You'd need to check that $\sigma \tau \sigma^2$ isn't in one of the other cosets.

Ah ok... Thanks!!

yes those are the right ones

Merry day of thanks to all

Happy Thanksgiving @KajHansen

Thanks Brody!

The cook (not me) got lazy this year and bought HoneyBaked Ham :p

But the food still tasted pretty good, thankfully

hahaha

hi @KajHansen :)

Hey there

Yes

Yeah.

Not a lot of ways to get groups of small orders.

Up to about order 6, you can literally just brute-force the possible operation tables

Without any special machinery

@KajHansen hi, missed ya. Do you feel ok?

Ah ok!!

Yeah, I'm just fine @Null

@Null, were you the guy from Germany?

I met someone from there in here recently

The order of the elements of $\{\langle \sigma^2\rangle, \langle \sigma^2\rangle\sigma , \langle \sigma^2\rangle\tau , \langle \sigma^2\rangle\sigma \tau\}$ is the same for all elements?

@AkivaWeinberger Nah. You can't do that for knots. If it lies on a sphere it bounds a disk by Jordan-Schoenflies.

Ah, hm, arcs, not closed curves.

@KajHansen Is every group of order 6 isomorphic to D_3 ?

No @MaryStar, there's a cyclic group of order $n$ for every $n$

But those two are it

no

@KajHansen yep, german/NRW. where were you?

If there is to be a function $x$ such that $\forall t, a(t)x(t)+b(t)=0$, what conditions must the functions $a,b$ satisfy? How many such $x$ will there be?

Georgia, southeast US

@KajHansen ah, i meant where were you in germany^^

@Brody there would only be one unique $x$

Oh, I've never been unfortunately

@KajHansen well, don't feel sad, as I was never in Gorgia^^

I meant I met someone from Germany in this chat room recently. Thought it was you but couldn't precisely remember

@AkivaWeinberger Yeah, you can do it for tame arcs I guess. Thicken it up a bit to an embedding of I x B^2. Then translate to make it lie on the boundary; that is the cylinder I x S^1. Now do the Alexander horned sphere construction.

@meow-mix true if $a,b$ are everywhere nonzero

You're missing basically nothing @Null

$a(t)$ must be non-zero, or both $a(t), b(t)$ = 0

@KajHansen maybe Saturated exo? (that was my previous name)

@KajHansen the grass on the other side of the fence is always more green i suppose.

in the case that both $a,b$ are zero functions, then there exist infinitely many $x$; a degenerate case, if you will

I arrived at the same @meow-mix. "degenerate" how?

Hi everyone.

This state's pretty backwards in a lot of ways, but I won't get into that on here.

Hello @BalarkaSen!

degenerate as in, it goes against the normal rule

Yeah

@meow-mix How can we check if the group is isomorhic to Z_6 or to D_3 ? Do we have to check the order of the elements?

I just don't know if this solution is equivalent to the author's or not :/

@Ted might have some choice words I'm sure ;)

@KajHansen it would turn this chat into politc.se i guess^^

@BalarkaSen Relevant (Fox-Artin horned sphere)

shit that's crazy

Remember the Fox-Artin wild arc I linked to earlier

@Brody what does the author say

Ya, that's the typical example of wild arc I usually have in mind.

Nice construction.

@meow-mix If $a(t)=b(t)=0$ for some $t$, then $x(t)$ can be chosen arbitrarily

@BalarkaSen, do you know a name for this topology on $\mathbb{N}$

(@meow-mix That's accidentally clever)

I said to choose $a,b$ everywhere zero

I constructed it for a counterexample a couple years back

(Because not only does it related to your conversation, but also I mentioned the Fox-Artin wild arc and there's a song called "What Does the Fox Say" @meow-mix)

Oh lord...

LOL

I'm from GA @KajHansen

@Brody i guess youre right

Not sure, @Kaj.

@meow-mix that the author's is right?

That's right! @Brody, you said you're NW of ATL right?

@KajHansen Uhuh

Rome area?

No, in Cobb

Ah, not that far NW

commutative diagrams are very pretty

I think my area is still considered part of metropolitan, not sure

@KajHansen can you imagine a language that fills all knowledge of math into a string of 10 letters?

Are you at Tech? GSU?

I wish. lol

I am at Kennesaw

@Null well we'd have $x^{10}$ possible statements, given that $x$ possible "letters"

@Akiva So it turns out that literally an $\epsilon$-neighborhood of the wild arc, $\epsilon$ varying as it should, is an embedded sphere. I wonder if that's always true.

@Null, guaranteed that violates Shannon's source coding theorem

so $x = \lceil log_{10}s\rceil$ to minimize the amount of letters, given $s$ statements to state

hmm, wait no

Oh nice @Brody. When you say "Kennesaw", do you really mean SPSU?

boundary of an epsilon-nbhd I meant

Nope @KajHansen, KSU. Although the math major is based on the Marietta campus (formerly SPSU)

@KajHansen wow, didnt expect THAT answer, you were faimilar with it beforhand?

@Null, I had to google to get the name right, but we did cover some compression in my cryptography course

Yeah, that's what I meant @Brody. I had a couple friends at SPSU during the change to KSU and they weren't happy

@KajHansen I entered immediately after the <s>acquisition</s> consolidation. I've heard of the discontent second-hand, but there's been no ruckus

Ah

lol, pretend that's a strikethrough

You went to UGA right? @KajHansen

is the statement $(b^{-1})^{-1} = b$ true?

mhm @Brody. They had a pretty good math department, so I'm glad I went. Probably wouldn't have liked it there for any other major though.

I guess the exterior complement of the Fox-Artin horned sphere is not the standard 3-ball either. Near the non-flat points of the thing the fundamental group is pretty messed.

@KajHansen That's precisely why I wanted to go there

All the other departments felt too hand-holdy and / or not enough rigor

Really, even the science ones?

esp. natural sciences

Kind of. I took two or three major-level physics courses. Other than that, I'm basing my experience off of core requirements.

Modern physics should've been a lot more rigorous

@KajHansen I was planning to take that had I gone. Indeed, my idea was that UGA was distinguished by its rigor

That's a shame it wasn't up to standards though

Mathematical methods in physics was actually decent, but overall the path for getting into tough upper-level courses was too long. Couldn't get into quantum until 2 or 3 semesters before graduating, e.g.

I'm not really in a position to speak @Brody, but I can say the math department let me dive into rigorous, bust-my-ass courses from day 1. I can't say that about any of the others.

I had Ted's multivariable mathematics course starting first semester first year, and that was one of the hardest classes I took in all of undergrad

@KajHansen I see. At least UGA has a good amount of upper-level STEM courses. Most colleges in the state just literally stop at a depressingly early stage

Ted throws some hard nuts to crack, for sure.

I was about to ask if you've seen Ted :P

Indeed @Brody. I have a decent level of familiarity with U. West Georgia's undergrad offerings, and they're pretty pathetic (my dad's a prof. there)

E.g. real analysis only taught every other year by their math department

Yeah, I had three courses from Ted. It's a shame he's retired. The level of community and brotherhood is course engendered together with his enthusiasm and support for his students convinced me to do math over physics.

@KajHansen Oh yeah, that's a real shame they do that

I'm guessing Ted didn't smack as much in person as he does online

They're basically forced to unfortunately @Brody. No one wants to take it, and uni's typically need ~10 people signing up to offer a course. From what I understand, they only get like 2 people a year interested in point-set, so they have to set up independent study w/ profs, if the opportunity for that is even present

No smacking, no. But he has a certain....way of motivating people. Disappointment is certainly conveyed when necessary :P

UGA seems to be starting to have that problem as well. Ted distance-taught point set last year since they couldn't find 10 people to sign up. They also had to stop teaching theory-based calculus from Spivak. It's a shame that more people don't want to take the path least traveled.

@KajHansen This is exactly why I probed all the catalogs. Glad I said "no" when I saw theirs. That's sad though, as I had the impression West GA was one of the larger / more popular ones

Have you given any thought to transferring @Brody ?

@KajHansen Yeah, I'm going to have to if I care about my mathematics. I'm thinking GA Tech or State. Either seems to be a step up

That's great. I don't know a whole lot about either of those

I have a ton of friends at Tech, but they're all engies

@KajHansen Yeah, that's the lot of 'em. But their courses in general seem to be more advanced and theory-based. There, I'll be able to take point-set then algebraic, which would be cool

:D

KSU just has one semester of "topology". And while it has "partial DEs" and "elementary number theory" in the catalog, they haven't even been offered the past two semesters :/

That's too bad

and

$$\phi(g^{-1}) = \phi(g)^{-1}$$

is it a homomorphism?

Hmm

We need to somehow verify $\phi(gh) = \phi(g)\phi(h)$ for arbitrary g, h

Looks the highest undergrads here are Real II, Abstract II, Complex Analysis, Numerical II, and Topology. Surely, there should be more? @KajHansen

Probably. I took something like 13 or 14 major-level undergrad courses

At UGA

Grad-level classes are also available if you've proven yourself to be sufficiently competent

I didn't even take the PDEs, algebraic topology, or graph theory courses either.

Granted, alg. top. is only taught every 2 years

Or probability

Or the two numerical courses

Wow...lol

^lol, im just drunk haha

Typical German ;)

but isomorphism preserves identiy right?

homomorphisms preserve identity

So we've got that checked off the list already

We just need to confirm multiplicativity in @meow-mix's case, but I'm not seeing it :/

You know what sounds German to me? Categories and morphisms...

@Brody fascism and Kafka?

Exactly one of those words is familiar

@Brody with Kafka i mean his book "metamorphosis"

Kafka is a writer who produced some rather kafkaesque work

@Null I know neither of those

I didn't like Metamorphosis much

I mean, it's very good, just not of my taste

it is tragic, so i agree: not my taste

Tragedy is fine. Eg, I am a big fan of Dostoyevsky.

Sort of like me and Kubrick films

@Kaj Watch Tarkovsky!

Holy crap, it's @JasperLoy. And he hasn't changed his handle since last seen

Mentally noted @BalarkaSen

lol, I do my best

I have become sick @Jasper

Well, I've been sick, but it got really bad in the past year

@BalarkaSen i like more drama (drama as by definition)

Got that depression @Jasper

has anyone heared of Stanislaw Lem?

Of course.

Hey @Balarka. Should I wait until I learn the Jacobian in Ted's book before we move onto the next mini-lesson?

@Brody Yeah, learn some derivatives.

Exercise is pretty good, I agree @Jasper. I struggle for the motivation oftentimes, but I get in calisthenics near-daily at least. It's something, and it helps.

does anyone here charge for lessons? just aksing, not judging..

@Null, I made a lot of money tutoring in undergrad. Lots of demand.

It's pretty common in the states

any hint to prove that $\mathrm{Aut}(\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}) \cong D_3$

@BalarkaSen I recall partials, directionals, the T-N-B vectors, and the Jacobian from Calc IV, but I've forgotten much of the latter stuff unfortunately

i can do it computationally

@KajHansen do you think 50 usd are fair or extraordniaray per hours?

@KajHansen I should take up on the Athletics department's request, then :D

@Null, extraordinary, but upper-middle-class suburban parents will often happily pay that for their kid getting SAT prep, lol

I usually do $20 per hour, and sometimes feel bad charging that much

@JasperLoy the algebra text im reading introduces categories and implements them within the text

@meow-mix Use matrices.

specifically, Aluffi

@KajHansen so 30 usd should be some kind of upper bound by your opinion? (for mid-low class)

I think that's what you need to do @meow-mix. An automorphism will be determined by the image of a generating set; in this case (1, 0) and (0, 1) works

Write down an isomorphism w/ GL2(Z/2)

Oh yeah @Null

oh wait i just realized

LOL

@KajHansen im not that good and im feeling bad for charging anything, so i feel ya

(for under highschool)

Are there any tricks to computing automorphism groups @BalarkaSen ?

Not in general that I know of, @KajHansen. Do you have something specific in mind?

Not really, but I see people ask about it on main a decent bit

Yeah, there are a couple of good exercises out there about automorphism group of various groups.

is an a->a by definition an automorphism? is A to B an automorhpism if |A|=|B|?

Why not just read Lurie's Higher Topos Theory? All of algebra is basically just infty-category theory, which is basically just infty-Grotherndieck topoi.

@Null false

@meow-mix So from my problem earlier, I think Spivak's solution is indeed wrong

1. automorphisms arent from one set to another, its just from one set to itself

@meow-mix eh.. is there any set that doesnt have an automorphism?

But I'm not really certain, I'll have to ask again...

no

the identity map

a->a

so automorphism is superfluous imo, just my humbly opinion

weird conclusion lol

@JasperLoy care to elobrate (link is sufficient)

@JasperLoy, why don't you have a math:SE profile?

I need help with the solution for this.^ (i.e. qualifying the textbook's soln.)

k, gotta run

@BalarkaSen run forest!

He's gone, lol

@JasperLoy not in specific cases

@JasperLoy it states certain properties without proof, imo handwavey

(up to cringy=

haha

@JasperLoy mmh, can you elobrate why you dont want to participate?

subgroups ma

p to subgroups in homomorphisms right?

@JasperLoy I see. didnt thought THIS way

The textbook says if $\forall t, a(t)x(t)+b(t)=0$, we may have $a(t)=b(t)=0$ for some $t$ and one may arbitrarily choose $x(t)$. This is surely false?

eh i could probably prove it

@JasperLoy what are you sick with?

@JasperLoy technicly probably correct, but "did not thought" is perfectly viable (altho i dont know the case for it haha)

ah

hes german, lay off :P

Adding -ly after -ic words can be tricky even for native speakers

@JasperLoy it is cool to talk with a english nerd :)

@Null not being that guy but "did not thought" is not correct grammar

It's not immediately clear whether to append -ly alone or -ally, unless there's some reliable etymological rule

@meow-mix ok, im probalby emberassing myself, but we in germany have the perfect/past tense

yeah; to negate "i thought" would be "i did not think"

which is dumb indeed

@JasperLoy s long as someone understand me i see no reason to change. do you see the problem?

@JasperLoy good habit imo

@JasperLoy what are their responses?

Well, there's always a 'why' to language. It's just that sometimes those 'whys' aren't very good

The English in mathematical literature is also sometimes weird, even by ordinary English standards

@Brody esperanso ft win

Esperanto?

@Brody si si senior^^

Same here, except Italian

@JasperLoy i think German doesnt help you in any way

umm

(just my humble opinion)

Doesn't have to be about utility

@JasperLoy i see!

@JasperLoy you dont take respect to the fact that i am in fact immortal ;)

Proof: i did not died yet

@JasperLoy I can't dissociate Italian from all the clichés unfortunately. Of the Romance languages, French sounds the prettiest to me

@JasperLoy but: the universe repeats itself sometime, so...

If I'm dead before I can experience recurrence, then recurrence doesn't mean much to me.

Ah, neat. Which is DK?

@JasperLoy can you spell MW, DK please?

@JasperLoy Jasper Jones confirmed imo XD

If the universe is reborn, I highly doubt a replicate of me will be too. Even if I was cloned at this very moment, would that be my conscious being in that body? If not, what's even the point

@Brody i dont give a single fuck, as long i drink my beer. see what i mean?

@Null But that's cheating

life is short, drink your beer

@JasperLoy that is fucking deep. ;)

Beer for some, coffee and theorems for others!

Why not beer and theorems? ;)

@JasperLoy a variation of carpe dfiem imo

@Null I thought it was pianist Kyle Landry at first

And then I saw it was a keyboard, not a keyboard. lol

@Brody why not both is acutallly a meme^^

@JasperLoy still, the gamer is unpayable

There's expensive anything in tech, esp. with gaming

@Null It's become a spectator sport. Be good/entertaining enough, and people will pay to watch you game

@JasperLoy I might be getting Lee's smooth manifolds, along with some others as an alternative to Spivak's diff. geom. sequence

@Brody Yeah. I mean, I won't, but people do. Plus, even if most of your audience isn't paying, you're still drawing in traffic/ad views and that counts for something

@JasperLoy Hmm, alright. I will keep all this in mind. Thank you!

I have no real sense of how that kind of economy works, though

@Semiclassical You're talking about gaming?

Yeah.

And Youtube content creation in general, I suppose

I mean, in some cases there's merchandising. Other times you get some degree of sponsorship; Patreon is also a thing