serious-discussion

[deep mango]

true! slurp is terrible at his native language

[odd narwhal]

Are you?

[odd narwhal]

Slurp's not a native speaker

[deep mango]

:?

[deep mango]

yes!

[odd narwhal]

Idk i'm pretty good at celeste

[deep mango]

i don't believe you

[odd narwhal]

@neat frost confirm

This is true about me tho

[bright hill]

yeah

forgot slurp is a jersey boy

[deep mango]

lol

and now u understand my point!

[odd narwhal]

He also talked about how he was bad at hebrew like 2 sentences up

[bright hill]

lmao, I didn't see that

[odd narwhal]

.

[bright hill]

birthright people?

[odd narwhal]

It's a trip non israeli jewish people are taken on by the israeli govt

Kinda propaganda vibes but it's also really nice

Like idk how much of it is trying to push an image of israel

But a lot of it is just

Hiking in nature

[bright hill]

ah

[deep mango]

yesss

i will be going soon

[odd narwhal]

I am waiting

(ryc has been saying he's gonna be going soon for like 2 years)

[deep mango]

it keeps being a bad time!

[odd narwhal]

I blame your friend

[neat frost]

Yes

[odd narwhal]

But now you're a free bird

Or rather

A free

[deep mango]

if my friend had been able to go

[odd narwhal]

Well, you know

[deep mango]

then i would have gone before i met you

[odd narwhal]

Chrew

[deep mango]

which would have made my life a hell of a lot better

but alas i must engage in charity

[odd narwhal]

Damn i've been on this discord for a little over a year

[deep mango]

i will do you the service of eating your food

[neat frost]

Ryc giving into the propaganda like a good brainwashed little jew

[deep mango]

yes absolutely

shawshank is gonna take me to the big dome

[neat frost]

[deep mango]

you should have taken him up on that offer!

mine now dummy

[neat frost]

The big dome?

[deep mango]

im not meeting shawshank idk who tf that guy is

[neat frost]

Israel has many domes ryc

[deep mango]

yeah but like

the defense one or whatever

[neat frost]

The iron dome?

[odd narwhal]

You mean the iron dome?

[deep mango]

yes the iron dome

[odd narwhal]

First off there's like, 20 something of them

[neat frost]

Smh my head

[deep mango]

what!?

[odd narwhal]

Second of all you're not going near thosw

[deep mango]

no!

fuck

[odd narwhal]

That's military ground

[deep mango]

yeah but you guys have military connections right

[neat frost]

They also now have the iron beam or whatever

[deep mango]

what's the iron beam

[neat frost]

Laser

[deep mango]

wtf

[neat frost]

Pew pew pew Star Wars

[deep mango]

i think i'm just gonna stick to swimming and hiking

[fair mural]

militaries have lasers now ryc

this is starting to be too close to science fiction

[odd narwhal]

Idk any aerial defence people

U can see the dome in Jerusalem

And pray at the western wall

[deep mango]

yeah

well of course i was gonna do THAT

[odd narwhal]

Put a note in asking god for a hot israeli bf

[bright hill]

the first message ryc ever posted was slightly over a year ago

[deep mango]

oh come on

[neat frost]

How funny would it be if ryc got arrested for trespassing in a military base or whatever and started an international whatever with the us?

[deep mango]

"sightly" it was like april 1st

[odd narwhal]

I talked with ryc about this and he told me he's wanted to go for a while

[deep mango]

it's been 1 and a third years

[neat frost]

Such a cliche, you gonna go to the Dead Sea as well?

[odd narwhal]

Wow I only joined like 2.5 months after you ruc

[bright hill]

spot on

[odd narwhal]

I didn't realise that

[neat frost]

Then lemme guess, you wanna go to eilat

[bright hill]

why tf do you have that date remembered? lmao

[neat frost]

It’s April first, important date

[bright hill]

I...

see

[mint canopy]

I have some family down there, I think near Eliat? Might be way off. I really ought to show my face there sometime

But honestly I don't want to go there at the moment

doesn't feel like a good time

then again, when is a good time...

[neat frost]

Usually it’s fine

[deep mango]

yeah i'm a big water dude

[neat frost]

Like even now it’s fine probably

[deep mango]

ofc i want to do the floaty thing

[obsidian bolt]

Hello I have two equations with 1 unknown variable . Can any one get solution for x?
equ 1: (p* multiplier) mod F4 = g
equ 2: (p* multiplier) mod F = g

I want to get one multiplier on this two equation. how can calculate this equations to get same multiplier?

[odd narwhal]

You've been told to read #❓how-to-get-help

[deep mango]

i've wanted to go since i was old enough to

[odd narwhal]

Don't post your question in several channels

[obsidian bolt]

Ok

Sry

[deep mango]

yes but i was relevant within a month and it took you like 8 months to become relevant

[odd narwhal]

No?

[neat frost]

Idk man doesn’t seem sincere, if you really wanted to go you would’ve gone already……..

[deep mango]

well i remember cause it was april fools day and everyone's names were numbers

[odd narwhal]

I got into ivory after like 2-3 months

[neat frost]

It took me like a year to become relevant

[odd narwhal]

Also I was a math baby when I got into this server

[bright hill]

shin is relevant?

[neat frost]

And I’m like the most relevant now

Slow and steady wins the race boys

[odd narwhal]

More relevant than you dark

I also see you gave up your white colour

[deep mango]

i am the most patient person you will ever meet

[neat frost]

So is like a worm

[deep mango]

darq came to me begging for his blue back

[mint canopy]

I'll wait until I meet someone more so

Geddit

bc

I'll wait

haha

[neat frost]

Yes

[mint canopy]

please laugh

[neat frost]

No… don’t explain the joke

[bright hill]

how baby are we talking?

[neat frost]

I was gonna react with a kek until you started explaining…

[deep mango]

literally not baby at all

[mint canopy]

why, the joke was awful

[odd narwhal]

I just finished my 2nd semester

[neat frost]

Pity

[mint canopy]

you should feel ashamed of your sense of humour

pity on thee, not me

[neat frost]

Samezies

So I’m like a newborn shin

Niiice

[odd narwhal]

I knew very basic group theory, some pointset, calc 2, linalg and combo and was reading algtop and real analysis in the summer

[deep mango]

patience is more impressive the more someone has to be impatient for

what's a worm gonna do in israel?

[mint canopy]

eat dirt

probably

[neat frost]

Dry up

[deep mango]

i.e. what many ugs know by the end of their bachelors

[bright hill]

that's like

literally me

wtf

[untold sapphire]

https://i.imgur.com/GulMrGr.gif

[odd narwhal]

I am baby compared to where I am now

[mint canopy]

Hey Clerk

[bright hill]

except AT ig

[untold sapphire]

sup

[bright hill]

but that's coming soon

[neat frost]

What’s the diff between real anal and what you learned in calc?

[deep mango]

(kindergarten)

[odd narwhal]

When I say real anal I mean measure theory

[neat frost]

Oh

Do you have good measure theory books?

[odd narwhal]

Ryc Calculate Tor_1^kx,y

[neat frost]

I kinda wanna check that out it seems interesting

[odd narwhal]

I loke royden

I also have my notes and ankther professors notes

Mine r in English the professor's are in hebrew

[neat frost]

They good?

[odd narwhal]

I'd say

[deep mango]

Tor is like tensor right

[bright hill]

I'd like to see some of your notes sometime

[odd narwhal]

Tor is the derived functor of tensor yes

[neat frost]

Would you be willing to DM them to me?

[odd narwhal]

Sure

1 sec

[deep mango]

hmmm

[bright hill]

me too please

[deep mango]

what's (x, y)

[odd narwhal]

I also have at notes but they are unfinished and also the course was bad so I wouldn't recommend

[neat frost]

Thanks!

[odd narwhal]

Ideal generated by (x,y)

[deep mango]

do you mean k(x, y)

oh

i see

[fair mural]

can’t even do kindergarten math

[deep mango]

kindergarteners do like

[odd narwhal]

Darq what notes do u want

[neat frost]

I feel bad though because I also need to learn ODEs and it’s just so much memorizing that I need to work on

[deep mango]

subtracting numbers!

[neat frost]

Ryc how do I do this

[deep mango]

what

[neat frost]

Memorize

[deep mango]

oh

[bright hill]

hmm

[neat frost]

All these shits

[mint canopy]

I gave my 10 y/o Serre's "Basic Arithmetic" and he says it's too hard, is my child simply stupid?

[deep mango]

well all the ODEs techniques are like

[bright hill]

how about measure theory

[neat frost]

And how come I can’t just laplace everything

[deep mango]

opaque versions of very clear intuitive ideas

[neat frost]

So much easier

[deep mango]

like "i want to turn this into a product rule"

[mint canopy]

Drat, I could've sworn it was called Basic Arithmetic but it's "A Course in Arithmetic" ffs

[odd narwhal]

Ryc the idea is to show by dimension shifting that this is the same as Tor_2(k,k) and then you just calculate directly using a free resolution

[bright hill]

you're learning cookbook odes?

lmfaoo

what a loser

[neat frost]

Wdym

Like it’s some proofs and some techniques

[odd narwhal]

It might be Tor_0 actually idr which direction the dimensions shift

[deep mango]

yes, very poignant. i see. i should have thought of it myself, my colleague.

[mint canopy]

Fuck I've forgotten all my homological algebra ;_;

[neat frost]

Though the proofs are terrible the prof is terrible the course is terrible the questions are terrible and I fucking hate it

[mint canopy]

Really should've thought of that

[deep mango]

Odes proofs are fun

[bright hill]

the ones that go: this is a DE, this is how to solve it, this is why that's a solution

and that's literally all of it on repeat

[neat frost]

They might be, but the proofs the prof gives are like half baked

[odd narwhal]

No I was right it's Tor_2

[untold sapphire]

Boytjie

[deep mango]

oh that's annoying

[mint canopy]

Clerk!

[odd narwhal]

Sad times

[mint canopy]

Please don't hurt me!

[deep mango]

what's tor 2, is there a nice way of thinking about it

i know people say ext 2 is not nice to think about

[bright hill]

clerk and boytjie!

[neat frost]

Some of it is like that, but a lot of it isn’t. Like the first few lectures were literally that but since then it hasn’t really been

[mint canopy]

If you find out, let me know xoxoxox

Idk maybe this is a dumb way to think about it, but like

ext1 and tor1 are like 'holes' but dual in a sense

[deep mango]

yes

[mint canopy]

ext1(A,B) measures holes in A that are shaped like B

and tor1(A,B) measures holes in A that can be bridged by B, I guess

[deep mango]

oh that's a nice way to put it

[mint canopy]

this is how I think about it

And then ext2, tor2 are just gobbledygook let's be honest

[deep mango]

yeah i completely understand that

[mint canopy]

Idk, say "higher dimension" and it makes vague sense

[untold sapphire]

There is the Yoneda ext interpretation.

[mint canopy]

Yeah ofc

of literal extensions, 2-ary 3-ary etc

[untold sapphire]

That's something relatively concrete

[mint canopy]

But tor, oh jeebs

tor's way out there

[odd narwhal]

Tor_2 is always 0 anyways

[untold sapphire]

Nice

[terse knoll]

Hey nerds I need interesting concept for planets and or moons, literally could be anything like a giant jaw breaking, and don’t worry about the physics

[odd narwhal]

At least Tor is in theory easy to calculate

[neat frost]

Isn’t Tor like a browser

[untold sapphire]

yes

[odd narwhal]

Also

Yes

[bright hill]

lmao

it's short for torsion I believe

whatever that is

[untold sapphire]

yeah, it is

[mint canopy]

Just like how ext is short for extension

[untold sapphire]

this is because Tor_1(A, Z/pZ) is the subgroup of elements x \in A, such that px = 0

so it counts the torsion elements of order p

I think also Tor_1(A, Q/Z) is just the torsion subgroup of A

[mint canopy]

Seems right to me

[untold sapphire]

I find myself revisiting the motivation for tor like

fairly frequently

it's always interesting to like

try and go back and justify the definitions computationally

[odd narwhal]

Yes

[deep mango]

what

[odd narwhal]

I just had a question on this like a few days ago

[untold sapphire]

see working through weibel does have some payoffs lol

if it's any consolation i also worked through a bunch of that stuff

[odd narwhal]

I should document mistakes that aren't in the errata yet

So I can get an acknowledgement

[untold sapphire]

nice

[odd narwhal]

Idk if I should email chuck about it rn cuz then it'll be weird when I email him again asking if he's taking students

Imagine if the book got a reprint

Crazy

[terse knoll]

I’m coming up with different types of planets and moon for dnd and I need some ideas for what these planets and moons look like

[deep mango]

alright my idea for a planet is that it's in the exact center of two binary stars, and doesn't rotate. so it's always either midday or dusk, but the sunset/rise is on both sides. the planet is covered in turing patterns of land which have nice beaches on the borders and are about 2 miles thick, the rest of the planet is an ocean which is split into two connected components

[odd narwhal]

U can prove this two ways fairly easily, one is with direct limits which ig is more hands-on and one with LES which is very nice

Like take the tail of the LES for
0-> Z-> Q-> Q/Z->0

[mint canopy]

Oh that's cute

I see that

[deep mango]

the inhabitants are humanoid but somewhat amphibian. there is a wall that stretches the entire centerline of the landmass

[odd narwhal]

The way weibel proves it is that Tor commutes with filtered colimits and Q/Z is the direct limit of Z/n under the maps sending x to m/n x for n|m

[deep mango]

on one side live the people bordering one of the oceans, and the people bordering the other ocean despise them

[odd narwhal]

You have to check that the transition maps become inclusions tho

[mint canopy]

Gross! Go away Weibel!

[odd narwhal]

Lmao

[bright hill]

a planet with a qurtz green/yellowish opaque crust and a yellow core so the core is slightly visible

[chilly hull]

hello ryc

[deep mango]

but the first group doesn't mind their adversaries aside from needing to protect themselves

[mint canopy]

Why's the LES not good enough for him? Smh

[odd narwhal]

That's one of the only times I ever explicitly computed the induced maps until now

[untold sapphire]

i actually like this proof

[chilly hull]

is #foundations just graduate level #proofs-and-logic

[odd narwhal]

With weibel the answer is probably that he didn't think about it let's be honest

[chilly hull]

or is there something more to it that i do not know

[deep mango]

sort of but also sort of not

[bright hill]

and a bioluminescent atmosphere emitting blue light

[untold sapphire]

@mint canopy A modification of this proof shows that the torsion free Abelian groups are flat.

[odd narwhal]

Both proofs generalise easily to any integral domain ofc

[deep mango]

bioluminiscent atmosphere

is the atmosphere alive

[bright hill]

maybe

[mint canopy]

OK fiiiine that's cool I guessss

[bright hill]

also, the crust might be made up of slime

[odd narwhal]

That one's nicer cuz everything is 0

So no need to check transition maps

[terse knoll]

Amazing

[untold sapphire]

It is kind of weird to talk about applying Tor to injective objects

Hahaha

[odd narwhal]

Wdtn

Wdym

Oh

[untold sapphire]

Tor(A,Q/Z)

[odd narwhal]

Cuz Q/Z is divisible

Yea

Fun fact: Weibel.uses this fact without ever stating it (nor in the errata or appendix)

[untold sapphire]

Q/Z is like the canonical injective object. my monad brain is like

[odd narwhal]

I guess you're just supposed to know that in Z-mod injective iff divisible

Even tho he defined injective objects earlier

Actually it might be an exercise but i'm 99% sure it's not

[untold sapphire]

I think you're supposed to use Baer's criterion

[odd narwhal]

Yea sure clerk but he never states this outright

Which if you didn't know this fact you might not get what he's doing

But yea one direction is baer's criterion

[odd narwhal]

Ok my bad then

In my defence

He usually cites this stuff if it's in the book

[mint canopy]

CAn we just take a moment to appreciate how epic Baer's criterion is

[odd narwhal]

Baer's criterion is very based

[mint canopy]

I really think it's a great theorem

[odd narwhal]

Does the corollary follow from the exercise or from baer's criterion?

[untold sapphire]

You can do a lot of homological algebra without using Baer's criterion using the monad approach to homological algebra

like, you can prove a lot of the same theorems about ext and injective resolutions but without using the axiom of choice or anything

[odd narwhal]

Interesting

[untold sapphire]

in short like, there's an approach to homological algebra which, instead of being based on these notions of projective injective etc which make sense in any abelian category

you can have a kind of customized approach that treats certain objects as projective/injective etc

like

the first time i came across this was in the acyclic models theorem

there's a notion of a projective/free functor C -> Ab but

it's not literally projective in the sense of being a projective object in the abelian category

similarly there's a notion of an acyclic complex of functors but

it's not literally exact in the sense of the abelian category structure

instead you're working with these relativized notions

[odd narwhal]

Complex of functors?

[untold sapphire]

Yeah like a chain complex in the functor category [C, Ab]

[odd narwhal]

Oh ok

[untold sapphire]

the acyclic models theorem gives a notion of 'free' functor and an 'acyclic complex' of functors which allows you to carry out the proof of the comparison lemma

but they aren't like, just projective and acyclic in the usual sense of an abelian category

which is cool

i should talk more about this another time but i don't want to get into it rn

[odd narwhal]

I'll be happy to hear about it

Sounds really cool

Maybe I can force howard to learn this with mr

[mint canopy]

if you add an s it becomes showard

this came to me in a dream

[steel wind]

• past tense or n = shownard

a to e so its shownerd

so its show🤓

wonder why dead chat

[fervent pebble]

wtf it's pronounced REEK????

[ancient flame]

LOL

rick

[fervent pebble]

im gonna keep on calling it rike or rice

[neat frost]

Rycie bycie-shmoochikins-gigglehead-muffin

[ancient flame]

rycicle

[neat frost]

Ugh gmod your nickname scared me for a sec

[ancient flame]

LMAOOOOO

[odd narwhal]

No, that post was a joke

Ryc also didn't join in september

[fervent pebble]

thank god

[odd narwhal]

Should pin this

Tbh

I could

[open aspen]

you have the power for a reason

power is meant to be abused

[odd narwhal]

Chrew

[bright hill]

can you ping this again while you're at it?

this deserves to be pinned more than once

[odd narwhal]

You can't pin the same post twice

[calm thistle]

So, sometimes fractions are written with the partial part direct,y adjacent to the whole part and nothing but a space in between. But this notation signifies multiplication.
1 1/4 obviously = 1•1/4 = 1/4

[leaden torrent]

the same notation can mean different things in different contexts

"bras" in french means your arm

"bras" in english does not

[calm thistle]

Its annoying because the basic notation of the ten digits and a slash for division is still their.

Why bother writing it with an arithemtic operation if youre just going to have the wrong operation somewhere else

[leaden torrent]

i have literally never been in a context where this posed a problem for understanding

if im reading a recipe, i know 2 3/4 means 2 + 3/4

since otherwise they wouldve just written 6/4 (or 3/2 or 1.5 or whatever)

[odd narwhal]

And "bras" in physics are a notational abomination

[calm thistle]

But 2 3/4 just = 6/4
Why use a slash for division if you're just going to use a space for addition instead of multiplication. The slash follows standard notation while the sapce does not.

Whats even worse is using a dash for addition.

[leaden torrent]

this just seems like a very silly thing to complain about

[calm thistle]

1-1/4 should = 3/4
But in measurement 1-1/4=1+1/4

Why not just type a plus symbol instead? It's not that difficult

[leaden torrent]

like youre smarter than all those recipe writers since you know what the "correct" notation is

and want to prove that by bitching even though this proves literally 0 barrier to understanding

[calm thistle]

Im not any smarter than them. It just confuses me

[leaden torrent]

i have never seen 1-1/4, though.

that just seems bizarre

[calm thistle]

Well I just got it on a Quaker corn meal corn bread recipe. It's sorta weird using a minus when they mean plus.

Anyways, obviosuly the minus must indicate addition in the contrxt of measurements, becuase it would be simplified to 3/4 otherwise

[bright hill]

why was that the first example that came to mind?

[leaden torrent]

googled french false friends and its the first one that came up

[bright hill]

ah

lol

[tender tulip]

wtf is a direct and inverse limit

[frozen merlin]

what a great sticker

[storm sage]

direct limit is when it is equal to kL, inverse limit is when it is equal to k/L

thank me later

[eager reef]

It’s when you invert the limit

[tender tulip]

Me about to ask what an inverse limit is knowing damn well it would be funny if you said it’s the dual of the direct limit

[odd narwhal]

Direct limits are (filtered) colimits

Inverse limits are (cofiltered) limits

Have fun with that one

[zealous garden]

Ultraproduct desu

[storm sage]

groups of order above five hurt my head

becoming an ultraultrafinitist brb

[odd narwhal]

Personally I don't believe numbers exist

[bright hill]

I don't know about you guys

but I don't believe anything exists

[zealous garden]

I don't believe in zero

Nothing doesn't exist

[bright hill]

nothing can't exist if nothing exists

[zealous garden]

Empty sets are a logical impossibility

[ancient flame]

I went to

[zealous garden]

Something exists, therefore nothing does not exist

[storm sage]

Personally I believe paradoxes don't exist

Any paradoxes are a result of your interpretation of my logical system, not my logical system

[zealous garden]

Cope

[errant ridge]

Lmao

[sick burrow]

Does anyone know if a good resource for conic sections

Like a proper formal treatment that's aimed at math students trying to fill holes not highschoolers that just want formulas and don't understand rigor

[wooden flax]

like high school level conic sections?

but deriving the formulas

[sick burrow]

Yes

[wooden flax]

oh then AoPS intermediate algebra does exactly this

[sick burrow]

Reviewing for the GRE subject test and want to actually learn them properly instead of memorizing formulas

Damn it's not free

Not sure if I'm ready to swallow my (probably unfounded tbh) malware fears and try ||libgen|| yet

Do you know of a free resource

[cyan goblet]

if website-who-shall-not-be-named has malware, then i would be dead by now

[sick burrow]

Ok but I'm stupidly paranoid

Also I don't need an entire 700 page textbook

[wooden flax]

i promise you its 0 malware

lol

also how have you gotten this far without it lmao

[sick burrow]

I haven't needed that many books

[open aspen]

ok what is algebraic geometry though

it doesnt look very geometric

no pictures

so is it algebra but differenter?

[errant ridge]

as far as I know, it's studying the zeros of multivariate polynomial functions using tools from algebra

but I think modern AG is much more general or something

with "schemes" and stuff

[vivid halo]

all the examples of complex algebraic varieties (sets of solutions to systems of polynomial equations in affine/projective space) are pretty geometric

[errant ridge]

didn't you hear what they said? "no pictures" no pictures of complex algebraic varieties

im kidding

[vivid halo]

there are definitely pictures of Riemann surfaces but yeah anything beyond that is hard to draw for dimension reasons

[errant ridge]

https://tenor.com/view/stimulated-3d-gif-19844268

I think that's a 2D slice of a 6D complex algebraic variety

but 3 complex dimensions

[vivid halo]

some of the examples more closely related to number theory that are accessed through the modern language of scheme theory are still fairly geometric and there are ways they behave analogously to Riemann surfaces and things like this

[errant ridge]

where does the algebra part come in?

[vivid halo]

I mean all of these spaces are built algebraically

[errant ridge]

like the algebraic varities are given a group or a ring structure?

[vivid halo]

in the language of schemes, these are all locally of the form Spec(R) for some commutative ring R

[errant ridge]

"schemes" what even are they

all I know is that they're super abstract

[vivid halo]

well do you know what Spec(R) is

[open aspen]

There’s just so much scary terms in alg geo. It’s like math 2.0

[errant ridge]

I only know what a ring is

[austere kettle]

Hey

[errant ridge]

spectrum of a ring? no idea what that is

[vivid halo]

do you know what a prime ideal of a ring is?

[austere kettle]

Is anybody good at stats

[open aspen]

#❓how-to-get-help

[austere kettle]

And understand box and whisker graphs

[fading pulsar]

oh no

[austere kettle]

Tried nobody could help

[fading pulsar]

No one here understands box and whisker graphs

[austere kettle]

Why tho

[errant ridge]

lol I guess I'll come back after learning more algebra, I don't understand ideals of rings yet

[vivid halo]

[errant ridge]

[open aspen]

I’ll take the free lesson

I know what a prime ideal is let’s goooo

[vivid halo]

provided you know what a prime ideal is, Spec(R) is the set of all prime ideals of R

[austere kettle]

Migillope do you know box and whisker

[vivid halo]

you equip this with a certain topology

[open aspen]

No I only know what a prime ideal is sorry

[errant ridge]

wait so a prime ideal is just the ideal of a ring that's similar to the prime numbers in the ring of integers?

like that idea but generalized?

[vivid halo]

yeah exactly, for the ring Z the prime ideals are (p) for p prime, and (0)

[errant ridge]

huh okay

[vivid halo]

where (x) means the ideal generated by x

[errant ridge]

[errant ridge]

I see okay

[open aspen]

Spec(R) is a topological space defined on prime ideals of a ring?

[vivid halo]

yeah you define a topology on the set of prime ideals of a commutative ring called the Zariski topology

[errant ridge]

it's a topological space where the points are prime ideals?

[open aspen]

I expected it somehow to have something to do with spectral sequences

[errant ridge]

that's pretty cool

[vivid halo]

the reason why this is a decent notion is like

for example if you take the polynomial ring k[x] in 1 variable where k is an algebraically closed field

like let's say C[x]

[errant ridge]

okay

[vivid halo]

the prime ideals are just like (x-a) for a in C

and (0)

[errant ridge]

[vivid halo]

so Spec(C[x]) really behaves like C again

with this extra point (0)

[errant ridge]

what the hell

I did not expect that-

that is a decent notion

[vivid halo]

so a scheme is just like

some space that locally looks like Spec(R)

[errant ridge]

oh so kinda like a manifold

but instead with spec(R)

[vivid halo]

similar to how manifolds are spaces that locally look like R^n yeah

yeah exactly

[errant ridge]

oh niceee

so

if you wanted to study 3 variable polynomials from the ring R, you'd use spec(R[x, y, z]) and look at the scheme locally homeomorphic to that?

[vivid halo]

Spec(C[x,y,z]) would be affine 3-space

[errant ridge]

or is that not what a scheme is used for?

[vivid halo]

and in there you could study solutions to polynomials in 3 variables

so varieties in A^3

[open aspen]

Any baby examples

Like S^2 for manifolds

[vivid halo]

a good example of a scheme that is not affine is the projective line P^1

[austere kettle]

@vivid halo can you help me

[errant ridge]

#❓how-to-get-help

[open aspen]

No they only know spec(R)

[vivid halo]

you get this by gluing two copies of the affine line A^1 together

[errant ridge]

and affine 3-space is just any space that's isomorphic to an affine subspace of R^4? (R^4 the vector space)

[vivid halo]

(complex) affine 3-space is like C^3

[errant ridge]

ah okay

[fading pulsar]

wth affine space is this?? I saw it for like 2 lectures in my first class in diff geo and all it was introduced as was a vector space with origin removed

[errant ridge]

,w graph y^2 = x^3 +2x + 3

[fathom swallow]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by Neamesis#6134

[errant ridge]

would that be an algebraic variety?

actually nvm

[vivid halo]

yeah that's a great example of an algebraic variety

[errant ridge]

huh

[vivid halo]

that's an affine elliptic curve in A^2

[errant ridge]

I thought varities were "sets of zeroes"

or yk set of solutions

[vivid halo]

I mean you can rearrange the polynomial

so that it's like f(x,y)=x^3+2x+3-y^2=0

[errant ridge]

hm right yeah makes sense

[vivid halo]

usually you would consider the corresponding projective elliptic curve in the projective plane P^2

[errant ridge]

okay so in that graph it's a variety in R^2, so using schemes would an algebraic variety be on the scheme?

[vivid halo]

so instead of the zeros of a polynomial in 2 variables, you look at the zeros of a homogeneous polynomial in 3 variables

you can homogenize y^2=x^3+2x+3 as y^2z=x^3+2xz^2+3z^3

so just multiply everything by z until the total degrees match

[errant ridge]

I see

[errant ridge]

that's also a scheme?

[vivid halo]

yeah you can make this into a scheme

the affine elliptic curve y^2=x^3+2x+3 is the affine scheme Spec(C[x,y]/(x^3+2x+3-y^2))

[errant ridge]

hm but why do schemes help you? like why can't you use just C^2 or whatever instead of the scheme which resembles that

[vivid halo]

so one reason for this is that the scheme has extra points that encode some additional information

for Spec(C[x]) we had just one extra point, (0)

for Spec(C[x,y]) we have a lot of extra points

[errant ridge]

[vivid halo]

we have the generic point (0), but also the generic points of irreducible affine curves

so like we have lots of points which are not closed

which is weird

[errant ridge]

so you have to learn commutative algebra in order to get to AG?

what even is commutative algebra? like algebra with rings but looking more closely at commutative rings?

[vivid halo]

yeah definitely, since the spaces are built from understanding, among other things, prime ideals of commutative rings

yeah commutative algebra is just studying commutative rings and modules over them

[errant ridge]

I see okay

thanks!

[vivid halo]

[errant ridge]

wait nG what exactly are sheafs? are they used in AG too?

[vivid halo]

oh yes they are very much used in AG and are the main thing that I didn't mention in the definition of schemes

you can talk about sheaves of sets, Abelian groups, rings, modules, whatever

a nice motivating example is sheaves of rings as sheaves of functions

[errant ridge]

[vivid halo]

so imagine we have a space X and we want to associate to each open subset U of X a ring of functions O(U) on U

[errant ridge]

a ring of functions? O(U)?

f : U -> U?

[vivid halo]

yeah some ring of functions, maybe X is like a smooth manifold and O(U) is the ring of smooth functions U->R

[errant ridge]

ah okay

[vivid halo]

so keeping standard examples of rings of functions like this in mind what sorts of properties should this satisfy

one basic thing is that if you have two open subsets V≤U≤X then you should get a map O(U)->O(V)

since you can just restrict a function f on U to a function f|_V on the smaller set V

[errant ridge]

right

yeah

[vivid halo]

there's some obvious compatibilities, like if I have three open subsets W≤V≤U≤X then the composition O(U)->O(V)->O(W) is O(U)->O(W)

and so on

so far this tells us this assignment O is a presheaf, meaning it's functorial in the above sense

the thing that makes a sheaf in addition to these compatibilities is gluing:

if you have an open cover {U_i} of X, so a bunch of open subsets U_i with union X

and if you have a function f_i on each U_i

if f_i and f_j agree on each overlap U_i\cap U_j that means you should be able to glue

so you should be able to produce a function f on X whose restriction to each U_i is the corresponding function f_i

[errant ridge]

and they agree on the overlaps so it smoothly varies over X?

[vivid halo]

it's not really about smoothness, just that the values of the functions are compatible with each other

[errant ridge]

ah okay

[errant ridge]

okay

that's called gluing?

[vivid halo]

yup

[errant ridge]

ah I see

[vivid halo]

the other condition is separation: if you have two functions f and g on X

and they agree on each U_i

then f=g

[errant ridge]

ooh okay

so that's a sheaf of rings?

[vivid halo]

yeah these two gluing conditions mean this assignment is a sheaf

so like the assignment of rings of continuous/smooth/holomorphic functions on open subsets of topological/smooth/complex manifolds, all these are sheaves of rings on the space

schemes have sheaves of rings of functions too

[errant ridge]

oh wow

nice

lol I'm assuming I'd need to know more math in order to understand why sheafs are used a lot

[vivid halo]

I mean they're used a lot because they're a convenient way to organize the data of being able to talk about functions on a space

[errant ridge]

hm I see

[vivid halo]

packages everything up very nicely

[errant ridge]

that kinda makes sense

thanks

[vivid halo]

[errant ridge]

so you can have a sheaf of rings of linear transformations (matricies in a given basis i guess) of a finite dimensional vector space with the default topology

[vivid halo]

hmm maybe this isn't the best example

[errant ridge]

no i mean im just asking if it can be applied to vector spaces

[vivid halo]

oh sure well like one thing you can do is look at modules over sheaves of rings, or sheaves of modules

[errant ridge]

[vivid halo]

these are very closely related to vector bundles

[errant ridge]

oh really?

[vivid halo]

well so in the smooth setting you have the Serre-Swan theorem

that if you have a smooth vector bundle E over a smooth manifold X

then you can regard the space Γ(E) of smooth sections of E as a C^∞(X)-module

where C^∞(X) is the ring of smooth functions on X

this module is finitely generated

when X is connected, every finitely generated C^∞(X)-module arises in this way

[errant ridge]

this just went from 0 to 1000 in 2 seconds

[errant ridge]

hm that's a ring?

[vivid halo]

yeah since you can add/multiply functions pointwise

[errant ridge]

[vivid halo]

so what this says is that vector bundles are very closely related to modules over the ring of functions

in the smooth case we didn't need the entire sheaf of smooth functions

so we didn't need to consider like, C^∞(U)-modules for all open subsets of X

we only needed to consider a single C^∞(X)-module

there's other situations like complex manifolds where you really need the whole sheaf of functions and you can't just look at functions on all of X

[errant ridge]

I see

[gritty copper]

<@&268886789983436800>

[brittle socket]

Damn im genuinely suprised you are not banned yet

[gritty copper]

Mods on a holiday

[errant ridge]

what happened?

[surreal sapphire]

there was spam, i dealt with it

[errant ridge]

[arctic grove]

i hacked ur account and am about to disable it

[errant ridge]

[arctic grove]

gonna force you to study

[errant ridge]

I hacked my account before you did

[arctic grove]

[sleek wing]

Riveting

[bright hill]

I dreamt there was a Neamesis category with three channels specifically for Neamesis

And then a bunch of news articles flashed with headlines like

"the pilgrim of discord servers"

[arctic grove]

[bright hill]

I opened one and it read

"no one knows why he's doing this"

And then I woke up

[errant ridge]

[arctic grove]

Go back to sleep

right now

and continue

[errant ridge]

That wasn't a dream DarQ

[bright hill]

Lol

[errant ridge]

that was just you foreseeing the future

[arctic grove]

it was the opposite of the future

[sleek wing]

The past

[errant ridge]

thank you for your future insight oracle, you may now rest again

[arctic grove]

i meant to say the opposite of what happens in the future

[errant ridge]

grammar fail

[bright hill]

Anyways

[errant ridge]

yes no one knows why I keep extending everything out to infinity

[bright hill]

That's how I found out I'm spending a wee bit too much time on this server

[sleek wing]

Fuck this hizzaouse dawg!

[arctic grove]

Spend more time with me

[bright hill]

[errant ridge]

same that's why I log out every now and then

to keep at least a part of my mind sane

[bright hill]

Lmao

[midnight shadow]

hello

[bright hill]

Hellollo

[midnight shadow]

i just got a question

[bright hill]

#❓how-to-get-help

[arctic grove]

u dont do it to study??

wth

[errant ridge]

both

[arctic grove]

[bright hill]

Shyshu backstabbed

[arctic grove]

im used to it now

both of my friends using books i dont believe anyone should use

sadge

u sullied me for no reason aenigmata

[errant ridge]

[arctic grove]

so now i return the favor

[bright hill]

Allufi is fine

[arctic grove]

I know

[errant ridge]

so is D&F

[bright hill]

(╯°□°）╯︵ ┻━┻

[arctic grove]

but artin is better!

[errant ridge]

[arctic grove]

and doing category theory early on is a sin

[errant ridge]

not comprehensive = not good

[errant ridge]

[bright hill]

Artin isn't necessarily better

[arctic grove]

Oh hey its yohan

hello yohan

[arctic grove]

it is!

coz im using it

😌

aluffi dont got shit on that

[bright hill]

No you're not

[arctic grove]

i am

im finishing the LA part for now

[errant ridge]

the only reason he likes artin is because he has a physical copy of it

[sleek wing]

*I am not

[arctic grove]

im just skimming the theory part and then doing questions

[arctic grove]

wew i am!

😭

[arctic grove]

i got the copy coz i loved the book

smh

[errant ridge]

[arctic grove]

im reading artin rn

i know most of the things he is saying in the LA chapters

[errant ridge]

I'm eating artin's book

[arctic grove]

who knew shitposting about LA for a year and a half would teach u a bit

[arctic grove]

wat

[errant ridge]

oh really? how do you find the spectral radius of a matrix?

[bright hill]

At any rate, I'm not sold on allufi yet

[sleek wing]

Eigenvalues innit

[errant ridge]

you will be

[arctic grove]

idk

[arctic grove]

good good

go back to artin

[sleek wing]

Plus who actually cares about the spectral radius lol that’s like numerical analysis shit

[errant ridge]

artin readers be like:

[arctic grove]

be based

[arctic grove]

[errant ridge]

[bright hill]

I'm not sure if categorical treatment of algebra for a first intro is some pedagogically

[arctic grove]

neam really wants me to block him

and still bully him

[arctic grove]

exactly!

[errant ridge]

[arctic grove]

reading it later can be beneficial

but absolutely dont recc it

coz category theory is evil

[gritty copper]

Not reading it is more beneficial

[arctic grove]

based

[errant ridge]

cocat

[bright hill]

And I'm not picking anything till I'm done with anal

So fret not

[errant ridge]

real anal or complex anal?

[bright hill]

Hope still remains

[bright hill]

Real

[errant ridge]

in R^n? or metric spaces?

[neat birch]

why do i see anal the first thing i come back to this chat

[bright hill]

Just R

[errant ridge]

nice

[errant ridge]

because mathematicians love real analysis

[arctic grove]

neam when something isnt infinite:

[neat birch]

that abbreviation tho...

[bright hill]

Neam, are you an undergrad?

[arctic grove]

neam is in the same year as me

[errant ridge]

im about to be an undergrad in like 1 or 2 months

hopefully

[bright hill]

I see

[bright hill]

Lol

[neat birch]

also any of ya'll know how to do the harder inequality proofs

[errant ridge]

nope

[neat birch]

aw

[errant ridge]

no one in this server knows

[neat birch]

damn

[bright hill]

I also struggle with real analysis

[errant ridge]

it's a secret

[bright hill]

[neat birch]

uwu

[arctic grove]

then go study for the exam that will get u a uni!

[errant ridge]

time to log off again

[neat birch]

[errant ridge]

bye chat

[neat birch]

👋

[arctic grove]

hoping to never see you again axler user see you soon!

[errant ridge]

[arctic grove]

[frozen merlin]

are you waiting for results or is there an exam you haven't done yet

[errant ridge]

the exam is in like 8 days

7 days

[frozen merlin]

isn't that kind of late for an entrance exam

[neat lintel]

JEE

[errant ridge]

yea it got pushed back due to the covid stuff and other exams' date clashing with it and stuff

[neat lintel]

after 2nd attempt i have barely studied anything

[errant ridge]

pure math is more fun

[arctic grove]

but pure math wont get u a college
as sad as that is

😭

[errant ridge]

😭

[neat lintel]

this is why i'm listening to this song for the last 20 minutes

to be sad

😭

[arctic grove]

😭

[errant ridge]

imagine being sad when you could be happy

or angry

[arctic grove]

imagine feeling emotions when u could be doing LA

[neat lintel]

imagine not imagining

bruv this song depressing af

[deep mango]

JEE Doomer hours

[neat lintel]

still listening to it

ye

[arctic grove]

wtf

[sleek wing]

Half past JEE Doomer hour

[cyan goblet]

50 past JEE Doomer hour

[arctic grove]

Im not a jee doomer!

😭

[ancient flame]

24 hours!!!

[arctic grove]

Yay!

[errant ridge]

morbism

isomorbism

homeomorbism

automorbism

[gritty copper]

Morbomorbism

[bronze pelican]

Morbiomorphism

[open aspen]

Just made scrambled eggs for the first time in a while in my cast iron and nothing stuck!

[errant ridge]

Nice

[open aspen]

Last time I did, just a week or so after I got it, everything stuck and it was hell

Not so this time!

I'm proud if Mr. Ciron

[errant ridge]

looks nice

[warped smelt]

how many egss is in that

[open aspen]

4

but i also put in like 3 tbsp of water

makes em fluffier

[warped smelt]

never heard of that

does it taste good?

[bright hill]

nani

[warped smelt]

tbh i would add milk or heavy cream

[bronze wedge]

i do this too

it works way better than milk

[chilly hull]

i add sour cream

[tender tulip]

Can you view finite extensions possibly via homomorphisms in some way

the whole basis thing reminds me of the square of an ideal but it's not directly related in any way

[tender tulip]

you can for algebraic extensions

along with saying every intermediate ring is a field

or is there just any equivalent statement using polynomials instead of considering it as a vector soace

*space

[full isle]

L eggs

[full isle]

this also dilutes the taste of eggs if you dislike it

its a good strategy

[full isle]

lol wdym

dimension of any extensions is determined by dimension of a quotient as a vector space

the way you phrased your question is funny though

[zealous garden]

what's wrong with Axler

[wooden flax]

down with determinants

[bright hill]

shyshu be like: I don't use it therefore it's bad.

[zealous garden]

I see

[deep mango]

@arctic grove people subtweeting... hmmm

[zealous garden]

what subtweeting, he was @mentioned when I replied

[deep mango]

Oh i didn't notice LOL

[zealous garden]

huehuehue

[deep mango]

alright shyshu you got owned

[zealous garden]

LMAO

[deep mango]

also good morning for when u see this :)

[sudden raven]

Are the symbols within the parentheses (not sure how to spell it). Does anybody know what they mean?

[bronze wedge]

For the future #❓how-to-get-help

[sudden raven]

Even for silly questions like that

[bronze wedge]

Probably best to go to the physics server

[storm sage]

solar constant

[sudden raven]

Oh

[storm sage]

er wait

that's not what it's called

solar radius lol

solar constant is something different

[sudden raven]

So let me see if i understood

One solar radius = radius of sun

[storm sage]

yeah exactly

(we define it to be exactly 695700 km)

(since the actual radius of the sun varies)

at least I think that's exact, I'm not sure

[sudden raven]

Does it ever get smaller

[storm sage]

actually it might not be exact but w/e

[storm sage]

it fluctuates yeah

also depends on where you measure from

if you want more detail I would ask on the physics server

check #old-network

[sudden raven]

Do things have to be "sphery" for you to measure its radius

Or does everything have a "radius"

[leaden torrent]

for the radius to be constant, it needs to be a sphere.

but you can certainly measure a radius-like thing for other objects

we call this "distance from centre to surface" or something similar

the definition of a sphere is as a body where this distance is constant

i.e. |(x, y, z)| = x² + y² + z² = r²

[sudden raven]

Another similar question

Does everything have sin, cos, tan

Ive only ever seen people use it in triangles

Im not very good at math ;-;

[leaden torrent]

Most Christians believe that everyone has sin, yes.

[storm sage]

Lol

[leaden torrent]

IDK about tans, my impression is that certain complexions are more prone to tanning

[storm sage]

sine, cosine, and tangent are usually introduced using triangles

but you can take sine, cosine, and tangent of any angle

in general

and even of complex numbers!

[sudden raven]

What is a complex number :^

[leaden torrent]

not using a norm st |Re(z)| = |Im(z)|

uncreative.

[storm sage]

nami what are you talking about lol

[storm sage]

it's a number system where you can have the square root of -1

[leaden torrent]

"the "

[twilit viper]

oe the square root of any negative number

[sudden raven]

Is the square root of -1 an actual number

[leaden torrent]

it's a complex number

"actual number" is not a mathematical term

[storm sage]

shhhhh

[sudden raven]

Or did we just kind of said its imaginary and left it at that

[leaden torrent]

we say it's imaginary, but this is a classification term more than a philosophical one

it's like the "rational" vs "irrational" distinction

just a convenient term to use

not anything with metaphysical implications

[zealous garden]

It's a bivector

[twilit viper]

talking about “i”, what would be the square root of “i”?

[leaden torrent]

sqrt(2)/2 + i sqrt(2)/2

[zealous garden]

yep

easy

[leaden torrent]

,calc (sqrt(2)/2 + i * sqrt(2)/2)^2

[fathom swallow]

Result:

1i

[leaden torrent]

1i
no need to get so pretentious on us TeXiT, just say i

challenge: try to see geometrically why this makes sense

(what does "squaring" correspond to on the complex plane? what does "square rooting" correspond to?)

[twilit viper]

:0, i’ll try to, but now i want to go to sleep Xd, enough math for me

[leaden torrent]

this tool might be helpful: https://ventrella.com/ComplexSquaring/

in particular, try the "circle" and "imaginary" options

[twilit viper]

thank you

[young rain]

@small mist brb sry

[arctic grove]

determinants

[arctic grove]

its the other way round, the book is bad, hence why i dont use it

😌

[arctic grove]

thonk

[neat frost]

Good morning shyshu

[arctic grove]

good morning slurp!

[river spire]

physicists call eigenvectors which share an eigenvalue to be degenerate, what do mathhead call it?

[neat frost]

Vectors in the same eigenspace?

[river spire]

yeah

[blazing atlas]

scratched line

[open aspen]

TAing my first class as primary instructor this week 🙌

Starting this week, anyway

[cyan goblet]

good luck!

[open aspen]

It's T/Th from 630 to 820 PM

What a nightmare

[cyan goblet]

oh

good luck staying awake

what are you TAing, if you dont mind me asking?

[open aspen]

Elementary functions, basically college algebra and trig/precal rolled into one

[cyan goblet]

gotcha

[hollow sundial]

dinner time? oh hells no

[open aspen]

For real, I'll be on campus from 8am to 820pm

Gonna have to pack a lunch and dinner

[spare topaz]

Is it true that in 1942, Einstein gave the same physics exam to his students as in 1941, because the answers had changed?

[open aspen]

My cat is chonky and I don't know how to make it stop :(

She's already on diet food and I've adjusted her calorie intake according to vet instructions. I only give her 2 treats per day

She's pretty active too

[spare topaz]

Does she go outside?

[open aspen]

No she's an indoor cat

[spare topaz]

Does the vet know that?

[open aspen]

Yes

[spare topaz]

Mmm then the diet he prescribed should work

How long has she been on it

[open aspen]

Not long, but its actually a little more than I was giving her before

[spare topaz]

Perhaps the cat is just a phat cat

[odd narwhal]

she might be stealing food

[open aspen]

She only has access to her bowl

I got her a leash though, maybe I can start taking her out

[bronze pelican]

Me too!

[tender tulip]

Let $F$ be a field, $K$ a subfield, and $\sigma$ an automorphism of $F$,
For all $x \in K$, it's not always true that $\sigma(x) \in K$, right?

[fathom swallow]

Mizalign

[bright hill]

New pfp!

Eyyy