hi chat

@TedShifrin When have I fought you on that? I probably have but I forget why.

(I guess I might argue that the right analog isn't Gauss's law but rather Ampere's law.)

what is your problem, SIR!?

Roger that

I have kinda not really sorta hopefully made progress on my current problem

Is there a general method for determining when a subset of the free group has no nontrivial relations?

For example, so that I could prove that $\{a^nb^n\mid n\in\Bbb Z\}$ has no nontrivial relations (though this particular case can be done through algebraic topology).

$\{aa,ab,ba,bb\}$, by contrast, does have a nontrivial relation, since $bb=ba(aa)^{-1}ab$.

@projectilemotion hi

@SimplyBeautifulArt Hi

Arf, I just discovered downvoting an accepted wrong answer costs me reputation. Life is tough...

How much? @pilko

@pilko Downvoting anything that has been upvoted or accepted will cost you 2 rep

Ah, OK

or 1 rep for downvoting questions I think

I thought that was for downvoting any answer

Of course, all numbers presented here have plus or minus 5000 error

…Of course, of course

@AkivaWeinberger Nope, downvoting sometimes costs no rep

I would like to get +4998 on downvoting an answer.

:D

imagine a +5000 accepted accepted answer and you find a counter example that destroy it all. downvoting it would worth that much !

You know, one could get 300 rep for free, by creating five new alt accounts, have the five new ones plus your old ones all upvote each other's answers, and then have the new ones all funnel their rep into your main one through bounties.

Probably illegal.

Lmao

definitely illegal

I got +100 for free in each site by having a +1 answer in 5 different SE sites. totally legal :-)

Or I suppose you could have two prolific friends just agree to look at each other's answers whenever they post something

and then they'd both benefit from each other through mutual upvotes

@AkivaWeinberger I would imagine people check other people's posts somewhat regularly if they like that person

@SimplyBeautifulArt True. I usually don't hang out on the main at all, which is why I rarely notice people that I know's answers.

WHAT IS YOUR PROBLEM, SIR?

Hello guys, does anyone know where I can find something related with quadratic rings/fields?

.gooogle

ORACLE

So soctratic idea , gives the most basic trivial answer to any question -.-

I just googled xD but didn't find anything really important except for two pdfs

how do you define important?

Well, I'm looking for an approach of quadratic fields without field/galois theory

It seems that it's possible but also somehow artificial

For example this: math.stackexchange.com/q/1198188/133781

Hey, it is all cool, just relax

The "definition" is right but there is no motivation to define it in that way. So how much theory in quadratic fields/rings can be made without being too artificial?

That's basically my problem xD

It's a nice-sounding question, but it doesn't sound like anyone here is able to answer it. But someone may notice it on the chat history.

dance

and fix the errors

One way tip to heaven

@Xam wiki points to this book books.google.fr/… It seems to present the historical approach, and as one teacher said me once, if you don't want to sound too artificial, always have a look at how it was done in math history.

@pilko oh, thank you. I'll check it :)

As the dimension of the unit hypercube increases, the volume of the hypercube stays at 1, while the volume of the inscribed hypersphere goes to 0...

@Alessandro Yeah, lakes of Wada really hurts the brain.

I like the Wada basins picture tho. That's a bit illuminating.

@AkivaWeinberger I think the kind of thing you'd want is ping-pong lemma

hi @Balarka: I guess it's actually morning morning for a change.

@TedShifrin Good day sir :)

Sir?

its a form of respect because you are older than me

@TedShifrin Yep, I think my clock's finally fixed.

Don't remind me. It's my birthday in a few hours.

@Balarka: Only for a day or two. How gullible do you think I am?

Oh happy b-day in advance :D

Oh, happy birthday

Thanks, guys.

anything special you gonna do ?

Golv ? bridge?

@TedShifrin Hah, fair

@Kasmir: Going to dinner with longtime friends (like guys I've known 40+ and 50+ years).

@Balarka: As they say in the South of the US, I didn't just fall off the turnip truck. :D

That sounds nice Ted :D I wish you good day :D

I had never heard of that phrase until now.

Interesting.

Now I have to keep working on my exam =p I got my re exam in 2 days =p

last time i got 11 points this time am hoping for 27 :D

I have a question on the meaning of x<y<z. Do we really mean x<y $\land$ y<z ? Or do we mean x<y $\lor$ y<z ?

x < y and y < z

Happy pre-birthday @Ted! And have fun!

@Balarka: I always do my best to educate you :P

Indeed

Thanks, @Daminark. I appreciate it.

@BalarkaSen Thank you

@Balarka: Earlier today, anonymous came in with a question he said was for you (on the AM-GM inequality). I told him that wasn't your expertise and he could ask all of us. Turned out to be an issue I'd never thought about before. It was cool.

Yeah, I saw. It's better that you talked about that with him instead.

Well, I learned something, seriously. He seemed stunned. But anyhow ... I didn't think you'd be too upset that I usurped your authority — unless it's for topology :)

I keep forgetting that compactness of the unit sphere is not a thing I'm able to rest upon anymore.

Haha, not at all.

And yeah, that was before he knew that Ted taught for so long, he was surprised just by the rep

"rest upon"?

It's fun to learn new things, @Daminark, even at my decrepit age.

Oh ... "count on" — 'cuz you're now in Banach spaces.

Yeah, my instinct is to say "Well, $\sup_{|x| = 1} |f(x)|$ is achieved by compactne... Oh right...

lol good luck with banach

God, I feel much better after having a good, long sleep. Needed that.

Thanks

It's pretty cool stuff

hi chat

I don't know much about functional analysis. Yeah, it's pretty good stuff.

Now I /really/ see why Spivak's proof of inverse function theorem is bad, he relies on said compactness so can't go beyond $\mathbb{R}^n$

That's right.

The good proof is Ted's, where all you need is completeness of the unit ball.

Then Banach fixed point theorem kicks in

Not my proof, @Balarka. It's a totally common proof, which I learned from Lang's book, actually. But I don't know why Spivak and Munkres do the yucky one.

Yeah at some point I'm gonna need to check that out because the proofs we used in 207 for inverse and implicit function theorems were... not the best

Implicit especially was really confusing

@Balarka: In all seriousness, you keep getting sick mostly because of your f***ed-up sleep.

@TedShifrin Fair enough. I didn't see your proof anywhere except your book, actually.

@Daminark: Implicit is easy from inverse. If you don't want to read it, look at my video.

How does Rudin do it?

We used Sally, actually

Sally actually uses inverse to do implicit

That was one time where our prof went freestyle

Me too, @Daminark. That's standard.

And did so much chained iteration that was impossible to follow

He was like "Solve these in terms of these, do this"

If you get to G&P, you'll see there are lots of things you do with a slight trick + inverse fn. thm.

@TedShifrin Agreed.

That was one of the days where he sped up to the point of being incomprehensible

Oh, that sounds like Hubbard and Hubbard, @Daminark. I'm not so fond of them.

Oh yeah we had a problem on our last pset which was pretty nice and did that

I was consistently incomprehensible :P

The rank theorem is another good one, @Daminark.

What I really like about G&P is they parse everything in terms of coordinate charts. Eg, inverse function theorem literally says that a smooth map with nondegenerate derivative between manifolds looks exactly like identity.

@Balarka: It's basically a precursor of wanting normal forms for mappings of various rank conditions.

Heya Karim

hi @TedShifrin

Right.

I am learning about Algebraic K-theory

Very cool

It was to prove that for $F:U\subset \mathbb{R}^n \to \mathbb{R}^m$, if $DF$ has rank $k$ in $U$, you have that for any $x\in U$, there exists a ball $B$ and a diffeomorphism $\phi:B\to N$ where $N$ is a neighborhood of $\phi(p)$, along with a diffeomorphism $\psi$ in $m$ variables such that $\psi \circ F \circ \phi^{-1}$ is a projection to the first $k$ coordinates in a neighborhood of $\phi(p)$

I know almost nothing. Maybe less than $\epsilon$.

@Daminark: Yeah, that's the rank theorem.

I still don't know how to prove that transverse manifolds locally look like two transverse vector spaces inside R^n

Ah

Maybe you give it a Riemannian metric and exponentiate stuff

Constant rank theorem is very useful.

@Balarka: That's not right if "look like" means a change of coordinates on the ambient space.

@Bram28 This site is so competitive, I can't answer questions quickly enough before someone takes the cake before me.

@user400188: Be thankful lots of us have quit answering most :)

@TedShifrin HAPPY BIRTHDAY!

@TedShifrin I am enjoying my algebraic topology class

i'm really starting to find integrals are tedious

oh its your birthday ?!

I just answered you on FB, @Pedro. But thanks. You're very sweet.

Happy birthday @TedShifrin

Give me a few more hours, Karim, but thanks :) You're very kind.

@TedShifrin What's an example? If $M \pitchfork N$ inside $W$, you can change charts to make $M$ locally like a vector subspace but $N$ messes up. You're claiming you can't make another change of charts fixing $M$ that straightens $N$?

@TedShifrin I am thankful; it's just that it makes it hard for us noobs with no rep to get any.

Sounds like a bump function argument should be very possible.

@Balarka: I'm not claiming anything. I've never thought about it. But your exponentiation comment won't work ambiently, I figured.

Ah, alright.

@Balarka: You certainly can take $W=\Bbb R^n$.

Sure enough.

Hi @PedroTamaroff

Hello.

How's life and math?

I have a problem for anyone that is interested. Consider the group algebra $kS_n$, say over a field of char 0.

shudders at group algebras

And consider the endomorphism on itself that assigns the identity to $\mathrm{id}+(12)+(123)+\cdots+(12\cdots n)$.

Prove the kernel of this map has dimension the number of derangements on $n$ letters.

@BalarkaSen Slowly coming together.

feels deranged

grabs scalpel

[runs]

hi

hi @PedroTamaroff

Isn't it past your bedtime, Karim?

I am trying to think of a problem @TedShifrin

I want to figure it out before sleeping.

Ah :)

I will give it extra 10 min

@TedShifrin I tried last week to sleep 4 hours like Balarka last week, but that I was getting sick.

yeah sleep is important

No, do not do that. I have yelled at Balarka for years.

Please don't take life-advice from me.

What does "sick" entail here? Headache? Or worse?

covers Daminark's ears

Headaches

Don't worry my sleep schedule is already steady, though a bit unideal.

For some reason I can never sleep before 2AM unless I'm really tired

@Daminark lol this semester I sleep like 9:30 and wake up at 4

That's part of college, @Daminark. I usually only got to do my own work starting at 11 or midnight.

4 am

So on better days, I go to sleep at 2-2:15, and wake up at 8:30

I am productive in morning I found.

I've always been more of an evening person, but some people do that, Karim, for sure.

Then on weekends I sleep for 10+ hours

yeah I do that too.

In reality I usually sleep at 3

I am a lot more productive if I sleep normally

This quarter has gotten a bit better because I don't have to wake up quite as early on Tuesdays and Thursdays, no classes then

The morning air helps

@Balarka: Define "normally" and tell me the last time you adhered to said principle.

I have office hours in math on Tuesday so I tend to try to wake up at 10:30-10:45, and going to sleep at 3 means that's actually a good amount

@TedShifrin you what I found sometimes when I work on a problem while walking I find solution faster.

I want to download Weibel algebraic K-theory have to start working on it.

Last quarter I had 10:30 classes daily, and my professor's homework strategy was "spam book problems", so sleep was much more of a luxury then

hides from all the categorical people

@Daminark you learn a lot though right ?

@Daminark: I don't think I've ever been guilty of "spam book problems."

I think solving tons of problems helps understand the subject.

I am doing it for commutative algebr

commutative algebra *

So thing is, I don't know exactly what constitutes reasonable

Who used the word reasonable?

Fair point

But yeah I'd say this quarter, average pset size was about 9 problems so far

Very tough problems, but still just 9

Last quarter it was closer to 50

Which was rather painful

And all of them came from Buck, Rudin, Sally, or Hoffman and Kunze

Yeah, that's absurd. I did typically 10-15 on-line computational problems and then about 10-15 proof-type problems to turn in a few days later. So I'm in the middle.

Yuck @Buck.

Yeah Buck became a thing at the very end of the quarter, when we did "integration"

Did you ever go meet Tori and say hi for me?

Buck sucks.

I still love how Soug stated the Lebesgue Differentiation Theorem and was like "Alright guys it'll take an hour to do even a fake proof of this so I just won't", and used it to prove change of variables

That was the first Buck thing we did, and we were like "Uh... Alrighty then"

Oh yeah ... fake Lebesgue integration in a first course. Ugh.

That's not in Buck.

He doesn't do Lebesgue.

Well, see we didn't call it Lebesgue integration, as far as we were concerned it was the same Riemann integral

What I totally dislike about Buck is the old-fashioned coordinates approach, rather than vector inputs and vector outputs.

We said that if you have an additive set function $F$ which is absolutely continuous, and you have a sequence of cubes $S_p$ converging to a point $p$, then if $\lim_{S_p\to p} \frac{vol(F(S_p))}{vol(S_p)} = f(p)$ exists everywhere, we say $F$ is differentiable. Then $\int_S f = F(S)$

Yeah that was annoying

Right, that's the Radon-Nikodym derivative.

I have a torturous proof of the change of variables with Riemann in my book. I thought I had every detail right, but there are some things fixed on my errata page.

So Buck kinda does that in section 8.2

Without calling it that

I think it was actually attempting to make that statement about Riemann integration

I no longer have Buck in my possession. He was not my favorite, as you can tell.

He does only Riemann, yeah.

Lolol, that's pretty fair

And yeah we had done it in 208 with differential forms

And Schlag was all "See? Change of variables fell from the sky, correctly"

The theorem is still missing. Differential forms are powerful, but they don't prove a theorem.

nights everyone

Night, Karim. Keep up the good work!

thanks @TedShifrin :) Happy birthday again !

I'll dig up Phillip's notes but I thought we actually did it

Thanks, Karim :)

I mean only for 2-forms since we stuck to the plane

You need the change of variables to prove that $\int_{f(U)} \omega = \int_U f^*\omega$ for an orientation-preserving diffeo $f$.

Lol, then I guess our "proof" involved some sleight of hand

Or maybe I'm misremembering something

nods

Hello

DogAteMy! It's late!

How do you pronounce Poncelet

Also, yes

Is it pon-se-LAY?

Not "pon" ... it's a nasal French "on"

But, otherwise, yes.

But yeah, I'm thinking at some point in the near future I'm going to need to figure out why that theorem is actually true. Might wait until I have all the measure theory down from 209

Ultimately, it boils down to some estimates plus understanding why the determinant gives oriented volume of a parallelepiped.

Hmm, yeah thinking about the determinant as the product of eigenvalues probably helps in that respect

/pɔ̃.sə.le/

But, yes, DogAteMy, the French accent is on the last syllable.

@Daminark: You still need to know that shears don't change volume.

Ah

Plus, you have a problem if (some of) the eigenvalues are complex.

Oh darn, yeah that makes life trickier...

Sorry to spoil your party :)

OK it probably is worthwhile to sit down and do this