

what's this jumble thing? I've seen it mentioned in chat quite a few times already

I think I'll think about it for a little while. You can go ahead and post it.

It's a word puzzle in the newspaper, @Alessandro. You have to solve for four words (that are anagrams) and then make a final answer from a hint.

I'm surprised @Dodsy knows enough topology to get us stuck on a problem :P

There are indeed non-manifolds $X$ such that $X \times X$ is a manifold

@BalarkaSen I really don't!

Are we talking manifolds in which category?

topological

TOP

I think you guys were talking about homeology once and I went and googled it and read a few paragraphs.

Hmm ... Some exotic crap probably.

@SteamyRoot What's an example?

@SteamyRoot A one dimensional connected manifold?

I have no idea, @Mats. Why can't you do the same numerical scheme?

@SteamyRoot Interesting, what's the lowest possible dimension of $X\times X$?

@TedShifrin Mathematica gives me rootsums.

I don't know what that means.

@TedShifrin While looking for the jumble so I could do it myself, I came across the answer- so I will not be participating in today's jumble unless it happens to be the wrong one...

:C

So I'd say from the mindset of introducing you to some theory in analysis, our class is alright. It's not like people are now foreclosed from a problem-based approach, when people take graduate analysis, either here or in grad school, they'll deal with the material again, and then they'll learn problems while having a few helpful pictures and theory in the back of their minds, so I'd say net total is a plus

A sum of roots that can be expanded into Radicals sometimes.

OK, here's the Jumble for @Dodsy, @Demonark, @Astyx: "Once her car was repaired, she said this in regards to the damage." (4/3-5) DIOGNTDESDOR

Meh, I found the title of the paper where they prove those things.

Products of cell-like decompositions, by Daverman

It might not the same one, Nate. I don't know if all papers print the identical one.

Anyway I think we're approaching this with too much of a different mindset to really come to any kind of agreement on this, so let's shift our attention to the jumble :P

@Mats: But if you're doing a numerical scheme, just evaluate the sum numerically ... and then take the reciprocal, etc.

@Semiclassical why do we need the average of the roots? Doesn't it mean that we have to have both at once?

@AlessandroCodenotti That's the important thing.

Howdy @MikeM

Hey skullpetrol

hi

@Daminark I'm not saying there's a theory vs problem solving dichotomy. I'm just saying that the course is hostile and poorly organized (also the order and emphasis of topics doesn't make much sense to me). I generally believe that uc math majors aren't being taught well, and most of your professors tend to agree but Dept politics makes change hard.

@Semiclassical wait a minute i'll upload a pic

I don't do jumbles man

There's a reason I'm asking for the average.

@EricSilva I thought UC was considered one of the best math schools in America.

Just add them together and divide by 2

@TedShifrin What does 3-5 mean?

Three letters

in one word

It's quality, just that the pedagogy of classes is rather controversial

5 letters in the next

Well, there's a hyphen there

@EricSilva: I noticeably changed our algebra course at UGA back about 20 years ago (doing rings first, primarily, because of the large future-teacher component of the students). Most everyone agreed it was a change for the better, but one or two of the algebraists (Herstein/Fraleigh types) resented it and still do.

Ah, ok, separated by a hyphen.

So this would be a 4 letter words followed by a 3 letter word and then a 5 letter world

Oh sorry I misunderstood

...

It means hyphenated ... and there's quote marks around the 8-letter word.

@Dodsy our students are really smart and motivated generally I think. I'm really critical of the school bc I think it could be doing way better and be more compassionate, as it is the math major is pretty hostile if you're not weird like me or Daminark.

Nate: I would say "best" for a certain type of student, but certainly not all.

@Ted I guess people who care about a subject always have stronger opinions on how it should be taught

pretty good chance I would have failed out there

@EricSilva Student satisfaction is important and a lot of Universities forget that..

Well, we know you're totally incapable of learning or doing mathematics, MikeM, so there.

Well, we overdo "student satisfaction" and "parent satisfaction." That isn't my point.

Oh sorry, didn't notice your comment, Ted.

There's a prevalent view that education is now a consumer-driven thing and parents who are paying should get what they want.

I'm not trying to self-aggrandize? I hate that. I mean that when I started school I certainly was not ready for the way they teach.

The times they are a-changin'.

No, no, @MikeM. I'm agreeing with you. I was just being a bit sarcastic.

@Dodsy No, the times are a-tellin', and the changin' ain't free.

@Semiclassical their avg. is also $-b/3a$

Interestingly, I think a lot of my colleagues thought I was either way too hard a teacher or way too hand-holding a teacher. Certainly, plenty of students avoided me because I had a reputation of demanding a lot of work.

Main thing I notice in the jumble is that I can rearrange the letters to DIGDOORDENTS

NO WAY!!! ;)

So what kind of student does the University of Chicago best serve?

@AbdullahUYU exactly. That's where the inflection point was

I dedicated a lot more time to students than most faculty. Mostly because I stopped trying to publish research over a decade ago (I decided well-written textbooks would do more people good than a paper read by 20 people). But there were plenty of other faculty who weren't doing much research who didn't give a damn about good teaching.

yes, what is that mean?

So one root is to the left of an inflection point and the other is to the right

Nate: very self-confident people interested in math grad school and not so much in applications.

@TedShifrin "But there were plenty of other faculty who weren't doing much research who didn't give a damn about good teaching." That's terrible to hear.

yeah

@Dodsy idk, my experiences have been good, even great, but Im also weird

Why is someone now starring the Jumble? Good grief.

@EricSilva: You're also a very self-motivated, strong student.

So the two critical points will have opposite signs for y''

So that we can work on it while chatting, I suppose.

What does that tell you?

This is true @Ted I think I tend to put a lot more work in than is normal, even here.

@Dodsy So there are many paths the math major can go. The standard one is to do a calculus course via Spivak + a bit on manifolds first year, analysis via Baby Rudin second year + a linear algebra class, and then algebra via Dummit and Foote

@EricSilva No, that certainly sounds like that type of place many people could excel!

I had an interesting chat at a party the other night with a woman who grew up in France. We talked quite a bit about education in Europe, and the need to specialize too early.

@TedShifrin That's what he means, I think

I don't know about Eric but Daminark is weird, yes.

Then they do some number of electives

@Daminark Oh that's awesome.

Daminark is a fucking lunatic but that's not what Eric was talking about

All of that is inaccessible to the average math major in the US, Demonark. Totally so.

No schools here teach by spivak.

lol

It's all Stewart

How am I a lunatic? :P

We're all mad here.

It's still odd to me, all the minor-major in the US undergraduate studies

you expect i will tell you "one of them is min and one of them is max" but,

If we defined a math major by that curriculum, math departments across the country would close down. Also, most departments have to remember that there are science, econ, and engineering majors who fill MOST of the seats in the first- and second-year math classes.

That's fair @Balarka

I'm not mad, @Balarka. I'm a singularity.

we might have two roots that have same sign and can give us min and max @Semiclassical

If I was to find a specific spectral representation of an integral operator, after showing there is one indeed, is there a good way to approach it?

Hullo chat

That was Alice In The Wonderland reference of course.

Zut, alors. Salut, Astyx.

"Zut" ?

Or was it Through the Looking Glass, Balarka?

Je peux repartir si je dérange ... :p

Il y a longtemps on disait "Zut."

Je t'ai laissé le Jumble, Astyx.

I don't think it was that latter. It was what the Cheshire cat said.

J'ai vu, il est même étoilé

(no result found in terms of standard mathematical functions)

Well, of course not, @Mats.

I thought you were doing numerical schemes.

@Astyx When I had that french exchange living at my house, we went to the McDonalds and she ordered a Coke, but pronounced it "Cock" Is that normal in france?

So, do we have an example of that $X\times X$ manifold thing? I have no idea about it.

yeah

We do?

It's how you pronounce "o" in French. Long o would be "eau" or "au" :P

dim X = 3

@Dodsy We would say "Un Coca" in France. Perhaps he meant "coke" but pronounced it badly

"dogbone space", due to bing

Does it create a one dimensional manifold?

no

@Astyx Well it was a girl, but she said it twice!

Once she said she likes it warm!

It's a 3-dimensional (whatever that means) space.

Get your mind out of the gutter, Nate.

Well then you've solved it

There's covering dimension, e.g.

Did you tell her it wasn't pronounced as such ? I mean, english pronunciation isn't that intuitive tbh

See my comment above about pronunciation of vowels ...

Yes, I certainly did.

@TedShifrin Products of cell-like decompositions by Robert J. Daverman gives a survey of cell-like decompositions, which can be used to construct such spaces.

Right, o followed by e makes it long. Most languages don't do that, Nate.

Oh, interesting.

Ah, Steamy & MikeM. Thanks. Not surprisingly, this is the sort of thing I've never thought about in my math life.

Good afternoon, nerds!

Nate: I would have been a linguist if I hadn't done math.

Heya, @Fargle.

(Or whatever time it may be where you may be.)

@Ted I actually know a couple math/linguistics people

How's it going this fine Friday, @Ted?

Yo Fargle

I guess today's Jumble is harder. No one's blurted out the answer yet.

Also hey @Fargle

Trying to get ready for my trip, @Fargle.

I didn't look

Sup @Astyx, @Dami

Actually I think $X = S^3/A$ works where $A$ is the Whitehead continua.

Where you goin', @Ted?

@MikeMiller So since there is no space X where $X x X$ creates a one dimensional connected manifold, this means that there is no space $X$ where $X x X$ is homeomorphic to $S^2$.

@TedShifrin not my taste of thing

S^2 is not a one dimensional manifold

Balarka already gave a proof for S^2

That's the one point-compactification of the Whitehead manifold $W$, which satisfies $W \times W \cong \Bbb R^6$.

@AbdullahUYU had to step away. But, huh?

For X to be homeomorphic to s^2, x must be a one dimensional manifold.

@TedShifrin What does 4/3-5 mean?

Along "obvious questions" lines, there's this.

DogAteMy: Two words. First is 4 letters. Second is hyphenated, 3, then 5.

@Fargle: Germany, France, Italy, Croatia.

for example $x^3+15x^2+18+70=y$ 's derivative $y'$ have two roots that is $x_1=2, x_2=3$

@TedShifrin My goodness, what an itinerary! I hope you have a good time, and most of all stay safe.

Er, nah, maybe take that back. $S^3/A$ probably wouldn't work. Maybe $\Bbb R^3/A$?

I dunno, I give up.

I have also given up.

Currently I'd think it's Ding Ted-Doors

@BalarkaSen, Isn't tomorrow result of an exam named Madamik around your locality published ?

Woah that's a good one dami.

@BalarkaSen Yeah something like that works

Just curious, would finding the Kolmogorov complexity of a string be reducible to the Halting Problem in the general case?

Because I believe that one could just enumerate programs by ascending length until one halted with the desired string

@Ted: I have the jumble solution.

@Balarka: The question I just linked asks how to standardly see $\Bbb RP^n$ as a submanifold of $\Bbb R^m$ for some $m$. Obviously, we know the complex case can never work in $\Bbb C^m$.

Good, @Fargle. It took me a few minutes.

@AlexKChen I think so.

Should I spoil it?

see pic i uploaded @Semiclassical

Yeah. One of my friend from India gave it.

LOL, Demonark.

That's what my neighbor's done to me.

Ah that's annoying Ted.

I was once siting in my car at work and a worker door dinged me...

He just started leaving.

People suck.

lol, that sounds like my luck

Yup, Nate. When the car gets to be 8 years old, I don't care. But it's still young enough that it pisses me off.

Those don't have opposite signs @AbdullahUYU

Lel @Ted, that's unfortunate

Demonark is slow on the Jumble today.

I had to write this one out because of all the repeated letters.

No one remarked on my detailed comments about math courses and responsibility to teach students other than math (very small percentage).

No comment

:-)

@ted in my defense, I was busy helping someone with math :P

$LATEX$, $LaTeX$

Do you mean, teaching students that are not math students or teaching things that are not math?

LOL, @Semiclassic.

@TedShifrin What's that ?

How the word $LATEX$ is like that ?

I mean that the majority of students in the US who take first- and second-year math classes are NOT remotely math majors or interested in being such.

@AlexKChen $\LaTeX$ ?

Astyx, if you scroll up, you'll see I also commented on a long discussion with a French woman at a party about the French education system. She hated it.

LaTeX

Close enough

She was interested in literature and in biology/medicine, but ended up having to major essentially in business.

uh ha, we can not have roots that have same sign since $a,c$ have opposite signs. @Semiclassical

Heh weird $LaTeX$ $\Balarka$ $\LaTeX$ $LoL$ $\Lol$

With physics there's a similar issue of people only needing to take the intro courses

This jumble is very suggestive of door(s) and ding(s) but what's left is $Sym(Ted)$ so...

Yes, Semiclassic, but a much smaller "service" component than math has.

I have to go and eat but I'll happily discuss this later :p

Right. Do you see how that connects to what I said re: the inflection point?

Or we can discuss over lunch in 2 weeks, @Astyx.

Bon appétit.

@ted agreed. A similar remark probably holds for gen-chem

So, to my understanding most math departments have only a single set of classes for 2 years, right?

@TedShifrin I think you have very valid opinions, and would know better than myself what the education in America lacks, and thus I can do much else than to agree with your opinion. :)

Right, Semiclassic. Premeds/prepharm/prevets in both cases. But math has way more.

$\mathfrak{Wow! \; this \; font \; is \; so \; weird ! } \boldsymbol{Heh \; Heh\; Heh}$

Good God @AlexKChen

Does math see a lot of pre-etc people as well?

@Semiclassical i'll afk. but wil come again

Not entirely, Demonark, but close. A few places do have an honors-type course like mine for super-strong students. But relatively few. (The Spivak course has vanished almost everywhere because of the over-abundance of 5s on the BC AP exam.)

Mmkay

$\mathfrak{My \; pleasure} \; \boldsymbol{ye} \; \texttt{Daminark}$ @Daminark

Yes, @Semiclassic, but also business, econ, physics, chemistry, and engineering.

Those dastardly engineers

So, that might be part of it. For what it's worth, the math department does have other classes that are basically built as service

@TedShifrin Most schools here have honours programs, but an emerging trend is "Honours Specialization".

U of T

But service is much smaller at UC because (a) it's a tiny school; (b) there are no engineers.

has the best mathematics honours spec course ever.

@Dodsy Which T?

University of Toronto.

Of course, engineering is itself a pretty broad brush. Mechanical, electrical, aerospace...

Ah. There are several. >_>

When I was a grad student at Berkeley, they had a Spivak course that had roughly 75 students in it. It vanished years ago. But Berkeley has thousands of students.

There's a very standard calculus class, and a 2 quarter sequence in math for physical sciences (which, in particular, means chem, geophysical science, and apparently stats)

Sure, @Semiclassic. But they all need 4 or 5 semesters of math.

$\frac{a}{\frac{a}{\frac{a}{\frac{a}{\frac{a}{\frac{a}{\frac{a}{\frac{a}{\frac{a‌​}{\frac{a}{\frac{a}{\frac{a}{\frac{a}{\frac{a}{\frac{a}{\frac{a}{\frac{a}{\frac{a‌​}{\frac{a}{\frac{a}{\frac{a}{\frac{a}{\frac{a}{\frac{a}{\frac{a}{\frac{a}{\frac{a‌​}{\frac{a}{\frac{a}{\frac{a}{\frac{a}{\frac{a}{b}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}$ $ \boldsymbol{This \; is \; my \; last \;} \text{LaTeX}\; \mathbb{EXPERIMENT}$

That covers vector calc, complex variables, and ODE

Toronto used to have a Spivak course, Nate. I don't know if they still do. But, again, for a tiny fraction of the students.

Oh, I'm not sure either :)

There's also one for econ majors (largest department at this school), with a quarter of linear algebra and a quarter of multivariable calc

That's standardly called Calculus for Engineers, Demonark.

Right. My point being that the scope of engineering means your point is even stronger than it might seem

Oh, OK, Semiclassic.

This school is like all econ majors tbh

The two diverged because econ majors need more stuff like optimization, while physical sciences use more vector calc type things.

The physics department kinda went off and did its own shtick because it's a complete mess

Physics in particular needs separation of variables and eigenfunctions

At UGA business majors are no longer required to do a semester of business calculus. They now take a stat course. This may be due to my ill-formed political move of putting a priority on fall precalculus seats to the science majors who couldn't take chemistry without the precalc. This pushed the business majors into precalc in the spring, and the faculty were furious with me for that.

Anyhow, I made my point. You can't run a math department just for the pie-in-the-sky math professors and majors.

Merp

goes again on spiel about how undergrad business major shouldn't be a thing

Though physics usually only does functional analysis stuff at an engineering level

I agree with that, Demonark, but we're outvoted bigly/hugely.

I have no input besides that I like numbers and letters.

Lol but yeah, I mean, basically, even UC has some service courses, enough that it takes care of the majority of the demand here

LOL, @Fargle. I'm not trying to disenfranchise math students.