@SanathDevalapurkar Well there is a question where you are suppose to show that if $s: |K| \to |L|$ is a simplicial map, then this map will induce a simplicial map from $|K^m| \to |L^m|$ for any m. I'm wondering what exactly is meant by "induced"? Clearly it cannot be the same map s?

@user128779 It means the "new" map is obtained from the "old" map.

Heya @SanathDevalapurkar

For example, every group homomorphism $g:G\to H$ induces a group homomorphism $g^n:G^n\to H^n$ with $g^n(x_1,\ldots,x_n)=(gx_1,\ldots,gx_n)$.

@SanathDevalapurkar Fine. What about you?

You mean math?

@SanathDevalapurkar Well, 1st terminal this year went bad. Especially at math =P

Exams. There are in total. 3rd being the final.

It's weird, the educational system in here.

@SanathDevalapurkar So what've you been upto these days? Fancy category theories?

@SanathDevalapurkar I am not a prodigy.

There's all sorts of free TeX compilers and editors.

I am a fool who fiddles with stuffs.

@BalarkaSen, either is fine

@Kaj Addicted.

Like I said.

@SanathDevalapurkar Not even a genius. Knowing a bit of galois theory doesn't make anyone genius.

@Sanath, you using PC, not Mac?

If I recall correctly, Arnold taught high schoolers galois theory of covering spaces in paris, @Sanath

@SanathDevalapurkar, Why not use TexWorks?

@Sanath Perhaps my advise counts as a grain of salt but I always use writeLaTeX for noteworks.

Just an online latex editor.

I think @Kaj's suggestion is good. I'm expert on Macs, not Windows crap :P

It is! I've used it a decent amount, and I like it a good bit.

If only TeXWorks would do the proof-writing, too, @Kaj would pay for it :)

@TedShifrin Windows is not crap. tries to smack @TedShifrin

@TedShifrin heh heh

My only complaint is that I wish you could see what you are typing in real time with these things instead of having to compile every time. There's a Tex editor for Linux that will do that, but none for Windows AFAIK.

@SanathDevalapurkar You study a lot many things togather!

TQFTs, Algebraic geometry, category theory, etc.

TeXtures for the Mac used to do real-time typesetting, @Kaj, but TeXShop doesn't.

no, @Sanath... I grew up on Mathematica.

@SanathDevalapurkar Have you dug in at the book about topological galois theory I referred to?

You might have some fun.

@TedShifrin Yuck mathematica

@Balarka: It's clear you and I disagree on everything. Go to bed and get healthy.

@SanathDevalapurkar It's written by Alekseev.

I feel like an old ignoramus amidst young geniuses right now. I think I'll get something to eat...

A compilation of Arnold's note.

How do you think I feel, @Kaj?

@KajHansen You know a lot many things I don't.

I am a foolish fiddler, nothing more than that.

@TedShifrin PARI/GP.

@SanathDevalapurkar Deserved it.

@Sanath Don't you have school today?

@SanathDevalapurkar It's online I think.

@SanathDevalapurkar Cool.

Don't you start back in August, @Sanath?

@SanathDevalapurkar [about the book] it's a piece of cake to you if you know about Riemann surfaces and monodromies.

in case you don't, it works through examples and excercises.

Were you a bad boy, @Sanath?

I doubt you need a headstart, @Sanath.

@SanathDevalapurkar Since you know about topology, you know about covering spaces right?

have a look at Deck transformation groups.

why bother?

@Sanath: Some people call them covering transformations.

@SanathDevalapurkar I only know that covering spaces cover stuffs. It's not necessary to learn that to get a general idea.

As I said : I know nothing of topology.

Hurts my head.

it's totally ridiculous, @Sanath.

@TedShifrin "Deck" sounds much good, like you are permuting a deck of cards over the floor and looking at the action of the cover.

Yes,@Balarka, I personally prefer "deck transformations." But some books don't call it that.

@TedShifrin I have thought about reading Armstrong someday. But thought better of it.

@SanathDevalapurkar Totally lame.

Call them monodromy. Sounds loads better.

@SanathDevalapurkar Stop fiddling with terminologies and do the math.

@SanathDevalapurkar Health?

I've been teaching longer, @Sanath. I actually have been teaching in some form or other every year since middle of senior year in high school. So that makes 44 years.

G'night, @Sanath.

@TedShifrin And I thought I was slow with it ;-)

@SanathDevalapurkar Yes. You slither out whenever there is a mention of galois theory.

@robjohn: I had to fiddle longer than you had to fiddle :P

This reminds me of finishing AP calculus when I was a sophomore. Had to take a lot of bs classes the last two years to fill in empty time slots, and we didn't have one of those programs where you could take college classes.

@KajHansen Yuck calculus.

I am joking.

When I was in grad school I was super-duper speedy with those things, @robjohn, but that was about 40 years ago.

LOL. I wish I had a Spivak-type approach when I learned it though.

well, @kaj, look how well Drake did with that :(

@KajHansen Yuck Spivak

@Balarka. Enough already.

@TedShifrin Piskunov is the best,

@TedShifrin, apparently he's taking four math courses in the fall? Including 4950, that is.

Guess what, @Kaj. It's gonna be crash-n-burn.

@KajHansen Yuck 4950. Highly composite.

2 * 3^2 * 5^2 * 11

@TedShifrin Ah, it's only been 30 for me... that explains it :-)

No doubt @Ted, even if it were someone else. I cannot imagine taking that many courses. Even Sarah with her three maths was stretched pretty thin last semester.

Very ... almost imploded.

@KajHansen I am taking three courses too. Cannot say I am not exhausted.

Next semester I will relax.

Well, by grad school everyone does 3 courses ... but usually past a certain point homework is "optional."

I'm with you @PedroTamaroff. Had 2 big maths, a smaller math, and math methods in physics last semester. I was worn thin by the end.

And it didn't help having a ton of snow days.

I seem to recall someone suggesting it was a bad idea, @Kaj.

@TedShifrin This is weird. 1 out or 3 times whenever I move link-to-link in wiki I come back at etale morphisms. I wonder what the hell it is.

Or having a solely T/R schedule, @Kaj.

It's algebraic geometry with the standard topology instead of Zariski/scheme topology, @Balarka.

That's fixed this fall. I was worried about you retiring before being able to take your geometry!

@TedShifrin Eh... what's this standard topology crap?

@TedShifrin HA.

Ha? @Pedro

It's an exclamation.

I take it you said it?

ohhh, yes.

I hope next spring I have only students who actually want to be there and learn, @Kaj. Very frustrating that class.

I only get topology when introduced a metric. Otherwise it looks crap to me.

@TedShifrin Aren't you retiring?

well, there is a metric, I suppose, if you think of it as a manifold ... but metrics aren't antural @Balarka.

@TedShifrin antural?

natural, dammit.

I really do think that happens because it's one of the three choices one must take to get their degree. Between that and real/complex.

Right, @Kaj, and we now have a significant number of people who shouldn't be getting a degree. The new applied major will, presumably, make that more interesting.

I suspect the real class next semester might be the same way for the same reason, but I have hope with Dr. Fu switching over to Rudin.

Ohh, there's a new applied major? When does that track begin?

we're teaching complex every term, @Kaj, so maybe not. Although Pollack won't be a pushover.

@TedShifrin I don't even know what a manifold is. If I keep going on like this I will probably be back at asking what a topology is.

It's begun @Kaj. Only proofs required are 3100.

3100 was FAR worse than your geometry course. It was fairly easy, and we had to take it super slow.

far worse in what sense?

It seems like 3000 level courses are populated primarily by people taking the easiest route possible. Granted, I decided to take 3100 for the A, but man...

no, they are populated by everyone ... since they're prereqs for further courses.

The problem is that too many of our profs are giving generous Cs in those courses.

I argue that the cream of the crop is filtered out because they get overrides from other professors to avoid 3100, 3200, etc.

I override 3200 a lot, but not 3100 so much ... Only the people who get A's in 3500-3510, pretty much.

I don't want to teach just cream of the crop, @Kaj. Not at all. I want majors to want to learn/understand courses in their major. Sigh.

Oh, and to work hard. Sigh^2.

I don't know. The difference between our class from 4010 and our class from 3100 was truly night and day.

Can anyone come up with a field in which every extension is galois?

$\Bbb F_p$?

@TedShifrin Duh. Yes.

Every finite extension, of course.

Yeah, finite.

Otherwise things get creepy.

What about every extension is Galois, but the Galois group of the extensions aren't necessarily cyclic? :P

@KajHansen I am not sure. $\Bbb C(z)$?

yeah i think that'll do, right?

I'm trying to think....how would one even extend $\mathbb{C}(z)$?

$\Bbb C(\sqrt{z})/\Bbb C(z)$ is for example, an extension.

Oh yes. I see what you're saying.

with galois group $\Bbb Z_2$.

but you can come up with non-cyclic ones too.

in fact you have all finite groups as galois groups over $\Bbb C(z)$.

I'll have to think about that one. Kind of got me thinking about the inverse Galois problem too. I wonder how many finite groups they've confirmed that for.

ok i think C(z) is just what you want. i am going to bed, byes.

@KajHansen in $\Bbb C(z)$? every group

Oh no, over $\mathbb{Q}$.

it;s just that covering spaces of riemann surfaces with a couple of branches removed is a free group and every group appear as a quotient of free groups.

@KajHansen oh.

i don't know. they've got the monster confirmed with some high-figh technology.

I know it's kind of OT, but I remembered asking you to find an extension with the quaternions for the group $Aut(K/\mathbb{Q})$. And then you start thinking of all the bizarre groups like $Q \times D_4 \times S_3$ and that sort of thing.

@KajHansen no, i was thinking of semidrect product.

Oh no no, that's not what I mean.

if it was $D_8$ i could have done it easily.

What property does $\mathcal Q$ have that's quite rare, @Balarka?

The inverse Galois problem conjectures that any such bizarre group will have a corresponding polynomial over $\mathbb{Q}$ with $G \cong Gal(f)$.

@KajHansen yes.

@TedShifrin erm... not sure.

Very much worth understanding, @balarka.

@TedShifrin by that curly Q you mean quaternions, right?

I mean the quaternion group of order 8, yes.

it has lots of rare properties.

for example, Q8 is the only group isomorphic to Q8

Well, every subgroup is normal, yet it is not abelian.

@Balarka: If you're going to be idiotic, go to bed.

Thank you to @Kaj.

@KajHansen Never heard of that one before.

pretty rare, yes. does that help finding the galois group?

@TedShifrin I was actually having a different line of thought. I was going to construct a field extension of $\Bbb Q(z)$ with galois group Q8 and then specialize with Hilbert's theorem to $\Bbb Q$.

Well, answer my question first: IS there a quartic with that Galois group?

Oh this is fascinating

@TedShifrin no.

Huh? @Kaj @Balarka

A4, Z4, S4, V4. These are all possible quartic galois groups.

Not correct.

@TedShifrin I will list them correctly after getting sleep.

It can be done with basic galois theory.

But rather than regurgitating ... try to prove $\mathcal Q$ can't be there.

Go to sleep.

Every group with that property has $Q$ as a subgroup according to Google.

@TedShifrin i am ignorant about specifically thinking of a group that a field.

yuck groups.

fields are loads better.

@TedShifrin, 7.6.13C is truly evil.

Ah, that problem is cool. I think I stole it from D&F.

I remember the frustration building in my mind trying to solve that one after solving all the rest on that particular problem set. :P

And that's my fault?

Dr. Graham's, actually

I usually assign that as a grad problem, but I don't think anyone's ever done it, @Kaj.

haha! That seems to be a recurring theme...I noticed that many of the problems Dr. Boe gave us were grad problems for your class. (I downloaded your problem sets last summer).

So apparently I'm not the meanest teacher on the block ...

@TedShifrin What problem?

Galois theory stuff, @Pedro. I gave you my analysis one earlier :)

@TedShifrin Agh, I don't know anything about that.

=)

It's interesting, @Pedro, because it shows the ring of smooth functions is very much like polynomials. If a polynomial $f(x,y)$ vanishes on the $x$- and $y$-axes, then of course it's in the ideal $(xy)$. Surprising that this holds in the very non-Noetherian ring of smooth functions.

@TedShifrin Aha. I meant I don't know about Galois theory and stuff.

right, I know ... you will soon enough.

hello, please see my answer and comment if there is something wrong math.stackexchange.com/a/860893/42370

Hi @Chris'ssis :D .. Greetings ! :-)

Greetings :-)

@r9m I was reading some e-mails. How are you doing?:-)

@Chris'ssis cooking sth :D ..

@r9m That's great! :-)

@r9m That limit is correct. How did you compute it?

@Chris'ssis Elementarily =P ... took me 4 hrs of work :| .. How do you come up with these crazy problems ? :D

:-)

@r9m I think I didn't come up with crazy problems yet, but they are about to come out in the next period of time. :-)

@r9m aren't they just candies?

:D

@Chris'ssis so the Book is coming out soon ??!! :D

@Chris'ssis that candy almost choked my throat from inside :P lol (cough)

@r9m No, there is still a lot of work to do for that.

hehehe :-)

@Chris'ssis bbl ... lunch time for us :-)

OK

I don't know what I am doing.

What should one do when I have developed the question so that the earlier answer no longer applies?

Ask a different question :-)

@r9m awww....oooo....

Hello, please if $f(t,u(t))$ is continuous and $f(t,0)=0$ can i say that $lim_{t\rightarrow +\infty} q(t) f(t,u(t))=0$ such that $1/q \in L^1((0,+\infty))$ and $lim_{t\rightarrow +\infty} u(t)=0$

@Sawarnik ? :o

@r9m eu .. were .. cu.. cooking!

@Sawarnik lol .. I can't cook kya ?

@r9m oo...what were you cooking?

maybe...fish?

@Sawarnik sausage :P *ps: don't like fish at all :P

@r9m umm...what is the hindi for sausage?

orly..i heard that all bengalis like fish, fish, fish

@Sawarnik idk ..

@r9m then i shud google sausage

@Sawarnik I'm not your typical bong :P lol .. in terms of food habits atleast

@r9m eew...this sausage doesn't look very nice

@r9m true....how good is ur cooking?

@Sawarnik so and so :P

:P

am I correct $\sum_{i \in S_1} \sum_{j \in S_2} a(i).b(i,j).c(j) = \sum_{i \in S_1} a(i) \sum_{j \in S_2}c(j).b(i,j) = \sum_{j \in S_2} c(j) \sum_{i \in S_1}a(i).b(i,j)$

@Euclidean assuming the sums are finite, yes

if they're both infinite you have to worry about conditional convergence

Thanks.

@blue are you familiar with motivic galois theory?

i was looking for an intuitive (yet not crude, like the one the paper in the comments give) explanation of this.

personally, if find it much more natural to consider the group simply as PSL(2, Q), but i guess there is more to it.

nope

why would it be PSL(2,Q)?

automorphism group of Q(z)/Q

wait, no, that'd be PGL(2, Q), right?

well that's just boring - that cannot tell the difference between different transcendentals.

@blue well, there ain't any other satisfactory explanation.

unless you can build up galois groups over formal laurent series constructively.

are you taking Gal(pi)=Z literally?

@blue so you're saying that i shouldn't?

\:

If we "replace" the quantity $\mathbf{w}\cdot\mathbf{x}_i$ with $f(\mathbf{x})_i$, where $f$ is a member of a Reproducing Kernel Hilbert Space, with associate kernel $k$, what is the rule we use to do so?

do you take the field with one element literally?

@blue why shouldn't i?

it's a placeholder for ideas that haven't been fleshed out yet.

@BalarkaSen srsly

@blue oh, you mean field with one element.

i misread.

i thought you meant field with on generator.

@blue this comment of Qiaochu makes me take it literally.

rather than taking it litterally.

@BalarkaSen but that emphasizes my point: Qiaochu had to expand to a broader scope where we interpret Gal(Q(pi)/Q) differently than with the literal, naive perspective.

@blue eh, fundumental groups generalize galois groups, and that ain't any litter.

we know that both very well.

If $f : G \rightarrow F$ is a group homomorphism, is $H$ isomorphic to the direct sum of the image and the cokernel of $f$?

@BalarkaSen not sure what you're saying

or is that not a statement that is generally true...

@Alex if Alice is 15 and Bob is 12, how old is Cindy?

@Alex what's the definition of cokernel?

do you recall?

codomain / image

Not enough information is given to determine Cindy's age

@Alex right. what part of "if f:G->F is a group homomorphism" says anything at all about the letter H?

Ah, I misread again.

what the hell is H?

ohhh lol

i meant

$f : G \rightarrow H$

sorry

then no

^

what @blue said.

you're basically asking "given any subgroup K of H, is H a direct sum of K and H/K?"

because any subgroup K<H can be the image of a group homomorphism (e.g. the inclusion map)

pretty much yeah

there are examples of nontrivial groups K such that H/K \cong H

those examples couldnt be finite groups right?

yes.

they aren't finite, as far as i recall.

@Alex for counterexamples, pick any K in an H that is not normal. then it cannot be a direct summand.

@blue pffft.

@BalarkaSen that doesn't prevent $H \cong K \oplus H/K$ though

oh right.

hell.

well, take H = Z_4 and K = Z_2. H is not isomorphic to Z_2 \times Z_2.

another instructive example

i have to think more than to spell big names to get examples like that.

thanks for answering my question by the way

no problem.

@blue abandon all hope he who answers this question

generalizing the Dummit-Spearman-Williams criteria for septics should be really tedious.

hmm, just realized my other computer has been making pinging noises for days while I've been on my laptop

@blue heh

i have chucked out my audio gadgets loads time ago

now, about the galois theory. @blue i don't think the fact that fundumental groups generalize galois groups is a completely non-literal statement.

the literal understanding of "Gal(Q(pi)/Q)" would be the group of Q-automorphisms of Q(pi), which is the projective linear matrix group you mentioned.

@blue depends on what you'd call literal. Gal(Q(pi)/Q) has lots of ways to make sense, it seems.

well, if the exotic, advanced understanding that relies on analogy and is understood more than anything by just elites is the literal understanding, then what is the understanding I just described?

[psst : i am not sure we can call "Gal(Q(pi)/Q)" the group of Q-automorphisms of Q(pi). do we have any notion for transcendental galois extensions?]

what i am saying is that it's just not very convincing to work with Q(z)/Q all the time. it doesn't add anything fun to the notion of galois groups.

wait a sec. i think i have an idea @blue. what's the riemann surface of w = arcsin(z)?

@blue yes, i did it.

i found a plausible explanation.

note that $2 \pi i$ is the period of the function $\exp(z)$ defined on the complex plane $\Bbb C$. Consider the multivalued function $w = \log(z)$ which has infinitely many sheets corresponding to $\log(z) + 2\pi i$, $\log(z) + 4\pi i$, etc. Now these sheets are all away from the branch point $z = 0$ and being cut through the branch cut and pasted upwards so that any point that loops around the branch point winds upwards to the sheets above instead of coming back.

So the monodromy group is infinite cyclic, i.e., $\Bbb Z$ and monodrmies generalize galois groups.

@blue

mmhmm

=D

perhaps this perspective worths to have it's place as a comment in here? i mean, it's much easier to understand and much less voodoo than the explanation of Qiaochu.

Finding last digits through modular aritmetic is fun

@Chris'ssis I learnt a new crazy way of dealing with $\lim \displaystyle \int_{[0,1]^n}\sqrt{\frac{\sum x_i^2}{n}}\,dx$ :D .. btw how does the asymptotics look like actually ? :o

@r9m What is that way?

@Chris'ssis I edited that method in my answer to Jack's Q 4m b4

@r9m Yeah, I saw it.

@Chris'ssis how does the asymptotics look like ?

@r9m I need to find my paper ... (this might take some time) :-)

@Chris'ssis okay :D .. thanks :-)

Hi. Assume that $..., X_1, X_1, ..., X_1, X_2, X_2, ..., X_2, X_3, ...$ is a strictly stationary process, where each $X_i$ repeats for an unknown number of times (at most $M<\infty$), $X_i \in \{1,2, ..., M\}$. Is then $..., X_0,X_1, X_2, X_3, X_4,...$ strictly stationary? I would like to show that it is.. Actually weak stationarity is enough for me. It is clear that mean and variance does not depend on time but I am not sure about covariance

I started a bounty for 100 reps, please be my guest! Thanks a lot! math.stackexchange.com/questions/839956/…

hello, can someone help me please {math.stackexchange.com/questions/861061/…}

Does anyone here understand the lie algebra?

@r9m there?

@DanielFischer does it bug people a lot when a speaker uses the wrong noun gender?

@G.T.R It bugs those who know which gender the noun has. That is not quite a negligible minority yet.

anybody here?

@VibhavPant are you here?

[^This question is (hopefully) not as stupid as it sounds by the way: it's very technical software.]

The GAP manual and tutorial aren't very helpful.

Using "LogTo" should work. Maybe it's the laptop/installation . . .

Yeah, SaveWorkspace(filename) and the command line -L aren't working either . . .