serious-discussion

[deep mango]

but i would never buy garden pizza on purpose

Ew

[snow ice]

oh yo same, except I eat it with the soil!

[deep mango]

Likeeee

[timid bronze]

Margherita pizza is so bomb though

[silent junco]

[timid bronze]

Wait is there an Italian food ranking tierlist

Lemme do it

[deep mango]

What im making is margherita

Idk never seen that

[neat lintel]

hot sauce in d 🥀

[silent junco]

actually i agree

[deep mango]

On pizza is nuts

[neat lintel]

Oh its a on pizza?

alr then i agree

[deep mango]

YES look at the other stuff

[snow ice]

margeritta pizza is the goat

[neat lintel]

idk i ain a chef i js thought its a food tier list

[limber thunder]

at least I agree with the SSS picks

[deep mango]

I have a billion food tierlists

[limber thunder]

pesto is goated

[neat lintel]

ngl tomatoes could be a bit higher

[snow ice]

@deep mango nice hs chem test btw

[timid bronze]

Ermmm idk the rest

Wait I lied

There

[silent junco]

ugh im so full i cant think about food rn

[snow ice]

im def getting blocked

[deep mango]

@timid bronze im gonna do this

https://tiermaker.com/create/italian-food-tasteatlas-2024-17320998

[deep mango]

Oh you did the shitty one without labels

🤣

yikes

[neat lintel]

hol up imma also do it cus why not

[timid bronze]

Thx

[snow ice]

when do I stop becoming a peasant and underling and I can post images?

[silent junco]

icl zohrans fake smile creeps me out

[deep mango]

This is super thorough

[silent junco]

he has a very "hello fellow kids" media presence

[snow ice]

who does?

[silent junco]

zohran

[snow ice]

ah

don't know him

[silent junco]

nyc mayor

[deep mango]

apparently we dont have a pizza cutter here

so ill have to use scissors

[snow ice]

he looks goofy

[silent junco]

???

not a knife?

[deep mango]

Welcome to politicians

[neat lintel]

use a knife

[deep mango]

Youve never cut pizza with scissors

[deep mango]

??

[silent junco]

not all politicians do this, some of them just aura farm

[neat lintel]

nope

[snow ice]

I cut pizza with scissors

[silent junco]

dude

[deep mango]

Woah. Wild

[neat lintel]

Who cuts PIZZA with SCISSORS like PAPER

[snow ice]

like trump?

[deep mango]

Kitchen scissors

[snow ice]

or nvm

forget what I just said

[neat lintel]

still using a knife is better

[silent junco]

trump doesnt aura farm. except for when he got shot i guess

[neat lintel]

more accuracy

[snow ice]

that's what I was referring to, other times hes just a goober

[silent junco]

yeah

[jagged forge]

goat cheese on pizza?!

[snow ice]

its popular

[jagged forge]

interesting

[snow ice]

I should have said that I'm afraid of everyone here

[jagged forge]

boo

[silent junco]

goat cheese is disgusting

cheese in general is gross

[snow ice]

😱

[deep mango]

Its great...

[jagged forge]

WHAT

[snow ice]

everyone is scary

[timid bronze]

@deep mango

[timid bronze]

Ong

[jagged forge]

W H A T

[timid bronze]

I don't like melted cheese for the most part

[snow ice]

😱 scary image

[timid bronze]

Tbf a lot of these Italian foods I can't try cz of pork

[deep mango]

Tiramisu #1 yep

[magic python]

I can't stand most cheap cheeses tbh

[snow ice]

AMERICAN CHEESE

[silent junco]

the fuck

[timid bronze]

I MEAN. LIKE LIQUID CHEESE

[magic python]

Never tried

[silent junco]

oh yeah ew

[timid bronze]

MELTED CHEESE IN PASTA CAN BE GOOD

[snow ice]

its just melty, thats all

[silent junco]

im just saying melted cheese is better than the alternative

[magic python]

True

[snow ice]

but very cheap and is synthetic

[silent junco]

sliced cheese is revolting

[timid bronze]

Depends on the cheese

[silent junco]

no.

[magic python]

I just cannot stand cold cheese (the block or preshredded bs)

[limber thunder]

https://tenor.com/view/garfield-gangsta-gangster-lasagna-blingee-gif-24006588

[timid bronze]

I could eat gouda and pepper jack cheese straight up

Rest no

[timid bronze]

WYM

[limber thunder]

lasaga

[snow ice]

lasagna is literally just a fattener, nothing else

[snow ice]

any other pasta is better

[timid bronze]

I'm not a huge lasagna fan but I want to try lasagna soup

It's so popular

[snow ice]

same

[limber thunder]

https://www.kkinstagram.com/reel/DSsidMjDFa9/

[snow ice]

I've tried lasgna but not lasgna soup yet

[latent edge]

Type shit

[timid bronze]

I do love myself some pesto

Pesto is bomb

[snow ice]

pesto is literally the jack of all trades of pasta

[snow ice]

its good at everything

[limber thunder]

lasagna soup seems like lasagna but in a ton of tomato paste

idk it existed

[silent junco]

this is good 👍

[timid bronze]

Albondigas is also a soup I wanna try but

A lot of places make it with pork 😭

[snow ice]

I for whatever reason don't get the obssession of miku and teto, I have no idea why

[timid bronze]

So I have to make tht myself too

[limber thunder]

albondiga to me isn't a soup

[timid bronze]

Oh rlly

[limber thunder]

it's the spanish word for meatballs

[snow ice]

Spaghetti all'assassina is my favorite type of pasta tho

[timid bronze]

Sorry if I was spreading misinformation

[limber thunder]

no prob

there's probably some soup that's based in these

[limber thunder]

https://tenor.com/view/miloie-cat-kitten-goobab-small-gif-11098941617683347434

[timid bronze]

I WANNA TRY IT

BUT I haven't yet

I hope it impresses me

[silent junco]

im feeling sleepy

[snow ice]

you can't burn it so it will never ever taste bad

[timid bronze]

Sleep

[deep mango]

GONNA FUCK UP HIS SLEEP SCHEDULE

Oops didnt mean for caps to be on

[snow ice]

well this looks odd

[timid bronze]

https://tenor.com/view/small-cat-angry-cat-cute-gif-24160051

You

[silent junco]

what does "chovying" mean

[silent junco]

no i must resist

[timid bronze]

Idk what TZ they are in but I assume everyone in the US is in EST 😭

[silent junco]

indeed

[timid bronze]

Sleeping at 8 pm nawt baddd

[silent junco]

anyway thanks

[latent edge]

no worries

[latent edge]

Real

This too

[vast wraith]

that would be a long hug

[deep mango]

Lmaoooo

[latent edge]

True

[timid bronze]

I hate it when I'm having a nightmare or dream tht I wanna finish but then I get rudely interrupted

Like last night I had a bad fucking unfinished nightmare

[old oak]

Erm, what?

[timid bronze]

But then I got interrupted by the cold air ;-;

[deep mango]

https://tenor.com/view/nursery-decor-baby-teethers-gif-22165028

[silent junco]

reze arc was so good

[timid bronze]

I didn't even see that 😭

[old oak]

How do you even milk hoaxes?

[silent junco]

aki is my favorite character now

[jagged forge]

omg did u see it

https://tenor.com/view/peak-panda-jjk-jujutsu-kaisen-gif-1206829322030625047

[silent junco]

yes

[jagged forge]

ME TOO

[timid bronze]

Me too*

[jagged forge]

i love this sticker

if only it was aki

[snow ice]

everyone is scary here, Idk why

[silent junco]

like aki is lowkey kinda weak and thats what makes him cool

[timid bronze]

Am I scary

[snow ice]

sort of

[ocean harbor]

i used to say like this too and im used to it now

[timid bronze]

._.

[latent edge]

I'm not :3

[gritty heath]

Eventually you'll be the horror too

[snow ice]

oh you are the complete opposite of what you think you are, your parents should've tought you better

[magic python]

Am I scary >~<

[jagged forge]

aki is the only responsible adult in part 1

[snow ice]

actually not too much

[timid bronze]

Yay

[snow ice]

ur cool

[jagged forge]

also he’s so real

[vast wraith]

https://tenor.com/view/scissors-seven-xiao-fen-gif-26487338

[timid bronze]

suremark is lowkey really scary

[snow ice]

ur kinda scary

[timid bronze]

He gets angry very easily

[silent junco]

this is very wrong

suremark has the patience of a saint

[timid bronze]

And he's rude to kids

[snow ice]

scary immediately

[magic python]

Nuh uh

[jagged forge]

roar

[timid bronze]

You need to see how mean he is to amukh and Arti

[vast wraith]

https://tenor.com/view/flirt-speechless-belle-beauty-and-the-beast-blushing-gif-17736299

[silent junco]

you need to see how much of chipotles bs he puts up with

[snow ice]

I don't trust you, you keep watching me

[timid bronze]

Ok true

[snow ice]

idk if its you, you aren't half bad

[latent edge]

Trying to decipher wtf this means

[silent junco]

👁️ 👄 👁️

[timid bronze]

bunker made me ragequit from life

[snow ice]

take your time

[ocean harbor]

real

[snow ice]

SEE

[latent edge]

I don't get it button

[snow ice]

its an alberti cypher, you just have to get the right key to solve it

[silent junco]

they should make more pills bubblegum pink

[snow ice]

for short

[vast wraith]

i made the sensible decision and left bunker

[silent junco]

and other fun colors

[snow ice]

you are very scary to me

[limber thunder]

i just lurk bunker

[silent junco]

have you seen his math? anyone would be scared

[limber thunder]

like i lurk a ton of other places

the loorker

[timid bronze]

Same 😭 I think I only stayed there for like 2 months

[snow ice]

whose math?

[timid bronze]

Then was like nope

[vast wraith]

real and true

[silent junco]

deltoid

[fresh comet]

that's been happening more frequently to me lately for some odd reason

[snow ice]

oh send it thru dms if you want

[timid bronze]

Nooo

[fresh comet]

both getting more nightmares and getting them interrupted

[vast wraith]

https://tenor.com/view/cat-kitten-handshake-deal-agree-gif-7134721403924803840

[silent junco]

just wait a few minutes hes probably gonna post some here

[snow ice]

alright

[timid bronze]

higher! is not scary

[snow ice]

ur the opposite of scary

[timid bronze]

Higher is definitely scary though

[fresh comet]

https://tenor.com/view/scarey-boo-rawr-cute-gif-22290620

[timid bronze]

"Higher" gives doxxer vibes

[random whale]

Me when mango (epic)

[random whale]

We're just saying words

[snow ice]

I hope they don't pin the message I sent

[silent junco]

higher is chill

[timid bronze]

I am not just saying words!

[fresh comet]

Higher is always watching 👁️

[random whale]

Then what else are you doing

[timid bronze]

Keep higher!

[snow ice]

higher! is goated

[timid bronze]

Nothing? 😭 I was just making a joke between variations of his user

[jagged forge]

BOOT = ING = higher

[random whale]

And you told the joke using...?

https://tenor.com/view/phoenix-wright-ace-attorney-game-take-that-text-gif-7989980

[vast wraith]

unless if you’re a seal

[timid bronze]

I'll make sure to stop doing my homework

[fresh comet]

it's alright
I hope your nightmares aren't the product of stressors within your life right now :c

[timid bronze]

. . .

[jagged forge]

im shocked i still haven’t gotten vactive back yet

[silent junco]

wait lemme check something

[fresh comet]

you can get it back right now if you pay me ten dollars

[timid bronze]

@latent edge They want to hear about your math

[random whale]

Have a 5h energy instead like a true red-blooded patriot

[fresh comet]

stop? oii, do your homework

[timid bronze]

;-; Fuck physics hw

[snow ice]

WHY THE F**K IS MICHEAL JACKSON LYRICS GOING ON IN THE CHAT WHILE IM PLAYING RAINBOW SIX SEIGE

[latent edge]

Say less

brb

[jagged forge]

i will pay u -100000 dollars to get it back

[snow ice]

physics is peak

[timid bronze]

Prepare for a number theory lesson with your instructor nDeltoid

(It was too difficult I couldn't grasp)

[snow ice]

I already learned the entirety of number theory for competition math

nothing too hard will happen

[timid bronze]

Oo

[snow ice]

is that slang or a typo Idk which

slang is so odd rn I don't get it

[silent junco]

okay yeah im technically not supposed to be telling you this but youre blacklisted from vactive for the time being because the bot thinks youre a bot. its a new feature that we're testing out where the bot is trained to detect suspicious activity and prevents certain accounts from getting active. since the rule for going from no role to active is the same rule for going from emeritus to very active, it applies to members like you too. ill probably get around to making a pull request adjusting its behavior for emeritus members soon.

[timid bronze]

I meant Oo

[snow ice]

oh

sorry

[timid bronze]

Nws

[jagged forge]

did u just call me a bot

[timid bronze]

You are a bot

[silent junco]

i didnt, the bot did

[vast wraith]

https://tenor.com/view/anya-forger-anya-spy-x-family-gif-17200077238442027522

[jagged forge]

is this fr

ur capping

[silent junco]

what 😭

[jagged forge]

r u pulling my leg

[silent junco]

you can check #『bot-and-website』 for the relevant logs

[latent edge]

tfw

[timid bronze]

Bot looking aahh Roblox avatar egg head

[timid bronze]

Wait you're dead ass?

I thougth you were jk'ing 😭

[vast wraith]

whats L

[silent junco]

(no but i bet suremark will be scrolling through that channel for a while now 😹)

[latent edge]

A finite extension field of Q_l

[snow ice]

like does the bot detect it if you are spamming a bunch of the same message

[jagged forge]

im going to kill u

[vast wraith]

i still havent at all gotten to look at adic stuffs

[silent junco]

it does but it just bans you, it doesnt blacklist you from active or anything lol

[timid bronze]

Calm down unc

[latent edge]

ok then dw let me delete

[jagged forge]

i was scrolling for like 5 minutes bro

[snow ice]

i see

[silent junco]

LMFAOO

[timid bronze]

LMFAO

[snow ice]

LMFAO

[latent edge]

Do you know how to relate a number field with its character?

[snow ice]

I'm so glad Super Matroid doesn't know who I rly am

[jagged forge]

u just don’t appreciate art

[vast wraith]

no and im way too sleepy to try rn 😭

[snow ice]

isn't that the egg for the hunt event?

[timid bronze]

No, I'm talking about this

[latent edge]

real

[snow ice]

I see it

[latent edge]

Essentially what you do is use Kronecker Weber

[silent junco]

is that... nvm

[jagged forge]

what’s writing with that

[timid bronze]

wrong*

[latent edge]

Well I forgot to say (abelian) number fields over Q

[jagged forge]

stupid autocorrect

[latent edge]

In general it's not so easy

[timid bronze]

Looks like an egg

Do I need to set you up a spherical integral

[jagged forge]

its a sphere…

[snow ice]

rn I'm just sitting in the event stage doing literally nothing lol

[timid bronze]

Still looking like an egg

[jagged forge]

this is a powerful egg phobia

[vast wraith]

that’s a topological egg

[jagged forge]

true

[latent edge]

Every abelian number field over Q can be seen in a cyclotomic extension

[snow ice]

I'm just bored @timid bronze

[latent edge]

Then the next step is the isomorphism between the Galois group of the cyclotomic extension and (Z/mZ)^x

from there you can consider characters from (Z/mZ)^x -> C^x

Now from the theorem above you essentially identify the abelian extensions with some subgroup of this guy

For example take Q(zeta_8)/Q

consider the character given by chi(1)=1, chi(3)=-1,chi(5)=-1 and finally chi(7)=1

Then you look at the kernel of such a character

In this case it's {1,7}

Now you look at the fixed field of this

7 corresponds to sigma_7 which is complex conjugation

This tells you that the field is real

[snow ice]

where's mi ipad

[latent edge]

and quadratic

[snow ice]

NO where's mi ipad

[timid bronze]

Learn math with him it's interesting

[latent edge]

since phi(8)/2 = 4/2 = 2

[snow ice]

I see

I'm listening rn

[latent edge]

With some effort you can check that the field is Q(sqrt(2))

[vast wraith]

lovely

[latent edge]

yeah that's field theory and some fancy character theory so far

okay now you might ask where is the number theory

[snow ice]

I see

I understood it

[latent edge]

Okay so a classic question you can ask given a number field K is classify its primes

Given a prime p over Q how does it change when you look at this number field K over Q

[vast wraith]

thats the class number thing right

[latent edge]

uhh what do you mean by this

class numbers tell you if the ring of integer is UFD or not

[snow ice]

it could break into several prime ideals.

[latent edge]

measures the failure of UFDness

[latent edge]

Right yeah but how do you pinpoint that exactly

[vast wraith]

oh ok so no then

[latent edge]

The idea the original isomorphism we had earlier

Galois group of zeta_n over Q

the isomorphism is given by sigma_k: zeta_n -> zeta^k_n right

[snow ice]

Factor f(x) modulo p in Fp[x] is what I do

assuming

[latent edge]

you have to elaborate on that

[snow ice]

where in K=Q(a), a has minimal polynomial f(x) that is an element of Z[x]

[latent edge]

not exactly

[snow ice]

I see... what was my issue?

[latent edge]

the issue is you don't know the ring of integer beforehand

so this might tell you false info when you take modulo p

Good thing is it works for almost all primes

[snow ice]

I see, it only works for most primes, those dividing the discriminant.

[latent edge]

yup

but what I'm showing is another way to get the same info

Let's take an easier example for demonstration

Take Q(zeta_4)/Q = Q(i)/Q

From earlier discussion this has Galois group isomorphic to (Z/4Z)^x

[snow ice]

mmhmm

[latent edge]

okay now let's look at Dirichlet characters

(Z/4Z)^x has 2 elements so we should also expect 2 characters

One of which is the trivial character and the other is quadratic in nature

Given by chi(1)=1, chi(3)=-1

[snow ice]

yep

[latent edge]

Yeah now the golden key is realizing that from the isomorphism you can relate Frob_p in Gal(Q(i)/Q) with p

so chi(Frob_p) = chi(p) for unramified p

[snow ice]

OHHH

[latent edge]

now we look back

for primes p = 1 mod 4

plug in the formula

you get chi(Frob_p) = ?

[snow ice]

chi(Frob_p) = 1?

[latent edge]

yes

what does this tell you?

[snow ice]

P splits in Q(i)

[latent edge]

right

how about p = 3 mod 4?

[snow ice]

-1?

[latent edge]

yeah and what does that correspond to?

[snow ice]

p is inert in Q(i)

[latent edge]

mhm

now the only missing thing here is p=2

[snow ice]

:)

[latent edge]

you can check this one is ramified

[latent edge]

but remember this thing only works when p is unramified

[snow ice]

as its not coprime with 4

[snow ice]

I see

[latent edge]

there's also a result to keep in mind if you get chi(p)=0 this tells you that p is unramified

and sure enough 2 is not coprime with 4 as you mentioned so there we have it

[snow ice]

got it

its fun learning with a post-grad person as a pre-uni :D

[latent edge]

yeah you can do similar things with more complicated examples as long as you are working over Q and your number field is abelian

If your number field is not abelian then you forget about it

You need more machinery to handle these

[snow ice]

does a laptop do?

[latent edge]

No I mean math tools

[snow ice]

OHH

I think we scared @timid bronze away lol

[latent edge]

Anyways

You might wonder what if we don't work over Q

[snow ice]

i quite do

[latent edge]

Just any arbitrary number field K and keeping stuff abelian

yeah for this the answer is not far off from the story over Q

[snow ice]

oh

I see

[latent edge]

yeah but it requires more care to get the right arithmetic object

Essentially you want to play the same game. Over Q we had (Z/mZ)^x and so the task is to somehow generalize this group

So that it works for our needs

I will not get into the details cuz it might be too much lmao

But essentially that's what you do in class field theory

[snow ice]

I got it

tysm

[latent edge]

[snow ice]

@timid bronze YOU CAN COME BACK NOW! WE ARE DONE WITH OUR BEAUTIFUL MATH DISSCUSSION!

[latent edge]

btw im curious did u study number theory before?

[snow ice]

yes, a little bit but yes

the one made for high schoolers

[latent edge]

No I meant number theory

not that

[snow ice]

ohhh

yes

[latent edge]

Things like decomposition group and whatnot

what year are u in

[snow ice]

nope

[latent edge]

ok so you were using chatgpt?

[snow ice]

to understand a bit yes

[latent edge]

[snow ice]

I know sets, coprime

elements

and int rings

...

sorry

I do know modular arithmetic too

[latent edge]

It's okay next time don't use chatgpt for mathematics

If you don't understand what I say sometimes just let me know I can explain

[snow ice]

I understand now however

isomorphism

[latent edge]

Do you really understand isomorphisms

[snow ice]

the concept yes

[gritty heath]

Its just equality if you squint hard enough

[latent edge]

ok tell me what you understand

[snow ice]

its a structure-preserving translation between two objects

sets with the same number of elements are isomorphic

[latent edge]

Have you seen groups before?

[snow ice]

yep

lots of times

[latent edge]

quotient groups?

[snow ice]

nope

can you teach me?

[latent edge]

yes but before that I want to make sure

Take Z to be our set

[snow ice]

yes?

[latent edge]

and take multiplication to be our operation

[snow ice]

yes?

[latent edge]

Is this a group?

dont use gpt

[snow ice]

yes.

[latent edge]

why

[timid bronze]

I can't contribute meaningfully

But nah, I went to go do something

[snow ice]

I realized I was wrong as it doesn't have inverses.

:(

its not a group

[latent edge]

that's right it is not

but say you are given one move to make it a group and that move is shrinking your set

[snow ice]

yes?

[latent edge]

Obviously if you shrink down to 1 element it is a group but that's not interesting

What's the largest set you can shrink Z to in order to make it a group?

[snow ice]

2 elements

1, -1

[latent edge]

can you check for me why is this a group?

[snow ice]

first three (closure, associativity, identity) work for all ints with mult operation

and 1 has its inverse that's an int (1) and -1 has its inverse that's an int (-1)

[latent edge]

yep good job

:3

[snow ice]

:3

is this knowledge good for a 13 year old?

[latent edge]

yeah groups are really nice

[snow ice]

:)

[latent edge]

okay consider the following thing

say you have a butterfly

🦋

[snow ice]

prettyyyy

but yes

[latent edge]

yeah now we want to classify the symmetries of this thing

you can see that if you draw a line in half

then you can reflect the butterfly

[snow ice]

right in the vertical middle

[latent edge]

and it still looks the same

right

[snow ice]

yes?

[latent edge]

alright anything else?

[snow ice]

nope?

[timid bronze]

You're 13?

[snow ice]

yes?

[timid bronze]

Damn

[snow ice]

anyways

[latent edge]

well you can do "nothing"

doing nothing still preserves the shape of this butterfly

[snow ice]

I guess so

[latent edge]

so it is a valid symmetry

[snow ice]

I see

[latent edge]

okay great let's use r to denote the reflection symmetry

and 1 to be the "do nothing" symmetry

[snow ice]

yes?

[latent edge]

so in total we have 2 symmetries right

[jaunty ibex]

ur cooked

[solar spire]

when i was 10 i lowk was on real analysis

so

like

he gotta lock in

[jaunty ibex]

ok buddy

[snow ice]

mhmm

[latent edge]

right let's put them in a set {1,r}

now observe this: what happens if we do r twice?

you know you take your butterfly and reflect twice

[snow ice]

its no longer symettrical

[latent edge]

huh

[timid bronze]

No need to compare my friend

[solar spire]

this is just ironic i was nowhere near that

hes very far ahead for 13

[latent edge]

okay so say this is the butterfly (left|right)

[snow ice]

yes?

[timid bronze]

Very well then

[latent edge]

if you do one reflection what happens

[snow ice]

the right is on the left

[latent edge]

yeah you got it good job

now do it one more time

[jaunty ibex]

u just get back to where u are

[latent edge]

shut up please

this is confusing to read

is it (right|left) or (left|right)?

[snow ice]

left right?

[latent edge]

yeah exactly

you see what happened here?

[snow ice]

I do...

[latent edge]

when you reflect twice it's the same as doing nothing

[snow ice]

oohh

[latent edge]

yeah the way you write this mathematically is like this: r^2 = 1

[snow ice]

oh

[latent edge]

where squaring here means doing twice

1 here is the "do nothing" move

[snow ice]

I thought it would be something like x = 0 mod 2

[snow ice]

mhm...

[latent edge]

lol this is also something similar actually we will get to that

[snow ice]

oh ok

[latent edge]

okay but here's the gist

in our previous group we had {1,-1}

you verified this was a group and said 1 is the identity

and (-1)^2 = 1

right?

[snow ice]

yep

[latent edge]

okay now in this new group we made let's call it the "butterfly symmetry group"

we have {1,r}

1 is the identity

and r^2 = 1

right?

[snow ice]

yes.

[latent edge]

so what we conclude from this

[snow ice]

r u saying

r^2 is the identity?

[latent edge]

yes

[snow ice]

and the set is {r^2, r}?

[latent edge]

yeah you can write it like this if you want

[snow ice]

I see...

[latent edge]

yeah but the upshot here is that

{1,-1} and {1,r} are the "same" group

[snow ice]

yep?

[latent edge]

same as in they have same number of elements

and the way the 2 groups interact are identical

[snow ice]

yes.

[latent edge]

a fancy term to say they are the same in math is isomorphic

[snow ice]

mmhmm

so that's isomorphism

[latent edge]

but you see the beauty in this?

yeah exactly

[snow ice]

i do a lot

[latent edge]

we had 2 different ways to make the same group

[snow ice]

this is very neat

[latent edge]

one comes from number theory

and one is just symmetries

[snow ice]

geometry and number theory intertwine 🤯

[latent edge]

yeah lmao

[snow ice]

lol

[latent edge]

okay you mentioned mod 2 arithmetic earlier

[snow ice]

yes

[latent edge]

when you take numbers and mod out 2

the resultant is either 0 or 1 right

[snow ice]

yes

[latent edge]

okay let's look at this under addition

basically i want to say that we can just take the elements {0,1} in this group

[snow ice]

mmhmm

[latent edge]

since anything will just collapse to these two numbers modulo 2

[snow ice]

yep

[latent edge]

okay great

what is the identity here?

it's either 0 or 1 so you have a 50/50 chance lmao

[snow ice]

0?

[latent edge]

yeah why

because if you add anything to 0 you just get that number

[snow ice]

a+0 =a

[latent edge]

yeah right

also what happens if we add 1+1 modulo 2?

[snow ice]

to what?

[latent edge]

just add 1+1 and take mod 2

[snow ice]

its 0

[latent edge]

exactly

we found a 3rd group with the same properties

[snow ice]

:)

[latent edge]

2 elements and one of the elements goes back to identity when you do it twice

so this group is also isomorphic to the other two groups we discussed

[snow ice]

i see

I really do

[latent edge]

yeah im glad

You can take this idea really far and do amazing stuff with it

[snow ice]

one equation is a regular one, one is from mod arithmetic, and one is a representation of geometry

[snow ice]

what college concepts run on isomorphism btw?

[deep mango]

@timid bronze i did it

[snow ice]

or use it?

[latent edge]

that would be abstract algebra

[snow ice]

i see

[latent edge]

if you are curious I can give you a book to read which I think is somewhat readable

[snow ice]

like linear algebra?

[latent edge]

yes you also take linear algebra

[snow ice]

I may read it soon

do send but

[latent edge]

http://abstract.ups.edu/

[snow ice]

I bookmarked it

it seems neat

[latent edge]

[snow ice]

I'll def read this sunday

[latent edge]

yeah have fun then

[snow ice]

tysm

:)

[torpid bay]

[latent edge]

now you

let me teach you about cohomology of arithmetic groups

jk

[torpid bay]

am scard

i do have a question tho

I was looking at functions that converge on the unit circle, like $\sum_{n=0}^{\infty}{z^{q^n}}:q\in\mathbb{Z}^{\geq2}$

[fathom swallow]

Yeatte

[torpid bay]

and I saw a few symmetries in it, but I was wondering, for a function f(z), with certain 'nice' properties, I was wondering if instead of going from f(z) -> symmetries(f(z)), If its possible to go the other way? and also if it's a one to one relation

[rough basalt]

campus has reopened

I lurk in the physics building once more

[latent edge]

idk what you are trying to say here

[foggy meadow]

Good one

[foggy meadow]

https://www.shadertoy.com/view/tltcWf
^here's one that converges.

[latent edge]

so you are basically looking at lacunary functions

[torpid bay]

like, If I have a group (probably a certain kind of group ig given that i have the restriction of complex numbers), can I always be able to find a function f over the complex numbers where that has those symmetries? like with the sum above, f(ze^(2ipi/(q-1))e^(-2ipi/(q-1)) = f(z) as a symmetry, and f(z^)^ = f(z) as another

[latent edge]

okay okay

so given a g in G you are interested in functions f(g.x) = f(x)?

g.x here is the group action on your set

[torpid bay]

ye

[latent edge]

right then I have a lot to say about this

[torpid bay]

[latent edge]

let's start with easy examples

You can view the cosine function as an example of this

Do you see how?

[torpid bay]

yeppers

translation as an exmaple

cos(x+2pik) = cos(x) :integer k

[latent edge]

you can do better than translations but you are on the right track yeah

[torpid bay]

cos(x)=cos(-x)

[latent edge]

yeah there is also this guy yup

[torpid bay]

also compelx conjugation

right?

[latent edge]

let's work with R for now

[torpid bay]

ok

[gritty heath]

I think complex conjugation doesnt work

[timid bronze]

Took you long enough

Bro I’m so sad

[latent edge]

okay let's use T for translation

and S for reflection

So we have these actions on R: Tx = x+2pi and Sx = -x

so far so good?

[torpid bay]

yep

[latent edge]

right if you play around a bit with these

you can see S^2 = id here

[torpid bay]

yep

[latent edge]

and STS = ?

[torpid bay]

T^-1

?

[latent edge]

yup

[torpid bay]

-((-x)+2pi) = x- 2pi

[deep mango]

https://tenor.com/view/chud-chudjak-wojak-soyjak-marcvin5-gif-13903021270190271130

[latent edge]

yeah exactly

So this group G is generated by two elements S and T such that it satisfies the properties we discussed above

this G has a name and it's called the infinite dihedral group for obvious reasons if you've seen dihedral groups before

[torpid bay]

i know the finite one ye

the 2 element, but i hadnt heard of ones larger than that

[latent edge]

yeah the infinite guy naturally gives you the sinusoidal functions

you can see that one of the sinusoidal functions is invariant on the whole group

but one of them is only invariant under translations

[torpid bay]

only one of the dihedral groups?

the infinite one?

[latent edge]

yes we are still talking about the dihedral group

[torpid bay]

ah ok

ye

[latent edge]

notice that cosine is even right

[torpid bay]

yep

[latent edge]

so cos(Sx) = cos(x)

and obviously it is invariant under translation

that's the whole thing

meanwhile sine is odd

it's very common to attach a factor in front of the resultant for these cases

Like sine gives you a minus sign whenever you do S right

[torpid bay]

yep

[viral yew]

cox(x) = x

[latent edge]

so this motivates you to cook up a character chi: G -> {1,-1}

where it outputs 1 for T and -1 otherwise

Then your sine function will look like sin(g.x) = chi(g) sin(x)

cool right

[torpid bay]

ye

[latent edge]

yeah this POV gives you more meaning for these things

[sterile bolt]

why are people so smart

[latent edge]

I find it really fascinating

[torpid bay]

I rmemeber needed an outside thing for the lacunary thing as well

[latent edge]

yeah there we go lmao

we can cook up a fundamental domain for these functions

you already know [0,2pi] works for translations

but in this case we need to shrink it a bit

what would that be

[torpid bay]

I haven't heard of a fundamental domain, what is it?

is that like a minimal one or smthn?

[gritty heath]

Analysts regularly exploit symmetry to study and break down functions

Fourier analysis is basically that

[latent edge]

do you know what is an orbit?

[torpid bay]

no

[latent edge]

intuitively yeah it's a minimal set where anything else is generated by just applying elements of G to the set

[torpid bay]

oh I think i heard of that idea then

[latent edge]

just like how with translations the fundamental domain is [0,2pi]

anything will just be a copy of this guy

[torpid bay]

h

if I combine the S and the T, I feel like i can shrink the fundy domain quite a bit

[0,pi/2]?

[latent edge]

why so

well hold your hourses for a min let's define a fundamental domain

[timid bronze]

I’m fr

My parents’ relationship has gone to shit

All my mom can talk about rn is a divorce

And she has nothing good to say about my dad

[latent edge]

Take an F in X. F is a fundamental domain if it satisfies these two properties:

1. for all x in X, there exists a g in G s.t g.x is in F
2. If x_1,x_2 are in the interior of F and g.x_1 = x_2 for some g in G then x_1 = x_2

[deep mango]

That is very sad

[timid bronze]

[torpid bay]

all right, makes sense ye

hold up

interior of F

i see why making interior of F distinction is necessary now

[latent edge]

yeah do you see why [0,pi/2] doesnt work?

hint: condition 1 fails

[torpid bay]

ye it doesn cover the pi/2 to 3pi/2 part

I would need the cos(x+pi) = -cos(x) thing to do that, which we didn't list as our syms

then should we change the translation thing to this and have another chi for it?

[latent edge]

yeah if you take 3pi/4 i think then it doesnt quite work

[latent edge]

if you want to make it [0,pi/2]?

[torpid bay]

yeah

[latent edge]

yeah you'd need to change how you set your symmetries

you can take T to be x->x+1

but then you have to add some constants to make stuff work nicely right

[torpid bay]

yeah

[latent edge]

but yeah it doesn't matter what you take it is going to be the dihedral group again

[torpid bay]

yep

[latent edge]

Maybe as a fun exercise think about other examples with different groups

Maybe try the cyclic group with n elements or the normal dihedral group

see what you get

[torpid bay]

I feel like z^k would be nice

(ze^(2ipi/k))^k = z^k when k = 1,2,3...

compelx nums tho, stretching can work

(az)^k a^(-k) = z^k

a fundamental domain for that, when allowing stuff outside f(z), feels like there isn't one?

if we don't, then z^2 would be the normal dihedral group then i think

as for cyclic, just z^k = f(z) then (over the ocmplex numbers)

whose fundamental domain would be an infinite pizza slice from r = 0 to inf and theta from 0 to 2pi/k then?

,, a^{x\partial_x}f(x) = f(ax) \ a^{x\partial_x -k}(x^k) = a^{-k}(ax)^k = a^{-k+k}x^k = x^k

[fathom swallow]

Yeatte

[torpid bay]

$T(f)(x) = f(x)$ where T is symmstery vs $ f(g.x) = p(g)f(x)$ such that g is symmetry

[fathom swallow]

Yeatte

[foggy meadow]

Maths

So, the derivative uses a modification of the mean value theorum.

[torpid bay]

ye

[foggy meadow]

So, did you figure out the disc thing?

[torpid bay]

me trying to remember what the disc thing is...

[foggy meadow]

Lacunary functions?

[torpid bay]

ah those, im still thinking about them, the thing in #math-discussion seems to be tantentially related

[foggy meadow]

I'm still working on these sign transforms. I've been working out derivatives. Which I losted the formula for and am finding it again.

I found it though, yay//

[torpid bay]

ima just call it lacunosa town functions

https://tenor.com/view/white-kyurem-legendary-pokemon-tao-trio-dragon-type-pokemon-ice-type-pokemon-gif-3719499543500738301

[grand wraith]

can someone solve this rq

4(2+1)/2+1

not division

a fraction

[neat lintel]

So it's 4(2+1) all over 2+1 if I'm reading it correctly?

@grand wraith

[slow pier]

I think thats wrong

[ancient frost]

i think add 1 after

[neat lintel]

So then it's (4(2+1))/2) +1 ?

[slow pier]

You dont need the extra paranthesis but yes

[neat lintel]

Right, I add them just to simplify it

[young fern]

U made ur account dec25th 2025

[snow ice]

yes?

I chat frequently here, I'm not an advertiser

I'll just delete the post if you want

[young fern]

Yea ion think u can sell stuff here. But if u deleted it it's fine.

[snow ice]

it's just funny cuz of the ram epidemic

[young fern]

Lol is there rlly

[snow ice]

anyways gn

[young fern]

Alr man gn

[sonic field]

@timid bronze don't tell me I don't season MY food...

[hexed tangle]

too less seasoning

[sonic field]

u cray 🤣

[hexed tangle]

https://tenor.com/view/continuous-linear-functional-functional-analysis-normed-locally-convex-topological-vector-space-linear-extension-gif-4860982402928570671

[real summit]

CHROMAKOPIA

[silent junco]

true

i only learned it for normed vector spaces but i guess you can weaken the condition and the proof holds yeah

[delicate plover]

4

YO. wait

I cant do simple arithmatics now

[brave flare]

[old oak]

Ah, the Hanna-Barbera theorem

[hexed tangle]

https://tenor.com/view/topology-complete-reflexive-nuclear-montel-gif-13792408670939158513

[hot lion]

topological vector space mfs coming up with 45 adjectives for their spaces instead of just studying banach spaces like normal people

[hexed tangle]

LMAO

[cyan karma]

Monkeys can regularly beat humans at certain tests requiring intelligence

Like memorization

You may think they are funny but they merely evolved in a different direction

[brave flare]

who is making these

[silent junco]

why dont they just start naming their spaces by the axioms themselves

like T_n spaces

[brave flare]

i dont think T_n spaces are a good example of naming scheme

[old oak]

Apes

[brave flare]

me when regular vs regular hausdorff

which one is T_3? noone knows

[old oak]

Yeah, point set topology is one of my favourite branches of mathematics, but this stuff is just deranged:

The worst part is that according to this, the Arens-Fort space is a perfectly normal topological space

[old oak]

The T_something ones are all Hausdorff, but I only know this when I've taught point set topology recently

Well, the T_0 and T_1 ones aren't Hausdorff, but you know what I mean

I wonder who even studies T_0 spaces specifically? The T_1 class at least has the Zariski topology as a representative that's actually used.

[brave flare]

i dont think i even know rn what T_0 is

isnt T_1 that {x} is always closed

[hexed tangle]

i got them from Ultra

https://tenor.com/view/metrizable-locally-convex-topology-bornological-mackey-gif-12304092366374550948

[old oak]

Yep, T_0 is "given any two points, one of them has a neigbhorhood not containng the other" (but you don't necessarily know which one).

T_1 is "given any two points, each of them has a neighbhorhood not containing the other", and is equivalent to singletons being closed

[sonic field]

I love these so much haha

[old oak]

T_2/Hausdorff is "any two distinct points have disjoint neigborhoods" and is the only separation property worth remembering unless you're some kind of sicko

(and yes, I am some kind of sicko)

[zealous bronze]

apparently the zariski topology on the spectrum of a ring is often not T_1

specifically when not every prime ideal is maximal

[sonic field]

I hadn't gotten far enough in alg or diff top yet to see but is there any areas where knowing a particular space has those properties is important? I mean my goals are knots and associates surgeries but I don't remember seeing anything regarding them in the texts Ive been digging in and out of and I've always wondered that

[hexed tangle]

in Spec R a point is closed iff the prime is maximal most primes aren’t so their closures contain larger primes singletons fail to be closed hence the Zariski topology is typically not T_1

[old oak]

I don't know much about algebraic or difftop, but I can't imagine separation issues coming up there very often.

[sonic field]

I kinda felt that way too. Ok

[old oak]

most of them don't come up very often anywhere, other than Hausdorff.

[hexed tangle]

You usually don’t see it because manifolds and knots in them are Hausdorff by assumption It only starts to matter when you study moduli or character spaces where non T_1 behavior actually carries information

[hexed tangle]

i got sniped

honourable for a reason

[sonic field]

Ah gotcha. I was wondering why there was a reason for those types of classifications. That was before I realized that topology permeates so many other areas of math lol

[old oak]

Yeah, other than Hausdorff (without which you have all sorts of very pathological unintuitive behaviour), the next big thing is metrizability, but that's often not something you can have.

So the other separation axioms are sort of intended to give you a substitute for the kind of things you'd easily get out of metrizability.

(the Urysohn Lemma being a big example)

So that basically means they come up in niche situations suited to a particular kind of spaces studied in a particular kind of area (and topological methods come up in a lot of areas)

[hot lion]

the next big thing after Hausdorff is normality/T4 id say

[old oak]

Fair

[hot lion]

Because of the significance of urysohns lemma

Oh yeah you mentioned that

[hexed tangle]

indeed

[sonic field]

I guess it's not surprising that at some point it would emerge from anything in number theory... Not that I know much about it.

Then I read how generating functions can be used to construct a topological space. But then I guess that's more algebraic anyway.

-sigh-

It reminds me a little of this talk I went to years ago by one of my old profs, I forget wtf she was doin maybe I can find the paper. But she used some sort of stacking technique with higher dimensional structures to also create manifolds. Her jam was fibrations (omg phone stop). There's so much topology to learn ._.

[jagged forge]

mfw closure of {0} is the entire space

[sonic field]

Now that I think about it, feels like the type of manifolds she was carving out was something similar to the grassmannian

... The more I think about it, the more that .makes sense...

Ugh y I doin this early in the morning. I want pancakes

https://tenor.com/view/kitty-cat-cute-slamming-gif-17531055

[hexed tangle]

interesting

[sonic field]

Yeah it was nuts it's like she took slices or something from a number of hypercubes and when she stacked them a particular way it created some sort of manifold structure. I could be very wrong tho, this was back in like 2912

2012 LOL

I'm so cooked

[hexed tangle]

you should eat something

will help ya focus better

[sonic field]

Fr lemme do that.

🌪️

[timid bronze]

I don’t see enough seasoning

[hexed tangle]

https://tenor.com/view/panda-spinning-spinning-panda-giant-panda-cute-panda-gif-4148261905191282079

[timid bronze]

@solar hawk You’re allergic to all of this

@jagged forge Am I right

[solar hawk]

I like the cucumbers or serving

Something*

How come there’s no slowmode in discussy 2?

[old oak]

2912 was a great year

[old oak]

Yeah, there definitely should be

6 hours or thereabouts

[timid bronze]

Not a lot of people talk here

[hexed tangle]

only ally the fish talks here

[timid bronze]

Only Achilleus bullies people

Also why am I a FISH

Are you calling me boring

[hexed tangle]

fish here symbolises a free person

[solar hawk]

Fish

[hexed tangle]

who can swim through the vast ocean

[timid bronze]

I feel like you’re tying this to a negative connotation.

I know your antics.

[solar hawk]

Crazy work

[hexed tangle]

i'm just being kind

[timid bronze]

Mhm…

[hexed tangle]

https://tenor.com/view/sung-jin-woo-gif-12908384644153191582

[solar hawk]

LMAO

[timid bronze]

Would you eat muffins that had veggies in it

[solar hawk]

Ewwww noooo

Why

Why would anyone do such a thing

[timid bronze]

You wouldn’t taste the vegetables

[solar hawk]

Oh

It tastes like a muffin??

Then yea

Ez healthy muffin

[timid bronze]

You need to get your vegetables in someway or another

Since you hate veggies

[solar hawk]

I could just eat one of those

Veggie smoothies Suremark said

But he said it’s as

Ass

[timid bronze]

I can confirm veggie smoothies are ass

Especially celery ones

Veggies are so good

Idk how kids don’t like them.

[solar hawk]

Bro

I like some veggies

Like

Broccoli, zucchini

[jagged forge]

i think those are zucchinis

[jagged forge]

i was talking about meal replacement drinks like soylent

[jagged forge]

zucchini bread is pretty goated

[timid bronze]

You need to try homemade zucchini noodles

They’re bomb

[jagged forge]

zucchini noodles? don’t think i’ve heard of that

actually i feel like that’s something jay would make

[silent junco]

zucchini noodles sounds like it shouldnt work but it does

[solar hawk]

Egg drop soup

[timid bronze]

IT DOESSS

I have it with pesto

And

Charred cherry tomatoes on top

[silent junco]

ooh

yes

[deep mango]

Like... sauteed?

[timid bronze]

@sonic field You both should try since it’s a low carb option

[jagged forge]

i think jay made it before actually now that i read chat logs lol

that looks awesome

[timid bronze]

Whatever you wanna do. I personally like them a bit charred

So like a tiny bit burnt

[jagged forge]

i had pad thai with konjac noodles last week but i think the noodles were overcooked cuz they were mushy sadly…