serious-discussion

[wooden flax]

there might be*

theres a theorem relating finite extensions to lattices which i dont recall at the top of my head

[tender tulip]

is there a category theoretic interpretation of seperable and normal extensions

i know there is one for algebraic ones but

[wooden flax]

i dont know of any but there might be

(not saying there isnt, just that i dont know of one)

[vivid halo]

in char 0 every algebraic extension is separable so you only have to treat the case of char p

and then there's a characterization in terms of the p-power Frobenius

[tender tulip]

I'm saying for arbitrary characteristic

[vivid halo]

a field F of char p is perfect iff all irreducible polynomials over F are separable, equivalently if the p-power Frobenius F->F is surjective.

so you just want a similar characterization of separability in terms of the Frobenius and then you're done since every field is either char 0 where this notion is vacuous, or char p

[tender tulip]

ah

alright, but what about normal extensions

[vivid halo]

normal extensions have a characterization in terms of automorphisms

that or like in terms of images of morphisms into the algebraic closure

[tender tulip]

hm

[wooden flax]

|Aut_extension(F)|\leq[F:extension] if the extension is normal

[vivid halo]

an algebraic extension L/K is normal iff every homomorphism L->\bar{K} has the same image, iff Aut(L/K) acts transitively on the set of homomorphisms L->\bar{K}

[tender tulip]

hmm

[vivid halo]

this feels fairly category theoretic to me

or at least as much as it can be

[tender tulip]

fair enough

I am going to see if I can construct the algebraic closure of an arbitrary field

[wooden flax]

good luck

[tender tulip]

i realized that this is a pain

[bronze pelican]

U need to use Zorn lemma at some point

[zenith tinsel]

hey yamin

[modest rune]

Its kind of hard for me to tell what a course-hour looks like

but just in terms of the list of classes

I think you could do it in two years

motivated by $30,000 I think its probably doable

That said, I do think that spending the extra year doing even more classes would improve your application

[azure nymph]

Also an extra year would give you a better chance of REUs, research with profs at your school, and building rapport with professors will be important for grad school

[south kelp]

Is getting into a better grad school worth 2 or even 3 extra years in undergrad? Cause I could either just spend like 3.33 extra years in undergrad by transferring back to my original undergrad (Caltech, I left for mental health reasons that I've mostly dealt with at this point) and probably improve my chances at higher rated grad schools or go to a lowish tier grad school but finish my undergrad at my local state school in 3 years and get a masters in 4. I figure I could just make up for the lack of prestige by doing post docs?

[vivid halo]

"doing post docs" is a big assumption lmao

just stick with what you're doing there's no point in trying to reroll and optimize like this

[south kelp]

alright cool

I mean it is a huge assumption but it helps me sleep better at night lmao

[vivid halo]

school prestige does play some role in postdoc applications but honestly the biggest factors are publications, knowing people, and luck

luck is a big one

[azure nymph]

There's also prestigious people at lesser known universities iirc Bella Bollobas is at University of Memphis which as far as I'm aware is not very up there in prestige

[errant ridge]

so you gotta publish as many good papers as possible when you're in grad school?

[vivid halo]

yeah 100%

[errant ridge]

ngroupoid did you finish your PhD?

[vivid halo]

no I'm in the middle of my PhD

[errant ridge]

when you were starting did you ever have the feeling that you wouldn't be able to do original research?

[vivid halo]

no

[modest rune]

(I think starting a phd when you sincerely believe you can't do original research is kinda insane haha)

[vivid halo]

ehh I guess the first two projects I worked on fizzled out so that made me question myself a little

but the project I started working on right at the start of grad school turned into a paper that I'm editing right now

[errant ridge]

no I mean not sincerely believing it but like yk getting scared that you might not be able to do that stuff

[errant ridge]

wow nice

what's it on?

[vivid halo]

I need to finish it up and I meant to finish editing this summer but I keep getting distracted by other projects

it's on single-valued iterated integration

so just like complex analytic geometry/Hodge theory stuff

[errant ridge]

[vivid halo]

bruhh my coauthor said he would write stuff and is not writing anything

(not for this paper, different paper)

[errant ridge]

[atomic meteor]

I need help in publishing a scientific research paper
by someone who has done it already

[errant ridge]

what's the best way to like not have it? like how do you go through undergrad in such a way that you have as little imposter syndrome as possible in grad school?

huh

really?

that is reassuring

[modest rune]

If your only concern is writing a graduate-able thesis

The bar is not that high

It’s a lot of work

But you certainly don’t have to be a genius or anything

[errant ridge]

I definitely am willing to work hard but I'm scared of hitting a plateau and not getting any smarter than that

im already pretty dumb

[errant ridge]

my concern is making research my career

[devout nacelle]

When/how do you find a niche for research?

Is taking graduate classes the stepping stone to narrow down your interests?

[open aspen]

that and REUs

REUs are very valuable for this

going to conferences is also great, even if you understand very little

[devout nacelle]

I see

[open aspen]

wdym

If youre looking for things you are interest in, I dont see how that is an issue

how did you know where you were going to grad school while in an reu

what you learn you like should influence strongly where you apply

so... what's that have to do with the reu

so in any event, that can help you find your interests. And even if it is in a very niche topic, you can explore related topics. once you get a general idea its easier to attend appropriate conferences, read more targeted survey papers, etc

[devout nacelle]

"Grad school" is not a classification that works well for me since my pathway is going to be a Masters followed by a PhD, very likely at different institutions

[open aspen]

I'd avoid reading modern research in essentially any field as it can get very overwhelming. Survey papers are perfect though for subjects specific enough to not have a textbook

[devout nacelle]

I'm near the end of my undergrad and don't have a good idea about what I want to do, but I do know that I'd like to keep learning more math for a while and explore

[devout nacelle]

I'll try to find opportunities for this winter

[open aspen]

if your school has colloquiums or reading seminars, sitting in on those can be valuable

this is how I learned I hate number theory

[devout nacelle]

My undergrad department is very small and doesn't do anything more than the bare minimum. I'll have to look outside but I'm aware of some opportunities for REUs and reading groups.

I've been considering approaching a prof for a directed reading course on measure theory and/or functional analysis (it is not offered as a course to undergrads)

Thank you!

[open aspen]

seems like a good idea

if youre not sure you like it, maybe something informal so you can comfortable cease operations if need be

[devout nacelle]

That makes sense

Thanks, I'll keep track!

[long epoch]

I can relate cus my ug program doesn't offer measure theory as a course either 😦

[devout nacelle]

[arctic grove]

[dreamy siren]

Is there a channel here covering Geometric Algebra? It is a different field from Algebraic Geometry, which I notice there is a channel for.

[neat frost]

Like high school analytic geometry?

[dreamy siren]

No

Geometric Algebra as in Multivectors and such

I may not have looked hard enough

[neat frost]

Oh yeah I just looked it up

My bad

[deep mango]

That could go in abstract algebra or differential geometry

But tbh not many people do anything in it here

Like

Specifically

[dreamy siren]

#algebraic-geometry seems to be it actually

[sleek wing]

Isn’t it just the exterior algebra lol

Cause if so then #groups-rings-fields

[deep mango]

Definitely not alg geo comm alg

The wedge product has like

[dreamy siren]

I think they are distinct

[deep mango]

One distinguishing property

Which is that it is not commutative

[brittle socket]

Chalk is a pigeon confirmed

Y the sully

Your profile pic is literally a pigeon

[dreamy siren]

geometric algebra's multiplication is the geometric product and not the wedge product

they seem to be somewhat similar though

[errant ridge]

ab = a \wedge b + 1/2(a . b) or some shit idk

i forgot

oh it's

$ab = a \cdot b + a \wedge b$

[fathom swallow]

Neamesis

[errant ridge]

they're identical if the vectors are orthogonal

[neat lintel]

Would u

Buy a physical copy of a textbook

Even if its available online for free

[odd narwhal]

It is commutative

Mod 2

[neat frost]

Retro wew pfp

[bright hill]

good morning slurp!

[neat frost]

Good morning DarQ!

[sleek wing]

Can’t we get a verified independent bipartisan fact checker on this please

[bright hill]

slurp, do you have any experience contributing to open source github repos?

[neat frost]

Yes

My own

Why do you ask?

[bright hill]

what's in them?

[neat frost]

My projects?

Like they’re just my projects, a bunch of dumb shit

[bright hill]

a bunch of dumb shit
naturally

[neat frost]

Yessir

[bright hill]

stupid unstable fuckinginternetconnection

[neat frost]

I was actually looking at some of my older projects the other day, some are actually kind of impressive. Like the code is shit, but the result is okay

[tight comet]

That is some respectable self-praise

Is your code all in ijk and uses functions fgh

[neat frost]

It’s mostly okay because it has pretty colors

[bright hill]

what results are we talking about?

[tight comet]

Complex set plotting?

[neat frost]

It uses functions because it was written in C, and no I’m generally decent about variable names

[neat frost]

Pretty colors

[bright hill]

[tight comet]

C

[bright hill]

what does that mean?

why are you being intentionally vague?

[neat frost]

For variable names I like going the Java way: AnIndexForIndexingTheArrayWhichIsDefinedOnTheLineAboveThisOne

[tight comet]

Did you invent PowerShell

Worst naming recommendation/conventions I have seen in languages, ever ever.

[neat frost]

I could show you a picture but then you’d be able to find my personal GitHub (with my name) ;-;

Meh okay I can show a picture without the title

[bright hill]

you can dm me if you wanna be extra safe

[neat frost]

Yesh good idea

[odd narwhal]

It mr

He wew

Do u wanna hear smth cursed

[neat frost]

Good morning shin!

[odd narwhal]

Good morning slur

P

[sleek wing]

what will u do if I say no

[neat frost]

Don’t say no wew, that’s wewd

[sleek wing]

kill

[odd narwhal]

I will say it anyways

To confound your expectations

[sleek wing]

as expected

proceed

[odd narwhal]

In an abelian category with all direct sums, homology doesn't a priori commute with direct sums i.e. it's not necessarily true that
$$H_n(\bigoplus_i A_i)\cong \bigoplus H_n(A_i)$$

[fathom swallow]

ShiN

[odd narwhal]

You need an extra condition

And I think that's hurbed

[neat frost]

That IS disturbing

[odd narwhal]

The extra condition is that the direct sum of monos is mono

[sleek wing]

this might be more disturbing if I fully knew what a homology was on general categories

seems spooky though

[dapper badge]

😮

[full isle]

lol this reminds me of funny

[dapper badge]

nooo I hate infinite direct sums

[full isle]

https://math.stackexchange.com/questions/903188/homology-commutes-with-direct-sum-and-product

stackexchange is just comedy sometimes

[odd narwhal]

This is true in Rmod since it's AB4

But for example this is false for the opposite category of sheaves

[odd narwhal]

They are trash

[ancient flame]

Rmod :)

[odd narwhal]

Tbh just consider calculating homology in modules

Freyd-Mitchell embedding guarantees this is ok

This is the same thing

[sleek wing]

hmm

it might help if I fully knew what a homology was

[bright hill]

hatcher book club when?

[odd narwhal]

ker/Im

[sleek wing]

hatcher? I 'ardly knew er!

see you can say ker/Im all you want but it doesn't tell me anything

[bright hill]

I thought they were pronouns for a sec

[odd narwhal]

I hate general abelian categories

[dapper badge]

i only know modules

[odd narwhal]

I just had a think and realised homology is a quotient object

That's terrible

Like it's obvious

But I hate that there's no canonical choices

Everything is up to iso/Equivalence

[dapper badge]

nah clearly one should think of homology as the cokernel of the embedding of the image into the kernel

[odd narwhal]

Yes but

Image is not unique

Neither is kernel

[dapper badge]

or as the kernel of the projection onto the image

you're not unique

[odd narwhal]

I am not

Like even if u freyd mitchell it that's not a canonical choice

Cuz once u change the category u may change the equivalence

So everything is always up.to isomorphism

AHHHHHHH

[dapper badge]

i only care about stuff up to iso

[odd narwhal]

At least cokernels and kernels are universal so it's up to unique iso

[dapper badge]

automorphisms begone

I always look at the skeleton of a category

[errant ridge]

https://tenor.com/view/wokay-walter-walter-dog-get-walterd-lol-get-waltered-gif-19284899

[dapper badge]

because I perpetually live in a state of halloween

[odd narwhal]

Spooky & scary

Shivers down my spine fr

[dapper badge]

omg back in elementary school

for my music class we sang this song called "the ghost of john" or something

and one of the lyrics is "long white bones with the skin all gone"

which is clearly a skeleton, not a ghost

song goes kinda hard though

[limber thunder]

wtf that's so metal for elementary

[dapper badge]

idk I think we were like 8 or 9 so basically adults by then

[errant ridge]

damn that's so old though

normally people become adults by age 2

and then get a fields medal by age 3

[untold sapphire]

please don't talk about sheaves in the discussion channels

I rarely check these channels and I resent being left out of the conversation.

I'm just kidding lmfao

[odd narwhal]

Lmao

I just said Sheaves^op don't satisfy AB4

[untold sapphire]

Wait is this true for modules??

Are infinite direct sums exact in R-mod?

[odd narwhal]

Apparently

Well, homology commutes with direct sums

So this should be true

[untold sapphire]

Wild

I presumably knew this but I forgot.

You know what's another great thing about R-mod?

Homology commutes with filtered colimits.

Oh wait lmfao

That makes it obvious

[odd narwhal]

Lol

[untold sapphire]

You can derive one from the other.

[odd narwhal]

Yw

[untold sapphire]

Anyway this comes up sometimes in algebraic topology actually

That's a cool application of this general nonsense

Check this out

[odd narwhal]

It just feels so natural that it's weird that it's not true in general

Also by direct sum is exact you mean as a functor from A^I to A right

It's always right exact also I think

It's one direction of exact always in abelian categories

[untold sapphire]

Let X be a topological space.
Every singular simplex in X has compact image, trivially. Therefore the set of singular n-simplices S_n(X) is the (filtered) direct limit of the set of singular n-simplices in compact subspaces of X, colim_\alpha S_n(X_\alpha)

[dapper badge]

I hear of filtered categories/limits/colimits sometimes but idk where they actually pop up

[modest rune]

right exactness is just the statement that it commutes with colimits and colimits always commute with colimits

[dapper badge]

oh thanks clerk

[untold sapphire]

And it follows that the chain complex of singular simplices in X is the filtered colimit of the chain complexes of singular simplices on compact subsets of X

and homology commutes with filtered colimits in R-mod

so

[odd narwhal]

Filtered stuff is a generalisation of direct limits

And inverse limits

[untold sapphire]

the homology of X is the filtered colimit of the homology of compact subspaces of X

[odd narwhal]

And directed sets

[untold sapphire]

I think that's pretty neat.

[dapper badge]

shout out filtered stuff

[odd narwhal]

Based

[untold sapphire]

Singular cohomology doesn't generally satisfy such a nice property. Although I think there are other cohomologies that are better behaved in that regard, like cech cohomology or the alexander spanier theory

[dapper badge]

filtered stuff is also where spectral sequences and double complexes and stuff pop up, right?

[untold sapphire]

Ah no different kinds of filtered here

Sorry lol

[dapper badge]

rip

conflicting terminology

[untold sapphire]

I mean not completely different but different enough that you'll get off to a very bad start with spectral sequences if we talk about properties of filtered colimits

[odd narwhal]

AB4 also means derived functors commute with direct sums

Since right(?) adjoints commute with colimits

Left derived*

[untold sapphire]

walter i was reading about filtered colimits fairly recently if you wanna talk about them

i was like

entranced

[modest rune]

left

[dapper badge]

I could be down

[odd narwhal]

Left derived is right adjoint

No

Waot no

[modest rune]

left adjoints

[odd narwhal]

Ur rifht

[modest rune]

Shiin

[odd narwhal]

It's left adjoint

[dapper badge]

but first I must do my daily character tables

[modest rune]

i love u

[odd narwhal]

Everything is backwards maxxx

[untold sapphire]

A simple theorem to keep in mind is that filtered colimits always commute with finite limits

which is nice

Also

[modest rune]

but the phrase "left adjoints commute with colimits" is so seared into my brain that you should not doubt me

[untold sapphire]

for nice categories like R-mod, the filtered colimit of R-modules is the filtered colimit of their underlying sets

[odd narwhal]

I got confused since left derived is for right exact

[untold sapphire]

like the forgetful functor preserves filtered colimits

[modest rune]

Think about derived as fixing the deficiency

[untold sapphire]

which isn't true for other colimits like the ordinary direct sum

[modest rune]

So that something already right exact does not need to be right derived

[odd narwhal]

Fair max

But left derived is still right exact

And not necessarily left exact

[untold sapphire]

hhhhhhm

[odd narwhal]

Since it's a left adjoint

[untold sapphire]

this conversation is making me dizzy

[odd narwhal]

I hate category theory

[dapper badge]

but they share same higher derived functors as original functor

[untold sapphire]

Wait left derived functors are left adjoints?

to what

[odd narwhal]

Wait

Am I being dumb

[untold sapphire]

This doesn't sound right to me but maybe it's right in some wild model category theoretic sense

Shin you're like half right

[odd narwhal]

Sorry

[modest rune]

Its not to my knowledge

[odd narwhal]

I got confused

[odd narwhal]

I meant to say left adjoints have left derived functors

[dapper badge]

I remember reading an overflow post a while back about if F is left adjoint to G, what can you say about adjunction between LF and RG

but it just linked to paper that I could not be bothered to read

[modest rune]

A left adjoint is guaranteed to be right exact so yeah

[odd narwhal]

And generally left derived is not right exact

I jsut got confused with the adjunction stuff

[modest rune]

although I will say the model category stuff really isn't that wild

[dapper badge]

does deriving additive functors in general actually tell us anything useful

[modest rune]

uh

it often does

idk what it would mean to "generally" tell us something useful

[dapper badge]

idk I've only seen it used for left/right exact to extend long exact sequence

[modest rune]

Oh yes

[dapper badge]

which I see the use in that

[modest rune]

that is more or less the right thing to do

There are also total derived functors

[dapper badge]

but idk how to recover homological information for functors which aren't exact, if there even is information to recover

[modest rune]

but the total derived functor is essentially just the same thing

[untold sapphire]

I'm so mad I busted my laptop yo

[ancient flame]

how

[untold sapphire]

I dropped it and now i don't have access to my notes while some of those tech guys take a look at it and see what's fixable

[ancient flame]

damnnnn

[dapper badge]

are the total derived functors just when you don't take the homology

[ancient flame]

this is why y'all gotta back shit up

[modest rune]

total derived functors are still sided

[untold sapphire]

Yeah.

[modest rune]

so left vs right

[untold sapphire]

I should have backed it up lmfao

[modest rune]

but you don't necessarily need some exactness hypothesis

[untold sapphire]

I didn't back it up

but i don't think the hard drive is completely fucked or anything

like i'd be willing to bet the info is all recoverable even if the machine is hosed

It doesn't even look that bad from the outside. It just doesn't turn on.

And the charging is intermittent

[dapper badge]

I think that's the way I learned derived functors in the first place Max

like you take resolution, and apply the functor

[odd narwhal]

They stay additive

[dapper badge]

that's the total derived functor (?) and then you can take homology for the long exact sequence

[untold sapphire]

Dold wrote a paper on deriving non additive functors

[dapper badge]

when you have exactness then you have (natural?) iso from the zeroth derived functor to the original functor so you can extend 🙂

[untold sapphire]

i can't remember if this was before or after quillen wrote homotopical algebra.

I'll say before.

[modest rune]

well i mean deriving non additive functors is well understood from the model category POV

[dapper badge]

oh for nonadditive stuff

scary

[modest rune]

total derived usually refers to like

[untold sapphire]

Yeah today!

[modest rune]

instead of thinking about Tor^n for example

you take all the n at the same time

The slight shift in persepective comes from like

Instead of thinking about derived functors on RMod

you instead think about them on the derived category

[untold sapphire]

there's also the notion of 'hyperderived functor' which is the same thing but you take a lot of adderall before you start computing it

[modest rune]

apparently hyper derived is just the same thing im saying but before they knew i was right

[ancient flame]

so much yellow

[dapper badge]

I read about hyper derived once when seeing cartan eilenberg resolutions

but it went over my head

[modest rune]

The big idea I guess is that you aren't really deriving a functor on RMod

instead, you are taking RMod, and putting it inside Ch(RMod) conentrated in degree 0

and then you are deriving the functors on Ch(RMod) which will agree with the usual thing

but the point is you can derive them as applied to any chain complex

[dapper badge]

so more generally you just take resolutions of complexes in general, right?

[modest rune]

Well

it turns out resolution in this context is better thought of as "cofibrant replacement"

[dapper badge]

meaning quasi-isomorphic complex of whatever nice objects, say projectives

[modest rune]

i.e., you aren't resolving A, but you are replacing A with a nicer chain complex (thinking of A as a chain complex concentrated in degree 0)

So to calculate a total derived functor, you replace your arbitrary chain complex with a quasi-isomorphic nicer one

[dapper badge]

makes sense ig

[modest rune]

The general idea is to force a functor to become quasi-iso invariant

The problem is like

What if it disagrees on two quasi iso complexes!

[dapper badge]

right, I mean it's the same story for usual derived functors

[modest rune]

You have to pick one of the two values to be the “correct” one

[dapper badge]

we use projective/injective resolutions because when they're quasi iso, that forces them to be homotopy equivalent, right?

add in "bounded" whenever necessary, idk

[modest rune]

Well, being quasi iso invariant on chain complexes concentrated in degree 0 isn’t very hard

[dapper badge]

fair

[untold sapphire]

I used to think that quasi-isomorphism was kind of a bullshit notion. But then somebody pointed out to me that via the Dold-Kan correspondence, chain complexes can be identified with simplicial Abelian groups, and a quasi-isomorphism of chain complexes corresponds to a weak homotopy equivalence of their underlying simplicial sets. By a theorem of Moore, simplicial groups are Kan complexes, which means Whitehead's theorem holds for them, and any weak homotopy equivalence is an actual proper homotopy equivalence.

[dapper badge]

what made you think quasi-iso was bs before that

[modest rune]

this seems

way more out there

[untold sapphire]

They don't have inverses lol. Just seems like a shitty notion of equivalence

[modest rune]

than quasi-iso being reasonable lol

lots of equivalences don't though

[dapper badge]

true, but I thought that was half the motivation for looking at homotopy first

[untold sapphire]

Well all i was going to say is like, it does have an inverse in a sense

So if you think of chain complexes as being spaces equipped with an Abelian group structure then a quasi isomorphism is a map between them which has an inverse up to homotopy, but the inverse and the homotopy equivalence may not preserve the additive structure.

[dapper badge]

but I also don't really have a say here because I'm new to derived stuff and don't know model categories lol

[untold sapphire]

It's a homotopy equivalence of the underlying spaces not necessarily respecting the abelian group structure

That made it more plausible to me i guess. i just have a preference for maps that i can actually invert

[dapper badge]

ohh that's a neat way to think of it

[untold sapphire]

shhh i'm going to stick my head in the sand and pretend simplicial homotopy is symmetric and transitive

[modest rune]

i mean

even topological weak equivalence isn't

[untold sapphire]

Yeah i'm not a fan of that one either! lol

[modest rune]

jeez lol

[untold sapphire]

Idk i understand there are technical advantages to those things but like

Working with spaces and maps is just intuitively more comfortable than working with their homotopy groups

I prefer Hurewicz fibrations to Serre fibrations and so on

[modest rune]

I don't really see the connection there

both fibrations are about spaces and maps

[untold sapphire]

Yeah, I guess what i meant there is that serre fibrations are mostly useful because of the long exact sequence of homotopy groups which in turn lets you do all kinds of wonderful things with serre fibrations as long as you're working with CW complexes. But if I want to consider spaces that aren't CW complexes than knowing that a map is a serre fibration might not be enough for whatever geometric purpose I have in mind

the connecting theme is that I prefer the stronger definitions that can be used to get more information than just what happens to the homotopy groups, what the long exact sequence looks like etc

again this stuff is super useful but less so once you go outside cw complexes

[modest rune]

I see

venturing outside of CW complexes is not something I consider often

in all honesty CW complexes aren't either lol

but I have to keep larping as a topologist for the NSF

[untold sapphire]

what do you work on, spectra or like, E_infty rings

[modest rune]

I work on stable infinity categories generally

mostly on stable infinity categories related closely to spectra

[untold sapphire]

ok.

[modest rune]

stability is one of the few notions that is kind of hard to make sense of without infinity categories though

I guess you can phrase it in model categories

You just need homotopy pullbacks to be homotopy pushouts (and for things to be pointed but you can always make that happen by taking a category over a terminal object)

"A stable model category is a 1-category structure used to present a stable (∞,1)-category in analogy to how a general model category encodes a (generally non-stable) (∞,1)-category."

based nLab

[deep mango]

I will superslap you

[odd narwhal]

Lma9

[neat frost]

Woke up on the wrong side of the bed rycie?

[deep mango]

I woke up at 5 slurp

[neat frost]

,ti —at Hawaii

[fathom swallow]

The time in US/Hawaii is 05:38, 08/08/2022.
Slurp is 13 hours ahead, at 18:38, 08/08/2022.

[deep mango]

I went to sleep at 12

[neat frost]

And…?

[deep mango]

Oh well, I can go watch another sunrise

I should probably get outside for that

[neat frost]

That does not excuse your poor behavior, young man

[wooden flax]

Smh ryc

[deep mango]

Its so pretty

Wow

[neat frost]

Almost as pretty…. as….. yourrrrrrr….rrr.r……rrrr……….. MOMMM

[wooden flax]

LOL

So funny I forgot to laugh !!

[neat lintel]

fact:you can never see a sunrise/set the photons you see took 12 minutes to reach you

you are looking at the past

[chilly hull]

what is that

[neat lintel]

Does photons have mass?

yes

[neat lintel]

How can u say that

[neat frost]

This is your one and only warning. Watch yourself.

[neat frost]

That is pretty

[neat lintel]

iirc rockets are designed with photons in mind so their trajectory doesn't change

[neat frost]

Photons don’t have mass, they have momentum

[neat lintel]

P=mv

[neat frost]

They also have friends, which you probably do not have

[neat lintel]

what makes you say that

[neat frost]

Your personality

[neat lintel]

wow ok

[deep mango]

Ok I found a nice seat

On a big log

[neat frost]

Nice and warm?

[deep mango]

No not really

[neat frost]

Good

[neat lintel]

But if they dont have mass they should not have momentum?

[deep mango]

Im in my pajamas and it's very windy

[neat frost]

Seats should be cold

[neat lintel]

I am confused

[deep mango]

Also I might have just given myself a lot of little splinters

Idk

[neat frost]

Classically that would be true, but this is some quantum weirdness that I have zero knowledge in

[neat lintel]

lol

[modest rune]

one time i was at a fancy hotel

and i saw a cactus

[neat frost]

Pain is good.

[modest rune]

and this cactus had no visible thorns

and i was like

[deep mango]

[modest rune]

"huh, this hotel is so bougie they took the thorns out of the cacti"

so i of course rub the cactus

for several seconds

[deep mango]

Oh god

[modest rune]

before i start to feel

a very unpleasant sensation

and i look at my hand and its covered in like

microscopic thorns

[neat frost]

[deep mango]

Bleeeeeh

That's horrible

[neat frost]

I guess the hotel wasn’t very bougie?

[modest rune]

in retrospect it wasn't even that nice of a hotel

[deep mango]

Removing thorns from cactuses isn't really bougie either!

[modest rune]

it is just dumb

but so was i

[neat lintel]

no it's easy

you just blow your thumbs looney too s style

and thorns shoot out

this is common knowledge come on

[neat lintel]

Why would u even think of doing that

[modest rune]

I was young

[neat lintel]

[modest rune]

there was no logic to it

[deep mango]

Like me rubbing my hand up and down an old tree trunk to wipe sand off

[neat frost]

Whenever I see a cactus I have an urge to touch it as well

[modest rune]

the worst part is like

i wasnt some tourist

[neat frost]

Speaking of I need to water my succulents

[modest rune]

i grew up in arizona

i am supposed to know about cacti

[deep mango]

I dont know if ive ever felt an urge to touch a cactus

[neat frost]

I hate Arizona

[neat lintel]

[modest rune]

everyone hates arizona

[neat lintel]

Why

I don't :)

Bruh

[deep mango]

I will fly through phoenix tomorrow morning

[neat lintel]

Arizona is cool

[modest rune]

PHX is a nice airport

I'll give it that

[neat lintel]

well not literally

[deep mango]

Yay

[neat lintel]

Huh

[deep mango]

I like airports

[modest rune]

I spent 18 years in arizona i have every right to insult it

[neat frost]

Then what kind of intrusive thoughts do you have. Probably boring ones

[neat lintel]

But why

[modest rune]

Its hot and politically insane

[neat lintel]

arizona is the state with death valley and the tea right

[neat lintel]

Did u live there in the first place

[deep mango]

This is a more cloudy sunrise than the last one

[modest rune]

the tea comes from california

[deep mango]

Oh well

[neat frost]

Oh I hate Arizona specifically because of its name

[neat lintel]

WHAT

[deep mango]

Uhhh

[modest rune]

i did not choose my place of birth

[neat lintel]

I HAVE BEEN LIED TO

[neat lintel]

If u dont like the place

[neat frost]

Arizona ice tea isn’t really from Arizona?

[neat lintel]

Oh lol u were born there ok

[deep mango]

"you shouldnt have left that chocolate on the train you human sack of shit"

[neat frost]

Also Arizona ice tea is meh ice tea

[modest rune]

idk how old you think i am

[neat lintel]

25 maybe

[neat frost]

[modest rune]

I'm 23, but still

there are not 18 years of my life

where I controlled where I lived lol

there are about 5

[neat lintel]

Wao

Here in india

You can live your whole life with your parents

[limpid oyster]

HERE IN CHINA

[modest rune]

arizona is very pretty though

[neat frost]

Jiyna

[modest rune]

and has nice outdoorsy stuff

[neat lintel]

Even after wedding

[modest rune]

that sounds uncomfortable

and i love my parents

[neat lintel]

Ello neighbour

[modest rune]

but i do not want to live with them lol

[neat lintel]

Pretty normal here tho

[modest rune]

tbf I hate living with anyone though

[deep mango]

I do not mind being very close to my family though I do enjoy my own separate living space that is not connected by a single door

Yeah

[modest rune]

my parents spend time in CA

and I enjoy seeing them

but i also enjoy not seeing them

[neat lintel]

CA?

Canada?

[modest rune]

california

[neat lintel]

Oh

Why tho

[modest rune]

@deep mango these contour integrals are hell

how am I supposed to come up w this stuff

[deep mango]

Oh yeah

So annoying

[neat lintel]

💀

[modest rune]

Like, give me a few hours and internet and I am sure i could do it but i feel like I am just expected to memorize this stuff

which is so lame

i guess the test is open book tho

[neat lintel]

Bro they r ur parents yo

You better not

[deep mango]

It's completely worthless knowledge

So silly

[modest rune]

I wish quals were just like

week long take home exams

that would be so much more reasonable

[deep mango]

I am happy to learn some of this stuff once and forget it once and be over with it

[neat frost]

What are quals?

[limpid oyster]

How do you learn number theory

[modest rune]

exams that you have to pass for phd

[deep mango]

But quals make you learn many things twice and forget them twice

[neat frost]

Ah

[modest rune]

lol

[neat frost]

You have to take exams? I thought you just need to defend your dissertation?

[modest rune]

I wish

No I have to learn a bunch of complex analysis

only to forget it

[neat frost]

That seems really stupid

[modest rune]

it is

[neat frost]

Didn’t you already learn it in like undergrad?

[modest rune]

Some of it

My complex analysis class in UG was not great

[deep mango]

Consider checking out richard borcherds' youtube channel. He's a fields medalist number theorist who does all these really good full courses on his channel about number theory, algebra, and related topics

[neat frost]

What’s their justification for it?

[modest rune]

I think the justification is that they want be like

broadly informed on basic topics in mathematics

which I think is a goal that only sounds good

[deep mango]

Make sure you have sufficient background for an advisor to be willing to sign onto you

[modest rune]

and isnt very useful if you think about it for more than 20 seconds

[neat frost]

But isn’t that the purpose of undergrad?

[modest rune]

yes

[modest rune]

but a bunch of the people taking quals already have an advisor

like if the qual was just in my subject area that would be very reasonable

[deep mango]

I kind of like our structure which is that there is a preliminary qual based on just real analysis, complex analysis, and hard linear algebra (complex analysis is a bit silly but the other stuff everyone at nyu needs to know very well)

[modest rune]

which a lot of schools do do

[neat frost]

What did you do your dissertation on?

[deep mango]

And then there is an oral which is very focused on what fields of math you need for your research

[modest rune]

neither of us has written a dissertation

Im only a rising 2nd year

[neat frost]

Oh I see

[modest rune]

this kind of exists here too for candidacy

[deep mango]

Your dad

[modest rune]

but afaict its like

fake

[deep mango]

Oh shit max outed me already

I see

[neat frost]

This is independent of your area of study?

[deep mango]

I dont think people usually fail the nyu one

[modest rune]

ryc goes to an analysis school

[deep mango]

But i do hear it's scary

[neat frost]

Oh

[modest rune]

well mine is more like

[neat frost]

Of course he does

[modest rune]

tell us what problem you want to solve

[deep mango]

Yeah basically everyone at nyu does pdes, probability, or numerical analysis

[modest rune]

and how youll solve it

[deep mango]

A few differential geometers

[modest rune]

and then everyone in the room pretends to nod along

[deep mango]

I see

Lol

[modest rune]

and your advisor already approved

like, i wont be allowed to do the candidacy thing unless zhouli thinks my idea is good

and no one else at ucsd is qualified to be like "no it isnt"

[deep mango]

Like 3 algebraic geometers / arithmetic geometers

[modest rune]

I guess courant is scarier

since like

everyone is at worst field-adjacent

i feel so bad for the non-math faculty i have to bring into my defense / candidacy exam

[deep mango]

Lol

Wtf

[modest rune]

My defense needs 2 nonmath faculty

which is very funny

[deep mango]

That is silly

[modest rune]

Normally people pull in scientists

[deep mango]

Sunrises are really not that great are they

[modest rune]

kinda lame tbh

[deep mango]

They are nice in their own way

Like

More quaint but

[modest rune]

the issue is that sunrises become progressively harder to look at

but sunsets progressively easier

[deep mango]

Somehow more realistic than sunsets, sunsets are very good at exceeding expectations in a kind of uncanny way and sunrises are good at meeting them

Yeah

[wooden flax]

[deep mango]

Is your plan to have nonscience people

[modest rune]

well

it would be funnier

so its on brand

[neat frost]

Sunsets are prettier than sunrises

[modest rune]

my thesis is probably going to just be a bunch of stuff stapled together anyway

[deep mango]

Mainstream!

[modest rune]

I'd rather focus on publishing projects than a thesis

I think sunrises would be equally pretty if you could actually look at them

In some sense the right way to look at a sunrise

Is to like

Face the opposite direction

And watch the light slowly flood over the landscape

Which can be really beautiful

[deep mango]

I guess so

Definitely depends on the landscape

And how like

Cloudy it is

[modest rune]

yeah thats another issue

sunsets are pretty regardless

sunrises need a better canvas

[deep mango]

Yeah

Ooh this is actually very nice

Nevermind it sucked

Whatever

[wooden flax]

Lol

[delicate knoll]

so guys, can you explain what is mean and mediam?

[sick burrow]

Median is what you are and mean is what I am in saying this

[sick burrow]

Cope seethe mald etc

[delicate knoll]

???????????????????????????????????^20

[neat lintel]

yohan has been here for many year

where

[deep mango]

It didn't actually suck

The sun just went in my eyes at that moment

[bright hill]

newton

apparently

LMFAO

fair

he really went extreme with it

he stared so long that his eyes hurt for days

[neat lintel]

When I was a kid I used to stare at the sun in the car ride bc people told me not to since they said it would hurt my eyes. I stared at it for like a minute every time

Anyway I have astigmatism

And I need glasses

I know how the sun really looks tho

None of u can say the same

[chilly hull]

man

i forgot to convert m/s^2 to ft/s^2 for g for this problem

31a/b

i used 9.8 for g

[bright hill]

Once he inserted a bodkin—a long needle of the sort used for sewing leather—into his eye socket and rubbed it around “betwixt my eye and the bone as near to [the] backside of my eye as I could” just to see what would happen. What happened, miraculously, was nothing—at least, nothing lasting.
@supple flame

[neat frost]

Wtf

I fucking HATE eyes

[solid snow]

newton wtf

[bright hill]

attempted accidental self harm
colorless green ideas sleep furiously

[bright hill]

what, slurp, wanna tell me you never touched your eyes?

[neat frost]

NO

DONT. TALK. TO. ME. ABOUT. EYES.

[bright hill]

laaaaaaaaaaaame

[neat frost]

COCK

[bright hill]

booooooringggggg

which?

lol

oh, I don't doubt that

I actually didn't think it was meaningless the first time I read it

let me pull up my interpretation

colorless = unimaginative
green = unripe
sleep = undiscovered
furiously = begging to be discovered

[storm sage]

I mean I think the point that there is some separation between semantics and grammar still remains

i.e. humans can instinctively tell what a "wrong" sentence is and what a "right" sentence is without any notion of meaning

[leaden sage]

chomsky said something like that

[sacred forge]

slavoy zizek is funny

[storm sage]

well yeah the sentence is chomsky's

I'm paraphrasing his point

even if that specific sentence is meaningful to some people, it doesn't disprove his point imo

[deep mango]

Jesus christ what the everloving fuck

I Hate eyes a whole lot

Disgusting

[bright hill]

ahahahahahaha

[chilly hull]

would it be better to do calc 2 -> proofs -> RA + LA (then AA) + calc 3 (then calc 4) or calc 2 -> proofs -> calc 3 (then calc 4) + LA -> RA + AA

[neat lintel]

tf

isnt calc 3 multivariable?

[chilly hull]

yes

[neat lintel]

so LA better first probably

what is RA?

[chilly hull]

real analysis

[storm sage]

whichever you want

I personally would learn at least a little linear algebra before learning multivariable calculus though

[bronze pelican]

You can get into a good grad school from your local state school

Grad schools arnt going to admit you based off the name of your undergraduate institution. One of the most important factors is your letters of recommendation

[bronze pelican]

Idk

State school just means funded by state government

It has no relation to whether or not its a research institution or has a PhD program

The UCs are state schools

But Caltech is not, since its private

[fading zealot]

What is the operation called when I am given a probability pattern and trying to reverse engineer how to get that pattern by, for example, throwing dice?

[leaden torrent]

i am not sure theres a specific term for that

naively i'd try to fit it to a distribution and then create a scenario that leads to that distribution

this, at the very least, seems like some form of fitting

[fading zealot]

Something like this

[leaden sage]

some kind of regression?

[leaden torrent]

it seems theyre given the distribution already

a regression is useful for approximating results as a distribution/trendline/whatever

not for converting a distribution into a scenario

i am not sure of a term for the latter, though

i mean, you could create some really contrived setups

it certainly doesnt seem like something you could consider a "function" as the attached image implies

[azure nymph]

There's also research schools that don't grant phds but do grant masters

[analog salmon]

tfw my professor emailed me on behalf of the textbook author to take my notes off github cause they're too good lol

[storm sage]

LOL

if I write a textbook, I want to make it a "pay however much you want" sort of thing

and hopefully I'll convince some rich student to give me their fortunes

textbook patreon

[leaden sage]

there were sites that did that before patreon also, iirc

[brittle socket]

[velvet dagger]

Wait like they don't like that you're competing? Or is it that it's too close to his book?

[ionic star]

that's the same thing?

[velvet dagger]

@ionic star well, I meant competing just like, maybe it's pretty different exposition of the same material but the author for some reason or other just doesn't like the idea that, oh maybe people will use the notes instead of my book

In which case this annoyance is noticeably more petty/bluff-y on the author's end

The other possibility is that it's very close to the book so the author is borderline saying this is copyright infringement

[manic garnet]

^

[surreal sapphire]

must be a pretty big book if the author is making any money at all

[odd narwhal]

What's the book

[bright hill]

oh, hello loch

how did the meeting go?

[errant ridge]

oh, hello darQ

how did analysis go?

[bright hill]

hello, neam

[bright hill]

slowly

I'm only doing an hour a day atm since I have other projects I'm working on

[errant ridge]

imo that's still plenty

that's like the perfect amount

[bright hill]

I don't know about that

I don't feel like I do much in a single hour

2 exercises at best

or a single section

[errant ridge]

do it faster

[bright hill]

I can't

[analog salmon]

3144Notes.pdf

Nothing too special but they match the book pretty closely I guess

I already see two typos on the first page lol

[manic garnet]

@analog salmon very pretty! how do you write these notes?

[analog salmon]

LaTeX and lots of time!

[analog salmon]

My notes are nowhere near as thorough as the books because I didn't want to have to typeset all the steps in matrix manipulation and everything

[manic garnet]

which software do you use to write it?

if any

[bold ferry]

notes on?

[neat lintel]

☠️

[analog salmon]

Used texstudio as my editor

[manic garnet]

thank you!

[neat lintel]

@storm sage yo can I dm you

[steel wind]

you can dm me

[storm sage]

yea sure, for what

[neat lintel]

math/cs

[storm sage]

just turned on DMs for this server

[tiny forge]

dangerous move

[storm sage]

😭

[neat lintel]

How are guys doing this morning you … drainers

I am finally leaving my 3 year job I had since high school

They asked for my email to send me gift for my service, so I am hoping for something good

[ancient flame]

oh damn

why are u leaving

[neat lintel]

Classes are staring so I moving 2 hours away from my hometown

Also the job fucking sucks even tho it pays well

It didn’t used to suck, just staff issues, one person does a lot of work
Guess who, this guy

[ancient flame]

oh ok makes sense

yeah

well, I wish you the best of luck in the future

[neat lintel]

You too potato

[frail lagoon]

thanks

[ancient flame]

potato

[arctic grove]

gonna dm u

with jee shit

[storm sage]

I turned it off

muahahaha

[neat lintel]

evil

The gifts fucking sucks

Anyone wants cook silverware’s

[arctic grove]

Nooooooooooooooooooooooooooooooo

😭

how will i send u jee stuff 😭

[storm sage]

muahahahaha

you will be forced to use the help channels

[ancient flame]

fuck jee

all my homies hate jee

[arctic grove]

Nooooooooooooooooooooooooo!

[arctic grove]

same tbh

[ancient flame]

lol

[neat lintel]

pls enable hep channels for me

[ancient flame]

???

[neat lintel]

potat

[ancient flame]

🥔

[leaden sage]

just take the SAT then

[ancient flame]

I did already

twice

[neat lintel]

Based

[neat lintel]

Hey

[open aspen]

Why do we use $Gs$ for orbit and $G_s$ for stabilizer? What a nightmare

[fathom swallow]

Migillope

[round panther]

hi

[sleek wing]

agreed. which is why I don't

[open aspen]

$Stab_G(s)$ is clearer, but that's so many letters

[fathom swallow]

Migillope

[neat lintel]

orbit-stabilizer: $$|Gx||G_x| = |Gx||G\frac{}{x}| = |G\frac{x}{x}| = |G|$$

[fathom swallow]

TTerra

[neat lintel]

good notation

[cold needle]

what the hell is this

[neat lintel]

proof of orbit-stabilizer

come on metal, you took algebra

shouldn't you know this?

[cold needle]

[frail lagoon]

So based

Reminds me of that proof that a+b=b+a by gradually rotating them around

[ancient flame]

lol

who has the most messages in this server

[neat lintel]

texit

[woven whale]

ann

she has even more than texit

[ancient flame]

oh wow

[open aspen]

I gotta steal this

[neat lintel]

credit me or else

[ancient flame]

wait texit has well over 500k and ann has less than 400k

unless there are almost 200k messages in channels I cant see

[woven whale]

oh I have a severe case of not being able to read a fucking number properly

[ancient flame]

lol

[velvet dagger]

@neat lintel unironically how a differential geometer prob would notate this

[storm sage]

does this notation actually mean something

and if so what

😭

[neat lintel]

the only factual part of what i wrote is $$|Gx||G_x| = |G|,$$ which is the statement of orbit-stabilizer

[fathom swallow]

TTerra

[neat lintel]

everything inbetween is meaningless

it's not a proof

[storm sage]

alright lol

glad this notation is not real

[muted totem]

Is there a name for a function between metric spaces that satisfies the usual definition of a continuous R to R function, but where we use the appropriate metric?

[leaden sage]

like isometry?

[bright hill]

bruh

[storm sage]

is it not just continuous

same thing, no?

[ancient flame]

yes, cobtinuous

[muted totem]

Ok, I wasn't sure if it had a different name when it was more general

[storm sage]

no worries lol I was just having an internal crisis

about whether it was just called continuous

[bright hill]

the usual definition of continuity is between arbitrary metric spaces

[muted totem]

Oh haha my bad

[muted totem]

I've never studied metric spaces before

[tall badge]

metrics are distance functions

in R, the usual metric is |x-y|

so replacing this by an arbitrary metric in the def of continuity on R gives the def of continuity on metric spaces

[muted totem]

Do we sometimes use different metrics on R?

TIL d(x,y)=0 if x=y and d(x,y)=1 otherwise is a valid metric on R

[tall badge]

rarely in calc classes

[muted totem]

True

[deep mango]

The usual definition of continuity is between arbitrary topological spaces

Metric spaces lets you use the epsilon delta formulation but I'd hardly call that the "usual" definition since it's subsumed by a broader one

[swift sun]

valid on any set in fact

[fading zealot]

It's a matrix

EUREKA

[arctic grove]

hello

[arctic grove]

gonna steal and not give any credit to u

[bronze pelican]

ORBIT

STABILIZER

[storm sage]

ORBIT STABILIZER

[fading zealot]

Is there a function that can “extract” what operation was performed on two numbers?

Say,
f(2,3,5)=1, since 2+3=5
f(2,3,6)=2, since 2*3=6

[storm sage]

nope, you could have multiple operations going to the same number

for example, 2*2 = 4 and 2^2 = 4

so what would f(2,2,4) output?

[fading zealot]

What if it outputs a set

[storm sage]

what operations are you including? like just the basic ones +, -, *, /, ^?

[fading zealot]

Yes, that would be sufficient for what I want to investigate

Mathematics!

How is that I don't have sufficient knowledge to write a rigorous definition for something that looks so painfully simple?

[storm sage]

I would write it like this:
Let a_1 = +, a_2 = -, a_3 = *, etc. for all the operations that you want, all the way up to a_N
Then, f(x,y,z) = {n | n in {1,2,...,N} and a_n(x,y) = z}

[full isle]

yeah but it will be wrong

like it wont tell you what you want

Also this notion isn’t particularly useful for a couple of reasons

[lost surge]

I'd like y'alls thought on a custom notation I thought up: inline bounds.

The traditional alternative is $\sum_{i=0}^{n-1}s^{n-i-1}f^{(i)}(0)$

[fathom swallow]

PhysMan

[lost surge]

But I think mine is more clear.

[storm sage]

my first reaction was oh god what is that

but now I see the logic of it

[lost surge]

It's clearer right?

[storm sage]

yeah I think it is pretty clear

the only thing is that it's less compact to write out

[lost surge]

Yeah.

I had to tweak the spacing in an image editor too, since LaTeX doesn't handle the construction very well.

The spacing was way too wide before.