Well in my case I am looking at the divergence of a vector field X.

So I define this normally with a volume form.

But the claim is that it is invariant in the local volume form I choose because the divergence is just the "trace of the linearisation of X".

Exam went well yesterday. They did call me once, but that was really a non-issue. Just a student wondering whether I really did mean to write $\sigma^{-1}$ when I asked them to find a $\gamma$ such that $\gamma\sigma\gamma^{-1} = \sigma^{-1}$. And yes, I really did mean that, since otherwise the exercise would be trivial.

In the same paper there is mention that $d(\alpha(X))_x = \alpha_x\circ A_x$ for a (contact) 1-form $\alpha$, a contact vector field $X$, and its linearisation at $x$, $L_x$.

@TobiasKildetoft it’d just be “find an element that commutes with sigma”, so yeah

@Semiclassical Right, and I really did not want the identity to be an option

Oof, yeah

@TobiasKildetoft just choose a sigma in the 2-torsion :P

It is not like this was a hard exercise anyway, since they had already written $\sigma$ as a product of disjoint permutations in a previous part

right

And the book has the formula for what happens when you conjugate a cycle

Were they given a particular sigma?

Yeah

Tomorrow the grading starts. 93 students did the exam, so lots of work to do

Godspeed

well our year has like 2XX people

and we sat 4 tests

*Gradespeed

two of the students handed in 25 pages each. I hope this is because they used a new page for each subquestion, but I didn't open the envelopes to check yet

Daaaang

@user193319 suppose it is, pick a point $x$ and a compact neighbourhood of said point, project the nbhd down to the different $\Bbb R$s, what happens?

That is a huge amount of material to grade

@Semiclassical As I said, hopefully they just wasted a bunch of paper. I mean, I don't see how you could meaningfully write 25 pages in just 4 hours by hand

@anakhronizein I am not sure... I’ve never seen that before.

Oh that does not sound fun

25 pages is 2 many pages

I will keep exploring on google then, thanks anyways!

"25 = 2" - Eric Silva, 2018

indeed

(not clickbait)

<--- is a crank

@EricSilva Actually one student did hand in just 2 pages. Which means it is either great or terrible

that's a lot of variance

Most seem to have used 4-6 pages, which is about what I would expect

What topic(s) was the test about?

@TobiasKildetoft What subject, group theory?

Interesting proof of Poincare's lemma: Say $\omega$ is a closed $k$-form on a ball $B$ centered at 0 in $\Bbb R^n$. Consider the flow $f_t : \Bbb R^n \to \Bbb R^n$ such that $f_t(x) = e^{-t} x$, which preserves the ball. This gives rise to a vector field $X$ on $\Bbb R^n$.

Define $F(\eta) = \int_0^\infty f_t^* \eta dt$. Then $F(\mathcal{L}_X \omega) = \omega$. By Cartan's magic formula, $\omega = F(d \iota_X \omega) + F(\iota_X d\omega)$. The second term vanishes as $\omega$ is closed; $F$ and $d$ commutes by differentiation under the integral sign. So $\omega = d(F(\iota_X \omega))$.

more like, introduction to group theory via permutations

@AlessandroCodenotti algebra. One question on CRT, one on permutations, one on rings (and unique factorization), and one on Sylow theory with a bit of representation theory

nvm

I see, sounds cool

What are $\gamma$ and $\sigma$?

$F$ is a cool operator. It's kind of like averaging $\eta$ over the flowlines

@AkivaWeinberger They had been given the permutation $\sigma$ and were asked to find such a $\gamma$

Ah

Hello, is it true: $f:X\to R$ is not bouded from bellow means that there exists $u_n\in X$ such that $||u_n||\to+\infty$ implies that $f(u_n)\to-\infty$ ?

@TobiasKildetoft right

@AkivaWeinberger hello how are you ?

@Vrouvrou I think you mean "and" instead of "implies that"

And I think that's right

yes

and

thank you for the answer

@Akiva Did you check out Trout Mask Replica?

but you are fine ?

Yeah

@BalarkaSen No

@Vrouvrou I'm in finals week, a little stressful

Er, weeks. Finals fortnight?

good luck for you

Hey everyone

Heyup

Currently I am on train

@BalarkaSen isnt this the normal proof

Either in or on, I'll let you guess which

it's certainly how i do it

@EricSilva hm, is it? i guess

my proof is H_{sing}^k(R^n) = 0 looool

i use any excuse to invoke my bud the magic boi

i like this proof

H_{♫sing♫}

H_{I CANNOT GO BACK TO YOUR PROUD LAAAAND}

i love this album so much

H_{covers ears}

H_{this is real music}

im 13 and this is REAL music

"Floral Shoppe フローラルの専門店" by Macintosh Plus is the best music because it's the only album i have listened to

what is that

the first popular v a p o r w a v e album

oh right

i think i actually like vaporwave

me too

i mean it's just slowed down 80's elevator music but it's nice

lol

i thought it was like a joke but it's just not bad

yep

ugh i need to read heat boiz

@BalarkaSen The Japanese word for "floral" is "furouraru"?

(I mean, they took it from English?)

idk man thats what the album label says

its a e s t h e t i c c

It's the word "floral" written in Japanese letters

I see you've turned crap music into a homology functor?

Heh... you said "functor" hehehee

@TedShifrin Trout Mask Replica is great music

DogAteMy is removing me? :)

ME NO TYPE GüD

also, it seems that I am so allergic to the idea of a functor that I type "function" every time I try to type "functor"

(it happened four times in writing out the above sentence)

You're just in your own category, @Xander.

YAY!

I need to learn category theory someday

you might need more time

Whoops, typo

Not today

damn ninja edit

now my comment makes no sense but that's ok

Someday in the far future

Eric, we rarely make sense in here.

We're rarely if ever expressible as the ratio of integers, after all

You've renounced your position as an algebraic integer?

Hello everyone!

sup nerd

I am not a Liouville number

Ah, the chief irrational one shows up on cue.

Or a Thoreau number @BalarkaSen

I love Walden Pond.

the writing or the place

Yes.

lol

With "Pond" in there it's really only the place.

i guess

idk if i like thoreau

idk if i dislike thoreau either

Well, what about the existentialists?

I'm not sure if I like Gödel; I have a sort of incomplete view of him

(ba-dum tlachp)

Neon meat dream of an octopus

- Beefheart

pure poetry

i like some existentialists ive read but i havent read that much work by existentialists

i remember reading walden and not being excited

i dont like sartre

$\exists$

$\nexists$

I spent high school and the beginning of college loving Sartre and Camus.

i like camus

If you love doing mathematics, you have to respect Sisyphus.

I am in the Dostoyevsky and Kafka clan

I haven't read the Russians enough to comment.

Yeah, Balarka does often wake up only to find himself transmogrified into a beetle waving his legs in the air.

:D

i think i dont know if i like many of the "classic" american authors

i always am left unexcited by the """"""great"""""" works

eg?

idk the general stuff people put on high school syllabi

I read "The Trial" by Kafka and quite liked it

I like The Trial

like walden lol

waits for Demonark to make one of his usual crummy puns

@Eric I don't know the American high school syllabi

I'm not sure you have to be excited by literature to appreciate it, Eric.

I like Catch-22

by excited i dont mean excited

We read Silas Marner :(

Hey @TedShifrin :)

Hard not to find Shakespeare exciting — we read a bunch of his plays.

i just mean i havent felt like ive gotten much out of it

hi Perturbative

@Ted i like shakespeare like a lot

@TedShifrin Yeah we have to read Shakespeare too.

I don't like him

Does anyone here define their one-point compactifications using disjoint unions?

Excuse me Ted but my puns are high quality

My first college French literature course was on the poetry of Baudelaire and Nerval. That was eye-opening.

@BalarkaSen the man has sick flow

@Perturbative: Well, you have to throw in a point, so ...

@EricSilva I feel his plays are mostly unexciting compared to his contemporary European writers like Dante or Cervantes

@EricSilva He makes the heat flow badum tss

@BalarkaSen so like moby dick, scarlet letter, anything by twain

i dont think shakespeare is good for writing good plays

I don't dislike Mark Twain

i think he's good because he spits like fire

meh

his bars are mad

never been a fan of rap :)

see ive always been a huge fan of rap

yup

so maybe that's well correlated with me liking shakespeare

uh-huh

I see we've reverted to the bad music homology functor.

i mean i think both of the writers you mentioned write better things but they're also not writing in english, shakespeare is eminently english and that's an important thing

@TedShifrin not a fan of elitism in music

Does one really need to define $X^*$ as $X^* = X \sqcup \{\infty\}$ as opposed to just a normal union?

@Perturbative: Right. You need to add a point at infinity, so of course you have to talk about that point.

well english is not really my native tounge so i dont care much

@EricSilva Gatsby, for example?

i am not very fascinated by the language

i hate gatsby

My musical tastes are very "old-fashioned," Eric. Classical and folk. And that's about it.

@Perturbative What's the difference?

that doesnt make other things bad tho :P

Game soundtracks are the best music

They're just emphasizing that this $\infty$ point is not in $X$

I always thought Irish people wrote better English than the English people themselves, because it's so weird

What DogAteMy said ...

Joseph Conrad is fun to read because of the total weirdness of the text

the content becomes more important and the narrative is without a lot of fancy beautification

@AkivaWeinberger There's some pedantic issue in the way the author defines a disjoint union (see here : en.wikipedia.org/wiki/Disjoint_union#Set_theory_definition)

@BalarkaSen conrad makes me feel weird

love em

@Akiva this is amazing

@Perturbative: That's done in case you want to define disjoint union of a set with itself (or with another set having overlapping elements).

Heart of Darkness is a really jarring novella

@Perturbative We don't care; this is topology, we only care about stuff up to homeomorphism. In fact,

i obviously tend to read a lot of lit from a staunchly anti-colonial pov so it always makes me feel weird

that isnt to say that reading things like that can be good

@Eric yeah that's the Chinua Achebe argument

but still weird feelings

yeah i mean i am a big big big achebe fan

Wow, most of this literary discussion is way over my head.

since it's only $X\sqcup\{\infty\}$ in the set-theoretic sense, we just care about it up to bijection.

Usually it's just the math :)

i reread the african trilogy last month and it made me feel a lot of things

I actually like Achebe! I recently read "Things Fall Apart"

@Daminark Lemme see if I can find the full concert it's from

@TedShifrin we're edgy kids who have read a lot of stuff to be edgy lol

idt im particularly edgy i just like reading books that fall out of the mainstream

DogAteMy: Did you have another snow day for school?

@Daminark You should listen to Trout Mask Replica

because a lot of things about my life happen to fall out of the mainstream by circumstance

There's nothing shameful about any of that, Eric.

@Eric fair.

@Balarka im actually weirdly into nigerian lit on the whole

a lot of really good english language writers

@Akiva I will probably play this as background for my psets

achebe being like the granddaddy of the whole thing

unsurprisingly. everyone else speaks better english than the english people

Ah, Eric is going to be concentrating on "shithole" literature. ... Quoting someone of no consequence.

@Balarka aight

lmao

@TedShifrin that comment actually made me so mad i wanted to break something, like i was furious

well i still am

@Eric how much Ergodic theory did Smart do with you guys?

I was past furious.

none @Daminark

Who said that??

at least idts

@Balarka who do u think

@Balarka: You really didn't hear that in your press?

the trumpet or someone of his clan?

Chief idiot and bottlewasher.

@TedShifrin Nah, just finals

@Daminark we mightve actually done like a bunch but i didnt go to class that much

Two finals today, actually, which is strange

Ohhhh. Your penultimate ones of those 'til college, DogAteMy :)

Yurp

@TedShifrin I did not hear this at all.

You excited to be moving on, DogAteMy?

What a fuckhead

And now his doctor is lying about his medical exam, too, Balarka.

Or so I definitely believe.

What do you expect, he's a shithole president

@Eric lel

@Balarka ive started rereading a bunch of stuff actually, currently rereading marquez and going to move onto borgia soon after

not just achebe

Good thing Eric was never my student. I would have growled at his missing classes....

@EricSilva Marquez is 100/10

lol i used to miss class a loooooot

i super agree

I love Spanish surrealism so much

Borges, Marquez, Paz, ...

marquez and borgia are hands down my favs

I love Almodovar's movies ...

jesus

borges

dumb phone

lol

Paging Lucretia ...

@BalarkaSen Something like this?

LMAO

good meme

@TedShifrin tbf though i dont miss classes where i wouldnt do well regardless

@Eric I like to say that Borges's Labyrinths, Dostoyevsky's "Notes from the Underground" and Eliot's The Waste Land are respectively my versions of Quran, Bible and Upanishad

lol

Borges and Eliot were actually very much influenced from Quran and Upanishad, respectively

so its kinda fitting

I once tried to read Dr. Zhivago (Pasternak) and gave up.

But Balarka wouldn't read it, anyhow. It's romantic drivel :P

i feel like none of the top tier latin american writers are brazilians

there's not much good portuguese reading that's like marquez level

@TedShifrin "Notes" is a romance novel :)

Oh.

in the same way The Fly is a romantic comedy film

lol

What's the Upanishad

(by that I mean it's edgier than pure Romance)

A religious text of some species, sounds like

@Akiva Ancient Sanskrit texts, part of the Vedas

I'm not sure whether I saw The Fly or not.

Ah

they fall in the category of Hindu religious texts

i guess

You can make a r

@TedShifrin It's a good movie

@balarka i read some of the upanishads my first year but i dont really remember anything

But only if you like a Freudian take on love and aging

@BalarkaSen works for me, though there’s also a solid thread of Western lit in The Waste Land (Dante, the grail quest, Tiresias etc)

otherwise it's a neat sci-fi hollywood

@Semiclassical Haha true

I couldn't make it the Bible because Dostoyevsky was taking that place for me

lol

eligion out of this

So it’s sorta a collision of east/west

I honestly expected someone to finish that quote

@EricSilva I don't know a lot of Upanishad.

And, heck, the third section is The Fire Sermon and the fifth is What The Thunder Said

there's a copy flying around in my father's bookshelves

there was this period of my life where i was reading a bunch of ancient religious stuff but it's all a blur now for reasons

@Semiclassical TWL is so full of references from everywhere that's it hard to mark it down as _______ influenced

yeah

Eliot was very much into Christian philosophy I think

personally

“These fragments I have shored against my ruins”

Is a pretty good summary of the poem’s relation to east/west as a whole

true

@BalarkaSen yes, though more so in his later work

Right, I was reading "The Journey of The Magi" this Christmas

rereading rather

When you say The Fly do you mean the 1958 one or the Cronenberg one?

Cronenberg one!

there's a the fly that isnt the cronenberg one?

TIL

1958's thing is fun to watch

@Eric yeah that's what inspired Cronenberg to make his version

he completely changed the plot of course

@EricSilva Cronenberg's one is a remake

<-- knows precious little about film

More like they are inspired by the same short story tbf

ah yes

Eliot references to the mysticism of St John of the Cross and Julian of Norwich in his later work

@Eric i mostly get to know about stuff by obsessing over certain directors

lol

reminds me that I still haven't finished Cronenberg's novel

it made me sick when i was like halfway through

@BalarkaSen the usual marking point in that regard is Ash Wednesday

Ah yes

With the Ariel poems coming after that, including Journey

I think

True

I think that is right

And eventually you have Eliot writing verse for a Christian pageant (Choruses from ‘The Rock’) which is explicitly Christian

hmm i havent seen those

Yeah

My book of Eliot’s collected works included that, along with a few of his plays

there’s a lot that can be said of Eliot’s religiosity, though, and I won’t try to sum it up from vague memory

I only have the selected poems that Eliot published himself

Returning to the Eastern influence, though, there’s a section from Four Quartets which is an extended reference to the Bhagavad Gita

I think section 2 of The Dry Salvages? Maybe section 3

Okay, yeah, section 3

davidgorman.com/4Quartets/3-salvages.htm

If $f:A \to B$ is a split monomorphism of rings, does it follow that the preimage of a maximal ideal is a maximal ideal?

Of course, how a Western Christian poet interprets/applies the Bhagavad Gita is different from how a Hindu would, so one shouldn’t overstate it

But that section is definitely in dialogue with BG

I have to read Four Quartets again

i haven't carefully digested it

Yeah

It takes a while

im reposting from hbar because it's so fucking beautiful

Let it be know it was I, Bernhard of Hbar, the finder of this Memé

Fuck you weren't supposed to know that I stole the meme

you Brazillian stalker

You bloody well told me you stole it you irreverent thingamabob

I have demensia

i forgot how to spell dementia

gr8