I understand the algebraic definition but how do you see that they are conjugate of each other geometrically @TedShifrin?

Because you look at the square and see what conjugating by a flip does :)

oh I see

oke good

@TedShifrin: Geometrically that's going to be maximized for $z = -i$. Doesn't work.

Right.

That's how I made it up.

G'night. Today's haul: first chapters each of Artin and Rudin :)

I'm nodding off, can't possibly continue.

first chapter of rudin !

@TedShifrin: But plugging in $-i$ gives 3...

good job

The second chapter though . . .

Winter is coming. (I said that before)

Oh, crap, @Mike, I see.

I didn't read artin before I should start though seems nice

I am doing DF

@TedShifrin: I think we might need something degree at least three, and we're going to need the symmetries of our set of points to suck. (Symmetries make it easier to find some point which maximizes it, I think.)

did you do the problems too @octatoan?

@MikeM: Slight perturbation works. $1+z+iz^2$.

@TedS: $(z-1)(z+1)(z-e^{\pi i/4})$ works.

I had to return my textbooks today.

hi @Owatch

However, I would still like to do problems.

Hello

Return them to the bookstore? School's over?

Today was the return deadline.

Yes, school is over.

This is why I hate the whole idea of electronic books and renting books ... Some books one really needs to keep.

@Ted: I guess we both got ours by symmetry-breaking.

I'll use yours. Easier to write down.

I'd go buy a book.

Yeah, @Mike: I thought I had done that the first time. Sorry.

But I'm not sure what to get.

@Ted Wait, $1+z+iz^2$ works? For which value of $z$?

That's the point, @DavidZ. It doesn't work :P

@DavidZhang: Its maximum is $\sqrt{5+2\sqrt{2}}$.

What are you trying to do, @Owatch?

I'd like to continue practicing the types of problems I did in Calculus II.

hey so @TedShifrin looking back at this problem of center of $D_8$ I think I can generalize it for example the center of that for $D_{2n}$ will be $\{1,r^s\}$ if $\frac{s}{2} = 2k$ for some k then s will be equal to k otherwise the center will be the trivial one but I don't know if my conjecture holds.

geometrically it makes sense

But I found it hard to sometimes understand the chapters I was reading, as the explanations were a bit confusing.

So I was wondering if anyone knew a good Calculus II book with problems I can buy, and practice with.

You need an $n$ in there somewhere, @Karim.

@KarimMansour it's true

What was your textbook, @Owatch?

oh, r^n, 'course

yeah r^n

Can you take books out of your college library, @Owatch? I'm a bit confused why you had to give the book back since presumably you're taking the next calculus course?

well yeah the reason this would work because we will have $r^s*s = sr^{-s} = sr^{s}$

Think geometrically, @Karim.

@TedShifrin I'm not going back to this school.

D_2n is actually the semidirect product of Z_n with Z_2

What's different with $n$ even and odd?

Hush, @Balarka :P

Oh, @Owatch.

So what book were you using?

I don't know about semi-direct products yet though but I am expecting this week I should be learning about them.

@TedShifrin k

@MikeM: Here's a good topology question, asked by one of my favorite MSE geometers. I actually met him a year ago.

when n is even we will have the symmetries will be kinda of different that is we will have symmetries along the lines that goes through each vertices and through the middle between each vertices while odd we will have symmetries along the vertices only

right

I've had good experiences on MSE with Jason.

He's a U Penn alum.

I'll have a hard time answering this. I don't really have a bag of, uh, 8-manifolds to play with.

I have no idea what the answer to his question is.

@TedShifrin It was Single Variable Calculus: Early Transcendentals

oh it will work always even for n is even but instead it will be always n/2

amazon.com/Single-Variable-Calculus-Early-Transcendentals/dp/…

Like thhat

Ah, Stewart.

but for example if n /2 is odd then the power of r that will be in the center will be odd

Very popular book. You might enjoy Simmons's Calculus book. It's well written and has reasonable exercises, not crazy ones.

however if n/2 is even the power of r that will be center will be even

I could always buy it. I would not buy the rental because it is in bad condition

hey @Owatch

you should also read pauls notes

Hello

Pauls Notes?

they were really valuable when I did calculus

1 moment

tutorial.math.lamar.edu

@TedS: I don't really see a good reason for it to be true. I think it's probably luck that it's true for n=4.

oke good thank you guys @TedShifrin and @BalarkaSen :D you know what I am doing now guys I am doing 1 week of reading the book and 1 week of just excerises so this way I capture more chapter and don't get bored

I found a book by George Simmons, for Precalculus.

blergh at this user's questions.

I need to finish grading, then I'll think harder, @Ted.

they have nothing whatsoever to do with algebraic topology.

lool

Which is why you should remove the tag...

this is just pure algebra

Yes, @Owatch, the Calculus with Analytic Geometry. Nice book.

But don't spend money you don't have, @Owatch. You can certainly find lots of problems on-line.

Yeah, it is expensive.

should suffice

khanacademy and pauls notes

Khan academy is nice.

Cheaper on Ebay

but ofcoz learning calculus is about solving problems and asking the right questions

@Ted: As far as I can tell his argument for $n=4$ works perfectly well in all dimensions

I haven't read it yet, @MikeM.

@MikeMiller What is the statement?

@TobiasKildetoft here

@MikeMiller Ahh, too much alg top for me :)

It's just a question about bilinear forms, really.

:P

Wait... you're serious?

and knowing like one fact about characteristic classes, yes

Poincaré duality in terms of nondegenerate pairings ...

@Ted: I posted it. His argument works fine.

And he already responded. Such math in real time :P

Now if only I can find an apartment so quickly ... :D

We found one last night, @TedShifrin :D

Gee thanks.

Sure, any time.

hm ok as I read this chapter I came up with this interesting way can you derive that center is subgroup of centralizer using group actions I will think about it when I come back need to go submit my assignment for physics

Group actions are awesome, @Karim.

OK, outta here.

Later @TedShifrin

@robjohn hi

@Vrouvrou how are you?

not well

@robjohn can you see this math.stackexchange.com/questions/1284540/a-counter-example

is the example given in this answer is right math.stackexchange.com/questions/1267849/…

?

interesting : I didn't know the category of covering spaces over $X$ preserved pullbacks (not that it's hard to prove, just a nice fact).

@robjohn Can you see deleted answers?

Hello, @Sawarnik

Hey, @BalarkaSen

Been doing any math lately?

Yes :D

Though the type you wouldn't like.

Right.

:D

@BalarkaSen Are you done with things like category theory?

I don't like category theory, though I certainly know a little bit about it.

:)

@Hippalectryon Hey.

@Sawarnik Hoi

@hipp Can you think of a simple solution to this? math.stackexchange.com/questions/1281770/…

@Sawarnik I see no obvious way.

@Hippalectryon Hmm, pigeonhole something!

@Sawarnik That wouldn't be too easy though, there would be nested pigeonholes

:O .. I think a really simple solution would be lurking somewhere.

Half pigeonhole theorem : if you put one pigeon in two half pigeonholes, then at least a rational fraction of a pigeon is in one of the half pigeonholes

I don't see why that^ holds? :D

Old pigeon magic

Quantum pigeonhole theorem : Any pigeon lies in any pigeonhole with positive probability.

@BalarkaSen Only in finite potential pigeonholes :P

Fair enough.

Also, if you have two pigeonholes and a single pigeon, whether probability of being in one is greater than being in the other or not depends on the spin of the pigeon.

@BalarkaSen heh .. seems like you have been studying physics?

Oh god. I'll get spinning pigeons for today's nightmares.

@BalarkaSen Here, a quantum star cluster for you

@BalarkaSen When did you start studying these things? :O ... don't tell me its in the course.

@BalarkaSen No two Fermionic pigeons can occupy the same pigeonhole

lol, that's classic @Rammus

Pigeon orbitals ftw !

Only two pigeons with strictly the same symmetry can interact :3

OK, let's write up a paper on bosonic pigeons.

That'll be more interesting

I bet we'll encounter the superpiegonity phenomenon.

@Sawarnik The starred message

@BalarkaSen We need to be wary that our pigeon curvature does not collapse into pigeon oblivion

Welcome to the Pigeon Mathematics chatroom

pigeonvalues messes things up.

OK, enough on pigeons.

bombard pigeons, to create new ones.

@hippa send

Is your pigeon getting messy at this late hour, @Balarka?

Hi @ted @BalarkaSen Have you got a pet pigeon ??

Right, it's bedtime, as you so kindly pointed me out, @Ted.

I never ...

salut, @Ramanewb

Just got out of an impromptu review session. I always seem to roughly double my office hours...

Well,@MikeM, maybe when you retire you'll get hours of speeches about your dedication and influence as a teacher ... :P

Actually, if your students take your office hours seriously and many attend, that's wonderful, and you're a rarity.

No, not really... I had about six today, out of 40. One or two that really needed to be there, but there are plenty more that did and weren't here.

Ah ... Yeah, frustrating

Why do students not come at office hours, though? I mean, it's meant for them and their questions. I have heard similar complaints from TA's and professors.

Somewhere between embarassment and laziness.

The majority of mine always have, @Balarka, because I push quite hard and most of them have fun at office hours.

oh, the grad students in this uni don't have fun doing office hours with my prof, I guess. he's usually pretty grumpy with someone if he/she can't figure something simple out (although equally friendly with someone when he/she asks/figures out nontrivial stuff)

Ok, some convex combination of embarassment, laziness, and the professor's bad.

hmm, must it be convex? :D

who cares if it is convex

go to sleep, Balarka :D

@TedShifrin: The scalars correspond to percentages...

Are you suggesting some professors are 300% bad?

I understood that, Mike.

Harsh words.

Maybe you're talking about me here. :P

No, I'm suggesting the percentages might add up to more than 1 :)

combinations in which topological vector space?

Go to sleep, @Balarka.

@MikeMiller And you were lecturing me about lecturing you about when to sleep :P

shrugs

Well, I got all my sleep already in my teens. I don't need it anymore.

Growing lads need it, you see

oh, a fine excuse.

i'm heading to bed.

and while I do so, this looks like pretty good stuff to me, if anybody is interested.

I haven't gotten much done today... got here, did some reading, got lunch, two office hours... yikes

Yes, Sullivan is a good writer. But you should probably learn cohomology before learning equivariant generalized cohomology theories, @Balarka.

Just a suggestion.

I think we shamed him off to bed.

He's back, @Ted

Oh, hell, apparently not :P

Stop pinging him in his slumber, Mike.

@MikeMiller Yes, sure, I am not reading it. I just skimmed through that stuff and it seemed good.

Localization is an essential tool in homotopy theory.

@TedShifrin conspiracies, damned conspiracies!

Does anyone have a (gentle) reference which introduces the connection between fundamental groups and Galois groups?

Those lectures were around my undergraduate days at MIT ... those be old!

@Ted: Those don't go through for like 15 minutes. He came back of his own volition...

hi, @AlexW

@AlexWertheim: Now you're going to get him going.

Hello @Ted and friends. :)

@AlexWertheim Well, Szamuely is good.

Connection or analogies?

Who, @Balarka, Mike? I confess I was interested in his response, but I guess there was all this sleep talk before I came in...

Also, Doaudy-Doaudy is nice if you don't want to dig that deep.

Just learn the fundamental theorem of Galois theory and the fundamental theorem of covering spaces ... and you'll see it! :P

@Balarka: I think you're off the deep end.

I'm not really sure, @Ted. :)

Ah, so there's a fundamental theorem of covering spaces.

nods

@TedShifrin Er, fundamental theorem of covering spaces isn't really enough to convince me of anything.

There is even a galois connection for posets :P

What is more interesting is the short exact sequence $1 \to Aut(X, Y) \to Aut(X, Z) \to Aut(Y, Z) \to 1$ of deck transformation group corresponding to the chain of galois covers $X \to Y \to Z$

I'm familiar with the basic rudiments of the FTGT, but someone remarked to me some time ago that Galois theory has a tight correspondence with the theory of fundamental groups. I didn't get it at the time, but I saw a question the other day where it came up, and I became curious again.

which, of course, resembles the short exact sequence of galois groups.

Oh, he'll never go to sleep now :D

I'll be honest, I don't know what a lot of those words mean. But I'll try reading a bit.

You don't know what deck transformation is?

No. I don't know what a Galois cover is either.

ok, try reading up some of those from Hatcher.

Ok, I'll try looking into some of the references you suggested. It's possible it might just be over my head at the moment, but I figured I'd ask. It seems like there must be some very beautiful ideas involved. Thanks!

@TedShifrin hey ted!

hi @Stan

Anyway, there really is quite a deep connection. I only "know" about it, don't really understand any of it. You can make an almost-convincing connection by considering branched covers of $\Bbb CP^1$ (equivalently, complex Riemann surfaces) and Galois groups over $\Bbb CP^1$. These two categories coincide nicely (keyword : function fields).

All of these are in Doaudy-Doaudy.

Now I am really heading to bed. See ya!

I am hoping to take real analysis this fall :D @TedShifrin I'm going to take the regular sequence. I heard honors is a lot of work.

hi chat

Ok, cool. I'll look into it. Thanks for your help again. Goodnight!

Even "regular" will be a lot of work. The honors is the crazy course, I believe. You'd have to remind me of numbers.

P.S. What's wrong with a lot of work?

You sound like the students who have dropped my class. GRR.

a lot of work is fine, as long as you've got enough power :)

don't horse around, @semiclassical :P

A lot of work is fine as long as the force is perpendicular to the speed :3

speed is a scalar?

a vector

no, speed is a scalar.

velocity is the vector.

@TedShifrin I love working. But I am taking honors econ and I worry it might be too much to do both.

(yes, i probably should thwap myself for that last pun)

Anyhow, @Stan, if by honors analysis you mean the honors sequel to the Spivak course, it's pretty ridiculous. But I'm not sure which courses you're talking about.

Ah ok I din't know there were separate words

@semiclassical, you have gotten quite revolting.

Yes, that's the one @TedShifrin. Why is it ridiculous?

the joys of being a physicist with a terrible terrible sense of humour

well, it's supposed to be the "next" course, so it should be multivariable whatever ... but it's off in Banach spaces, etc.

@Semiclassical Surely you didn't see out pigeon quantum physics

mmm? can't say i know what that is

@TedShifrin I like Banach spaces :P

heh, nice. superposition of spinning pigeons

yes, @Stan, and presumably you already know multivariable stuff concretely, but those students often don't.

surely those pigeons are beurrés.

i actually hadn't had to deal with any banach stuff up to this year. didn't run into it in undergrad, and definitely not in grad physics

well, grad physics should be full of Hilbert space ...

true, but the amount of formalism/rigor varies

and while the word "Hilbert space" is certainly par for the course, "Banach space" isn't

yes, I suspect physicists don't care much about $L^p$ for $p\ne 2$.

aye.

more precisely, the phrase 'Banach algebra' is what i've been seeing

arxiv.org/abs/math/0612487

for a typical example

@robjohn there is a problem with the answer

@TedShifrin I always feel there is more to learn, but yes the multivariable I think is safe to say solidly under my belt.

I just started worldscientific.com/worldscibooks/10.1142/8501 and it has introduced a term "gap functional" which is not in Wikipedia nor MathWorld. I am not trying to learn nonstandard mathematics. Should I be concerned?

I can always send you my finals, @Stan ... but you're still busy :P

That would be great! I have to take the placement test over the summer. I'll ask you for them in a few weeks and I will take it for real with time and everything.

@TedShifrin oh, btw, my mother said if you ever come to Chicago, you are welcome to dinner at our house.

She said she would gladly make dinner.

LOL, that's very sweet. I'll be in Chicago next spring, actually. :P

Yes, I am looking forward to meeting you in person hopefully :D

Well, you may live to regret that :P

Hahahaha

Doubt it. Unless I am required to sustain a conversation on real analysis :P manifolds I think I could muster a few comments

well, I'm sure you'll ketch up.

ugh

avoids slap

hi all!

Salut @Gato

@Gato o/

@TedShifrin i love complexe analysis :P.

@Hippalectryon ça va bien ?

@Gato Ca va :-)

T'as bien réussi à l'examen?

@TedShifrin C'est demain matin, mais je suis prêt.

Formidable.

So much for getting anything done. At least I have a good excuse now: Bhargava is talking this week.

LOL, yes, goofing off is always a good excuse to prepare for an exciting colloquium, Mike.

@gato Examen de quoi ? Que des maths ?

@Ramanewbie Analyse complexe.

@Gato Beuh... Trop compliqué pour moi !

@TedShifrin Le plus difficile c'est de ne pas se tromper sur le calcul de l'indice.

Pas grand'chose ça. C'est typiquement géometriquement évident.

@Sawarnik yes.

@Ramanewbie Pour l'instant, mais c'est de loin (avec les groupes) la matière la plus intéressante.

@Gato Tu fais aussi de la physique ?

@Vrouvrou I don't think so. Look carefully at the definition of $C_\theta$

@Gato :c ne lui parle pas de groupes il est au lycée xD

@TedShifrin Oui mais si je me trompe de signe maladroitement je suis foutu.

@gato Oui je ne sais pas ce que c'est !

Il faut tout simplement regarder :)

@Hippalectryon haha, autant pour moi.

@gato Je ne fais même pas de complexes encore

@Ramanewbie Seconde ?

@gato Oui...

un de ces jours, @Ramanewb

@Ramanewbie Tu as le temps, mais ça arrive vite.

@ted 3 or 4 more years to wait... It's long ...

@robjohn $C_{\theta}=\{u\in C, \sup(|x|^{\theta} u(x))<\infty\}$

@Ramanewb: Maybe you'll decide to do littérature instead of mathématiques.

@ted Arg Noooo !

LOL.

@ted Yes, 'lol'...

Nothing wrong with literature >.>

Moi, je me suis spécialisé en littérature française aussi bien qu'en maths.

@hippa I couldn't do litterature all day long. Two hours is fairly enough for me...

@Ramanewbie You probably couldn't do maths all day long either. I bet you can program weird stuff all day long though :3

@hippa But I would be more able to do maths all day long than doing French all day long ;(

@hippa And I would be able to do more maths than litterature in a row.

@Ramanewbie Je préfère la littérature. C'est juste que la littérature enseignée au lycée est nulle.

@hippa Yes, that, I could...

@Vrouvrou Yes, so $|x|^\alpha\in C_\theta$ as long as $\alpha+\theta\ge0$. That is, as long as $\alpha\ge-\theta$.

@Gato Pourquoi c'est nul ? Ce n'est pas inintéressant je trouve

@Gato Les matières enseignées au Lycée sont nulles FTFY

@hippa WRONG

@hippa The WAY they're teache is bad

Le programme est intéressant, mais la manière dont c'est enseigné est horrible

@Ramanewbie On y étudie par la littérature, mais juste des connaissances brèves, qui ne mènent pas à la réflexion, et les fameuses figures de styles..

Et avec la réforme du collège ça ne va pas s'améliorer

But $u(x)=|x|^{\alpha}$ $\alpha <0$ is not in $C$

@robjohn

@gato Non. Tu dois confondre avec le collège francais... Au lycée on voit le commentaire et la dissert...

@Vrouvrou $C$ is not mentioned in the question, is it?

@Vrouvrou Oh, it is...

@robjohn no but it is a part of the definition of $C_{\theta}$

@hippa Heuresement que j'ai fini le college alors...

Wow, lots of you math-y people here. I'm not one of you but I thought you might be interested in this question over on ELL.SE

@Ramanewbie Oui des méthodes, pas de réflexion je suppose que dessert pour toi c'est thèse anti-thèse foutaise ? :p

What is ELL.SE?

@gato lol non, on est ammené à la reflexion je t'assure...

English Language Learners @TedShifrin

@TedShifrin It's English Language Learners. :) We help people learning English.

@Ramanewbie e_e

@hippa Yes yes it's true (sometimes a bit)

@Ramanewbie Quand je lis "Chaque phrase est une forme linguistique indépendante, qui n'est pas incluse dans une forme linguistique plus large en vertu d'une construction grammaticale quelconque." Là je dis, oui ;).

@gato Rien compris.

Je doute qu'il y ait beaucoup plus de réflexion en Français qu'en mathématiques on l'on s'attend à ce que vous appliquiez comme des robots des formules qu'on vous balance sans contexte @Ramanewbie

@Hippalectryon +1

ah, thank you, @Catja