Mathematics - 2021-01-14 (page 1 of 3)

Balarka Sen

12:20 AM

@Thorgott yo

Thorgott

sup

Balarka Sen

you can unambiguously defined an abstract smooth manifold as a smooth submanifold $X \subset \Bbb R^n$ with the diffeology being the sheaf $i_X^* C^\infty_{\Bbb R^n}$

upto isomorphism of sheaved spaces

Thorgott

wtf is diffeology

is that like a differential space

Balarka Sen

i just made that up

Thorgott

ok, so what you're saying is that a manifold is up to diffeomorphism determined by its sheaf of smooth functions

Balarka Sen

12:30 AM

which you can define apriori as restriction sheaf

no need to do charts

user2103480

@BalarkaSen en.wikipedia.org/wiki/Diffeology you also wrote a wiki article to make it more convincing

Balarka Sen

lmao

its an actual thing nice

Thorgott

xposed

Balarka Sen

LMAO a diffeology is a differential space but backwards

Thorgott

but a manifolds already determined just by the ring of smooth functions on it, right

user2103480

12:32 AM

@Thorgott the sister site of xhamst...

Balarka Sen

$[\Bbb R^n, X]$ not $[X, \Bbb R]$

@Thorgott thats for ad hoc reasons

Thorgott

heh

its for partition of unity reasons, isnt it

Balarka Sen

prolly right

user2103480

12:45 AM

To anger both the algebraists and topologists here

A physics professor's notation, not mine

Balarka Sen

Insane

Thorgott

im shook

Balarka Sen

$$S_2 = \{\mathbf{u} \in \Bbb R^3 ; \mathbf{u} \overline{\mathbf{u}} = 1\}$$

user2103480

@Thorgott stunning and brave

Balarka Sen

$S_1, S_2, S_3, S_4, S_5, S_6, S_7, S^8, S^9, \cdots$

Thorgott

12:52 AM

${}^2S$

Balarka Sen

$T_2$ Ad(Rees) $C^u$

user2103480

@BalarkaSen still can't get over this one's beauty

you should change the $\mathbf{u}$ to $\vec{\mathbf{u}}$

Balarka Sen

You're right

$$S_2 = \{\vec{\mathbf{U}} \in \Bbb R^3 ; \vec{\mathbf{U}} \overline{\vec{\mathbf{U}}} = 1\}$$

2

user2103480

perfect

Balarka Sen

you are allowed to star this now

user2103480

1:02 AM

that was the plan all along

Balarka Sen

god's plan, not mine

the lord sayeth mysterious things

user2103480

lmao gotta resist the urge to make a giant floating physicists $S^2$

the evil twin of the almighty torus

Balarka Sen

lool

$\underset{\sim}{X} \underset{\sim}{X}^\dagger = 1$

probabilist's sphere

like the topologist's sine curve

user2103480

whut

what's with the tildes and the cross

is that just stupid notation

Balarka Sen

in my experience probabilists write X undertilde all the time

user2103480

1:13 AM

who uses a cross for a transpose that's just trying to use cool symbols

Balarka Sen

they also like dagger for dual

on the board

not necessarily on papers

user2103480

never saw anyone do that. thankfully

probabilists use awful notation at times

Thorgott

undertilde should be illegal

Thorgott

1:29 AM

what's the "geodesic opposite" of a geodesic

Edward Evans

cisedoeg

user2103480

2:01 AM

@EdwardEvans I see you ensure the presence of quality contributions in my absence

Edward Evans

Truth

user2103480

*truf

Edward Evans

Just ate sweet popcorn drizzled with honey out of a pan with a spoon

depression is

user2103480

was the pan coated?

(not with the honey, I mean teflon)

Edward Evans

nah

user2103480

2:03 AM

thank god

Edward Evans

it's an old pan with a layer of burnt junk across the bottom

Balarka Sen

lol oof

Edward Evans

Guys I'ma quit my master

user2103480

F

Edward Evans

F indeed

Balarka Sen

2:04 AM

whats the exit plan

Thorgott

this deserves a big one

$\mathcal{F}$

Edward Evans

Applying to jobs in software development

user2103480

I'll now light a doobie in your honor

Edward Evans

@Thorgott you mean $\mathfrak{F}$

Balarka Sen

@EdwardEvans sounds good to me

Edward Evans

2:04 AM

p-adic Hodge theory finished me off

it's super effective

exit plan was getting a big fat "he too psycho" letter from my psychiatrist

user2103480

I mean, you could also try not doing number theory, since that is godforsakenly hard I think

Edward Evans

so I don't get a fail, I can just.. take an indefinite sabbatical

user2103480

Ah yeah that's totally fair. Also, not a problem at all at german unis

Edward Evans

aye, can just finish at another time when I'm more of a stable human being and read about stuff I like as a hobby

Thorgott

$\digamma$

Balarka Sen

2:06 AM

yeah

this is a good idea

user2103480

Also, extra money surely makes life easier

Edward Evans

I'ma still hang around here and make quality contributions though

also yeah, a trainee software developer earns like 40k+ on average

Leaky Nun

sus

Edward Evans

lmao

Balarka Sen

iirc @Krijn quit for some time and did crypto and then joined back academia

totally do that i mean

Edward Evans

2:08 AM

Yeah my plan is kinda to work in software development for a while and see what I can get in crypto after I have some experience tbh

Thorgott

isnt crypto just less cool number theory

Edward Evans

literally

user2103480

step 1: wait for crypto to crash

step 2: invest in crypto

step 3: wait till it goes up again

step 4: profit

Edward Evans

you forgot ???????

Thorgott

this guy solved economy

user2103480

2:09 AM

I wanted to do the ??? but it was just too clear how it works

Edward Evans

I need 5 more years in Germany to get citizenship and then I'ma apply to the Bundesnachrichtendienst

user2103480

@EdwardEvans proper patriot, reading the citizens' mails

Thorgott

you should've thrown in a feint

Balarka Sen

lool

Edward Evans

Bundesnichtverdächtigendienst

user2103480

2:11 AM

my brother does his stats phd on crypto and he's pissed that he doesn't instead have money to invest in crypto

Ted Shifrin

Edward, I wish you the bestest in the next chapter.

Edward Evans

Thanks @Ted !

I hope to finish in the future though, now just isn't my time hehe

plus I just got Washington's book on cyclotomic fields which I've been wanting to read for a while

but for which I have had 0 time

Ted Shifrin

Too many people do grad school as default. That said, not having sound academic advice and doing ill-advised courses is shitty.

Edward Evans

Tbh my undergrad course was not at all good enough preparation for the courses here at Heidelberg

Ted Shifrin

We had lots of students at UGA in that position, way more at Berkeley.

Thorgott

2:14 AM

@user2103480 @Edward todays fun fact: there's a mathematical object called "eierlegende Wollmilchsau"

wtf

Gesundheit!

indeed, Ted, indeed

@TedShifrin I think I'll go back over prereqs when I can be bothered and then finish my master's part time when I'm in the position to do so

user2103480

@Thorgott why

Edward Evans

2:15 AM

Feels like a much more sane plan

user2103480

what object earned the right for that name

Edward Evans

@Thorgott and was zum Deifl isch des

Ted Shifrin

Part-time is harder than you think, but take your time and for sure get mentally strong.

Thorgott

I quote from a paper
"Acknowledgement: The results in the first section were obtained together with Martin Möller. We would like to thank him for allowing us to include this part. When he visited Karlsruhe and we discovered one remarkable property of this origami after the other, we called it “eierlegende Wollmilchsau”. We still call it W in this paper."

Edward Evans

@TedShifrin Yeah I'll be wary haha, but I think one can take as long as one wishes here

Ted Shifrin

2:17 AM

My deutsch isn't good enough.

Edward Evans

I think "W" is the highest voted answer on math.stack

Thorgott

@Edward de.wikipedia.org/wiki/Eierlegende_Wollmilchsau

Edward Evans

amazing

user2103480

its the german pendant of a perpetuum mobile.... in farm animal form

Thorgott

@EdwardEvans yeah, should be that way. ive heard stories of people that never took a course and just kept being students to get the cheaper train ticket.

user2103480

2:19 AM

@Thorgott approximately half of my friends who finished their studies

Edward Evans

There's a dude in my block who does that

Thorgott

nice nice

Edward Evans

174

A: Construct a function which is continuous in $[1,5]$ but not differentiable at $2, 3, 4$

$|x|$ is continuous, and differentiable everywhere except at 0. Can you see why? From this we can build up the functions you need: $|x-2| + |x-3| + |x-4|$ is continuous (why?) and differentiable everywhere except at 2, 3, and 4.

user2103480

tried to do the same after being enrolled in berlin but living in NRW due to covid. But they just hand out große zweithörerschaften instead of full enrollment so people dont get the ticket

Edward Evans

woops I meant the answer below that

Ted Shifrin

2:20 AM

Man, that's a tough one.

Edward Evans

@user2103480 sad

user2103480

@EdwardEvans indeed

Thorgott

yeah, that answer's famous

Edward Evans

große Zweithörerschaft is very German

Ted Shifrin

Idiotic votes.

Edward Evans

2:21 AM

Oh man I have to register at the Ausländerbehörde now

Thorgott

hey @Ted, do you have any clue what a "geodesic opposite" might mean?

Ted Shifrin

Thanks, Brexit?

Edward Evans

Right

Ted Shifrin

Opposite point on a given geodesic?

Balarka Sen

run backwards along the geodesic

Ted Shifrin

2:23 AM

Assuming we know the base point.

user2103480

@BalarkaSen you can't tell me what to do smh

Ted Shifrin

Sure he can!

Edward Evans

@user2103480 do you know the phrase "sorry ard"

that's supremely English

user2103480

nahh I forgot all the vernacular I was taught

Ted Shifrin

Ard?

Edward Evans

2:24 AM

it's "hard" without the h

Balarka Sen

@user2103480 your mom can tell you what to do

Thorgott

I don't think that's it, let me add context

Balarka Sen

o w n e d

Edward Evans

it's used when someone says a smartass answer

Ted Shifrin

Oh oh ... I hope copper isn't here.

Edward Evans

2:25 AM

or when someone pretends to be tough

Like "you can't tell me what to do smh" gets the reply "alright sorry ard"

Balarka Sen

lol

user2103480

@BalarkaSen it hurts so much

Ted Shifrin

ignorant

Thorgott

oh nvm

user2103480

@EdwardEvans alright sorry ard

Ted Shifrin

2:26 AM

Nvm?

Thorgott

I'm too stupid to read symbols

Balarka Sen

@EdwardEvans that comeback will get you dogpiled in the school yard by all the kids

Ted Shifrin

Grrr....

Balarka Sen

and you'll be known as sorrytard for the rest of your life

Edward Evans

My answers bring all the kids to the yard and they're like "alright sorry ard"

user2103480

2:28 AM

@BalarkaSen that's what mom calls me

Thorgott

it was supposed to be the geodesic opposite to an angle inside a geodesic triangle and my reading comprehension was just bad, for the record

Edward Evans

@Balarka just wrote a nice DSBM riff

gonna upload for your enjoyment

Balarka Sen

yes, please do!

@Thorgott lol opposite geodesic

this is what happens if your functors are all contravariant

Edward Evans

soundcloud.com/edmate/matesad

Ted Shifrin

Lol ... i would have said opposite side, so Thor gets partial credit

Balarka Sen

2:31 AM

why cant i reach this site, @edward

Edward Evans

soundcloud.com/edmate

what about this

user2103480

@EdwardEvans I'll never find the version again that ends with "I can teach you, but I have to charge MAH LAZER"

Balarka Sen

wait i cant reach soundcloud

what the hell

did india ban soundcloud

lol

Edward Evans

wtf hahahah

Thorgott

lmao

Balarka Sen

2:32 AM

switched from mobile hotspot to router and works fine now

very strange

Ted Shifrin

India banned a @Balarka

Edward Evans

India Bans the Ownership of Dolphins - SoundCloud

2nd result on google

user2103480

digitalmusicnews.com/2019/02/13/soundcloud-india-block

Balarka Sen

Oh wow what

stupid idiots man

they're banning chinese stuff because they want to show off

if you ban china you're banning half of technology

Edward Evans

lel

and half the world's population

Balarka Sen

2:35 AM

at this rate ill need a VPN someday

user2103480

@EdwardEvans the other half being in india?

right

lol

how do i focus

music

2:37 AM

i waste so much time aimlessly working on weird but interesting problems

Edward Evans

magnifying glass

frogeyedpeas

instead of sitting down and reading textbooks like the real deal mathematicians

Balarka Sen

listening now @EdwardEvans

frogeyedpeas

i am going to go study magnifying glasses

Edward Evans

write weird interesting problems down on little bits of paper and throw them into a fire

user2103480

2:37 AM

@frogeyedpeas "ctrl" + "+"

Edward Evans

weyy

Balarka Sen

@frogeyedpeas find textbooks which has weird problems

frogeyedpeas

any recommendations?

i like unexpected coincidences

Balarka Sen

what do you want to read

frogeyedpeas

and unusual cross field connections

user2103480

2:38 AM

@frogeyedpeas play CSS

frogeyedpeas

CSS?

user2103480

ok enough gonna sleep now instead of trolling sorry

gn

goodnite

Guats Nächtle

2:40 AM

@EdwardEvans ?

user2103480

to you guys as well

Balarka Sen

night

Edward Evans

It means Gute Nacht in farmer

Ted Shifrin

Oh, farmer.

🙄🤷‍♂️🙄🤷‍♂️🙄

Edward Evans

My german is that of a farmer

Balarka Sen

2:41 AM

That was good, @Edward!

Edward Evans

tyty

Ted Shifrin

Your English is only slightly better.

Edward Evans

Oioiiiii

Ted Shifrin

LOL

Edward Evans

I'm never sure what to capitalise in English

anymore

since the germans just capitalise everything

Ted Shifrin

2:42 AM

Every noun?

nothing lol

yeah

CAPITALISM

Happy Trump second impeachment day

send him to gwonton-oh-moe

Balarka Sen

do you think trump matters anymore

Edward Evans

2:45 AM

Ofc I do, why else would thousands upon thousands of knuckleheads have stormed the Capitol

Thorgott

what do discs in the hyperbolic plane look like?

Balarka Sen

this will continue to happen even after he's gone

@Thorgott in the Poincare model? just normal disks

Thorgott

by plane i mean half-space model

Balarka Sen

still a normal Euclidean disk, right?

the center can be off

circles are circles in all these models

its normal disks in Poincare and Poincare maps to half space by a Mobius transformation

which preserve circles

its normal disks in poincare because if origin is center it obviously is and you can push disks around by changing center using Blaschke factors, which are Mobius transformations

Ted Shifrin

Balarka be right.

Centers are very off.

Balarka Sen

2:51 AM

yeah

Thorgott

hmm ok, so in which directions are the centers usually off?

Balarka Sen

towards the boundary circle.

Ted Shifrin

LOL

Balarka Sen

I meant not of the disk, of the ideal boundary of the Poincare disk! whoops

lol

robjohn

@TedShifrin I've been off-center for quite a while.

Ted Shifrin

2:56 AM

Think distances on the half-plane, Thor. Draw a vertical ray.

Thorgott

but the ideal boundary of the Poincare disk corresponds to the x-axis in the half-plane model, no?

Ted Shifrin

We know, @robjohn, trust me. But we won’t impeach you

Balarka Sen

and infinity

Ted Shifrin

Yes. But think about vertical distances, Thor.

Balarka Sen

it makes sense that if your disk is reasonably close to the x-axis then the center will be skewed towards the x-axis, because... what Ted says

Ted Shifrin

2:59 AM

$\log b/a = \log c\b$.

Thorgott

what I'm thinking right now is that the Euclidean center of the disk can't lie on the same vertical ray as the hyperbolic center, because the ray is a geodesic and you can move the same Euclidean distance along it both up and down while staying inside the hyperbolic ball, so if the Euclidean center was on the same ray as the hyperbolic center, they'd have to be the same and that cannot be

Balarka Sen

that is nonsense right? what about disks on the y-axis?

the euclidean center and the hyperbolic center lie on a geodesic for any disk. just shift that geodesic to be a vertical line by an isometry

Thorgott

what about them?

Ted Shifrin

Huh?

Damn categorical Thor.

Thorgott

ok, but do you agree that the hyperbolic center of the disk is the midpoint of the intersection between that disc and the vertical ray it lies on?

Balarka Sen

3:03 AM

No what you said is false and I wont read anymore

Do the geometry correctly

Ted Shifrin

Midpoint how?

Thorgott

ah, it isn't, I see now

Ted Shifrin

Midpoint hyperbolically,,of course. That’s the equation Ted wrote.

Balarka Sen

Right ^^^

Thorgott

right, I was mixing up hyperbolic and Euclidean distance on the ray

Mike Miller

3:06 AM

You two have the weirdest BDSM relationship I have ever seen in print

5

Thorgott

ah ok, the Euclidean center lies on the same ray as the hyperbolic one, because you can just reflect geodesics across the ray, right?

Balarka Sen

@Thorgott Sorry to tell you the diagram ray -> H^2, ray -> H^2, R^2 -> H^2 doesn't commute

Leaky Nun

@MikeMiller you haven't seen much in print then

Balarka Sen

lol

Mike Miller

the 51st shade

Thorgott

3:08 AM

and going off to infty along a vertical ray, the Euclidean distances behave logarithmically, so of course the Euclidean center has to lie below the hyperbolic center

Balarka Sen

wait let me find 50 shades of (J P) may

Thorgott

bruh

Balarka Sen

cant find it

Mike Miller

It's been made private

Balarka Sen

:(

Mike Miller

3:14 AM

Amin might have access dunno

Balarka Sen

yeah ill ask

Thorgott

3:25 AM

wait no, that was wrong

the Euclidean center has to lie above the hyperbolic center, not below

Balarka Sen

this depends on where the disk is man

it could be near infinity, or it could be near x-axis

Thorgott

on the vertical ray, the distance between two points with fixed Euclidean distance becomes smaller and smaller as you move them up the ray, so the part of a disk lying above the hyperbolic center is longer in the Euclidean metric than the part lying below, whence the Euclidean center has to lie above

Balarka Sen

Again, this is as you move to infinity. What if you move to $0$?

The hyperbolic disks at $i$ for example all have Euclidean center $i$ as well

If you move down the hyperbolic center shifts towards the $x$-axis, so goes below the Euclidean center

Thorgott

the picture should be the same, log is monotonic everywhere, so Euclidean distance above the center will always be bigger than Euclidean distance below the center

I've done the computation too, the intersection of the $r$-ball centered at $ix$ with the $y$-axis ranges from $ie^{-r}x$ to $ie^rx$, so the Euclidean midpoint is $i\cosh(r)x$, which always lies above $ix$

Balarka Sen

I can tell you, doing 0 computation, that what you have done is hot garbage

Does the Euclidean center lie below the hyperbolic center?

Ryan Unger

3:37 AM

@BalarkaSen why are you so mean towards thor?

Thorgott

sure looks like it, but I have more reasons to believe in my calculation than to believe in some random picture

Balarka Sen

You think the Euclidean center there lies below the hyperbolic center? Are you blind?

What does below mean to you

The hyperbolic center is the red fucking point

Thorgott

wait, but then you're not even disagreeing with me

I've been saying the Euclidean center lies above the hyperbolic one all along

Balarka Sen

That's not true if you move above $i$

At $i$ they're the same. Above they exchange positions

Just reflect along a geodesic you'll see it

Your computations are hot garbage

Sorry

Thorgott

so tell me whats wrong with my reasoning

Balarka Sen

3:46 AM

No but do you understand mine? Take a huge fucking arc from $-10^{10}$ to $10^{10}$ and reflect the above picture

The red point is mapped way above the image of the Euclidean center of the circle in the image

Because reflections switches order along the $y$-axis

Thorgott

I get it and it somehow should be that way, because $i$ corresponds to the center of the Poincaré disk and there should be some symmetry or whatever

Balarka Sen

Exactly

So your computations are nonsense

Snapdragon-X

Yes

Thorgott

whatever, but tell me what's wrong with my reasoning, cause I don't see that

Balarka Sen

I don't know how to compute, @RyanUnger will help you

Only pure geometry here

Thorgott

3:48 AM

I mean the reasoning, not the computations

take a point on the y-axis

take two points the same hyperbolic distance from the given point on the ray, one above, one below

the Euclidean distance from the one above to the reference point is bigger than the Euclidean distance to the reference point from the one below, agree or not?

Balarka Sen

No, depends on where you are. I think it's log but offset by $i$.

$d_{Poincare}(i, iy) = |\log y|$

Thorgott

maybe that absolute value sign is fucking me up

Balarka Sen

Yeah corrected it

That's what's fucking you up

$|\log(1/2)| = |\log(2)|$

Ted Shifrin

@BalarkaSen Not really.

Balarka Sen