4:00 AM

CR means partials equal not that they exist

You can be discontinuous along a parabola or something
Then all the partials exist

OH true fine

by all partial I guess you meant all directional derivatives
which is what I meant too :p

So are there functions that are discontinuous but still satisfy Cauchy–Riemann?

4:00 AM
Yeah

Aw :(

Sure, easy to construct

I think the canonical example is something like, $\frac{\overline{z}^2}{z}$?

this is why I don't like analysis

Oh and it maps to 0 at 0 ofc

4:01 AM
So Cauchy–Riemann only implies holomorphic if they're continuous? Or is there some other condition
@arctictern Yeah, I thought complex analysis was the nice one :(

@arctic Yeah same, though complex analysis isn't /really/ analysis

Even weaker than continuous

How do I check if $\mathbb{Z}_{18}$ is a ring?

@AkivaWeinberger No, that's it

At least if you only work with holomorphic functions your life is good

4:02 AM

It's not as garbage as real analysis

pretty sure you need continuity for the guarantee @Zee

That's what I'm saying

If you go deep in complex analysis, it becomes very very much like real analysis

@DemCodeLines the definition of "ring" should include a list of properties to check

4:03 AM
Yeah like, if you're holomorphic then you're obviously continuous and satisfy CR, so since the back direction is enough, these are equivalent

@Zee A serious answer: Complex differentiable functions have really nice properties (they're infinitely differentiable, for example). Real differentiable functions, and their derivatives, can be horrible.
(Unless you know all that already?)

Yes, but if you open any research book on the subject, you will notice it is as analytical as real analysis
All kinds of estimates

Not in the same way, though.

@arctictern `A set, two binary operations (+ and *) and the operations satisfy: Distributive Laws and * is associative. I have no clue how any of this applies here

@Zee Ah, OK, I don't know a whole lot about the subject

4:05 AM
I mean, yes, estimates.

Am not an expert but I have a few advanced books on the subject that I found, the philosophy dept was throwing them away

It just seems like you have so much more structure that you can throw at things in complex analysis that just makes life better
Wait philosophy?

@DemCodeLines you need to work with the details of the definition of $+, \times$ on the ring as well

@DemCodeLines You literally have a checklist of things to check to verify that it's a ring. For example, does addition defined in Z18 satisfy the associative property for example? It's like, if I define a giraffe to be an animal with properties A, B, C and D, then in order to determine if an animal is a giraffe, all I need to do is verify that it has properties A, B, C and D.

4:06 AM
Yes, lol, I was walking and I see these complex analysis research books offered for free on the floor

@Daminark math is just applied philosophy

Only for the partials to exist, seemingly, unless I misread it
Oh, wait, never mind.

@arctictern "addition defined in Z18" I don't know what means...

@DemCodeLines do you not know what Z18 is? if you don't know what it is, you should have led with that question, instead of asking how to prove it's a ring.

It wants $u$ and $v$ to be differentiable, which means partials exist and it's continuous... though I thought the latter was weaker...

4:07 AM
I wouldn't quite say that... @Araske
I'd be amenable to saying that they're both, perhaps, branches of logic

math is applied logic is applied philosophy

Wait hold on what? I'd peg philosophy as being applied logic

something something relevant xkcd

Anyway, I remember reading that a locally integrable function that is CR is actually holonorphic which is mind blowing, but I didn't know enough to read more

Someone comment on what I wrote, 'cause I'm confused

4:10 AM
Wait hold on, so what you have is that a function is holomorphic if and only if it is continuous and satisfies the CR equations

Is continuity and existence of partials not enough? Or do we want the real and imaginary parts to be differentiable functions of two variables as well?

CR says if the partials exist then they are equal, it does not say they exist

(Cont'd) Pretty sure that that's stronger

@Zee That's not how I interpret it, it means they exist and match up correctly
Like this isn't a condition you can vacuously meet

@AkivaWeinberger look up looman menchoff theorem , maybe it will help

4:11 AM
@Zee It must requires they exist; else a function of random points could get called CR
or algebraic indicator or something else everywhere discontinuous

But yeah anyway, turns out @Akiva that this actually makes the real and imaginary parts harmonic

Balarka pointed that out to me once upon a time

But yeah this should force $u$ and $v$ to be real differentiable
Or at least like, this condition should be a thing

(Thanks for the reference, Zee)
> Looman pointed out that the function given by $f(z) = \exp(−z^{−4})$ for $z \ne 0$, $~f(0) = 0$ satisfies the Cauchy–Riemann equations everywhere but is not analytic, or even continuous, at $z = 0$.

Yeah
you do need continuity though

4:16 AM
Oh, and the partials need to exist everywhere (or at least in an open set around the point)
Existing at one point, and satisfying Cauchy–Riemann there, is not enough.
> The function given by $f(z) = z^5/|z|^4$ for $z \ne 0$, $~f(0) = 0$is continuous everywhere and satisfies the Cauchy–Riemann equations at $z = 0$, but is not analytic at $z = 0$ (or anywhere else).

What I always found weird in the realm of complex analysis is stuff like $f(z)=z+z^2+z^4+z^8+\cdots$.

makes sense, analytic is only defined w.r.t. open sets

Meh just assume arbitrary order of smoothness and you're fine

Which is convergent inside the unit circle but also has the unit circle as a natural boundary to analyticity

4:18 AM

So not only does that series not converge outside the unit circle, there's moreover no way to analytically continue to outside the unit circle.

something lacunary this way comes

Yup.

@semi it goes boom really fast

Quite.

4:20 AM
Huh, I had never heard that term before, but I like the etymology.

Another weird thing about that series which was pointed out to me, and which is apparently due to Mahler: It's transcendental for any algebraic value of $z$ (besides $z=0$).

@Fargle For a second I thought you were referring to "going boom"

'lacunary' is a very good term

@Daminark nah, I totally meant "yup"

And I was like, wait hold on doesn't everyone use that term anyway?

4:21 AM
/s

yes, they were referencing my highly developed and formalized terminology

ohiseewhatyoumean.jpg

It's in the comments here: mathoverflow.net/a/178561/55904

I think I want to start learning some concrete number theory and combinatorics soon enough
don't want to get #rekt on my admissions

t-rekt
Number theory and combo are pretty dank though, you def should

4:25 AM
Yeah

Still gotta go through Weil, oops

oh you need to learn nt from weil too?

Well, "need" might be a little strong
But in the way one needs to know, oh my goodness yes.

Oh @Araske is in on it too, I have declared
@Akiva not sure if I suggested this to you, but you want to join in as well?

Suh-weeeeeeet

4:28 AM
i think i and daminark was talking about getting into it at some point
we should make a study group

Yeah he talked with me about it too.

Yeah I also brought Fargle into the mix

Absolutely!

kewl

I hate number theory

4:29 AM
@Zee >:(
I kid.

A number theory study group?
I'd love to, but I don't think I'll have a whole lot of time over the summer

@Zee Your preference is factually incorrect

oooh number theory

I respect it but am horrible at it

Oh OK that's more fair

4:29 AM
Same, to be honest :/

I still don't know long division

In fairness, to be good at number theory entails being good at a lot of somewhat disparate and advanced topics.

yes we should make a study group
@Fargle but thats part of the fun!

lets do eeet

Who's making the room? finger on nose

4:30 AM
we going to start from Grothendieck, "SGA I" right?

Being good at analytic number theory seems like a very different beast than elementary number theory

@BalarkaSen leers

@Araske Remember that analytic number theorem thing Laci mentioned?

have fun with that.

4:31 AM
Baby Weil, not papa.

makes sense semi

How if you have an arithmetic progression with $(a,b) = 1$, then it contains infinitely many primes. Like dear lord

indeed

That's Dirichlet, isn't it?

Oh I was thinking we'd do class field theory via papa Weil

4:32 AM
yup

Yeah @Semi

Right.

:P Jk

all of laci's things are cool

If memory serves that's a result that's not too hard to prove via analytic number theory.

4:32 AM
I mean, all things in due time.

(I remember reading about the proof, but I won't say I understood it.)

so we have 3 UChicago dudes? man

Plus, with analytic number theory you can solve the RH and become famous and loved

Who's in UChicago? I know Daminark

me

4:33 AM
(and Eric)

Wtf 3 from the same school???

I guess I'll make the room, unless you're already on it @Dami

I'm on it

Eh, once you've got one it's easier to get another.

### ♦Very Secret Number Theory Study Group

Actually Number Theory Study Integral Domain but whatever (Weil'...

4:34 AM

We need to stop this Chicago invasion

mwahahahaha

> Actually Number Theory Study Integral Domain but whatever
lel

literally just a small list of applied math problems

4:35 AM
I, uh, mean, "haha"

thats all it would take to stop us

Is Chicago algebraic?

Why do you say such things @Araske?

we are everything that isn't applied math @Zee

I wouldn't say algebraic, like we've got a few PDE people around coughs Soug coughs

4:36 AM
lol

we don't even have engineers

I mean we actually have a bunch though

also lot of good topologist/geometers

That is true
We've only got one person in algebraic topology though

Does UC not have an engineering part? I'm so used to having that here.

4:37 AM
I have an eye for figuring out school biases , let me see the website

There's a molecular engineering program
Though it's not really engineering

we have analysts and topologists and algebraists

It's more research than anything
And I mean there are a few people whose work can go kinda applied-ish

its materials science

Makes sense.

4:38 AM
some of the cs systems people are a bit engineery
but thats really as close as it gets

Well actually you seem to have many who applied work

I mean, a lot of PDE people as a matter of course do applied stuff anyway
And I think you've got a couple people like Weinberger who do applied topology or smth

they exist
they just aren't well represented

For a bit of context, here's the list of science/engineering majors at UMN

But even then it's like, their work isn't primarily applied

4:43 AM
Is there an easy way to prove that $\mathbb{Z}[\sqrt{2}]$ is an integral domain?

You have a healthy mix, that's good

$\Bbb C$ is a domain, subrings of domains are domains

@arctictern huh?

Or it's not, applied for the sake of being applied. The Schrodinger stuff just happens
Then there's Lawler...

Ehh. I doubt that's quite true. I know UC has some good physicists.

4:44 AM
@DemCodeLines no two real numbers multiply to zero.

@Semi I was thinking more in the math department

Yeah, figured it out right after I said it

Lel

@arctictern To me, that seems pretty random. Why did you conclude that after looking at the question above?

@DemCodeLines $\Bbb Z[\sqrt{2}]$ is a bunch of real numbers, and you're asking how to prove no two of its (nonzero) elements multiply to $0$

4:46 AM
too many TCS though

Lol, I mean the school in general is a bit niche

In what way?

Specifically that the compsci and math departments, while having some range, do focus more on the theory
CS has got systems, and math has applied people floating around, but it's not a generic department so much as, theory

Well if it means more funding for math departments then even better

It shows up at a curricular level very strongly. For example, we've got a version of first year calculus which is more computational (but even they have to learn some proofs)
Then we have a 2 quarter "Math for social science" sequence, and a 2 quarter "Math for physical science" sequence

4:51 AM
our compsci dept in particular is quite rough if you aren't interested in theory
even our systems classes are somewhat theoretical

But that's about it, as far as I know. Physics was not happy and went and did their own math classes, so the latter is really more for geophysical sciences and chemistry
Math majors just do Spivak/IBL -> Rudin -> Dummit and Foote
But it's not enough, I still hold that algorithms in finite groups should be a core class and that Peter May should teach the 160s @Araske

My god

you said some words that make no sense

I hold algebra should not even be a grad core course

He'll probably explain continuity via a commuting diagram by just drawing it and saying "Do you see where I'm going with this?" :P

4:55 AM
no non-math person would even appreciate the beauty that is Split-or-Johnson
much less Unaffected Stabilizers

@Zee I would have a really hard time (meaning I would just fail at) taking a grad student who didn't know algebra seriously

@Zee but you need algebra to make analysis/topology work

Ehh. As with all research, it depends on what you're doing.

Nah, you need very little algebra for analysis
Algebra is important but tou can pick it up as you go

I mean it's one of those general knowledge things

4:57 AM
And that's where I disagree

depends on what you're doing I guess?

@Zee I mean, should anything be in the core?

Groups maybe, but rings and fields are just useless, unless your doing some abstract nonsense
If it was up to me, we would not have core courses

I have no opinions when it comes to math grad core tbh. Not in my purview.

4:59 AM
@arctictern Makes sense

But I think I agree with groups / rings / fields being something a math major should know about, simply as part of general knowledge.

if you don't know topology, analysis is hard-to-impossible
and vice versa

I mean if you're an analyst, modules have to come up, for example

Topology is important I agree

What the heck is even $\mathbb{Z}^2_2$? Is that some sort of "integer squared mod 2"?

5:00 AM
Spectral theory of operators is a fusion of linear algebra and analysis

Probably it's the Cartesian product of Z/2 with itself.
So ordered pairs of elements in Z/2.

As in, $\mathbb{Z}_2 \times \mathbb{Z}_2$?

\times

To be fair, that's a specific direction you can go which is heavier on algebra. But like, there's a general principle of breadth at work

And you need algebra to make a lot of things in topology work

5:01 AM
Right.

Sure, am not saying algebra is no good, just how much of the core it should be
Topology books usually cover the algebra too

I'd say everyone should have seen some amount of analysis, some amount of algebra, and some amount of topology
Am I asking that people learn advanced k-theory and algebraic geometry? Nope

I think there's also an argument to be made for certain topics as "general math knowledge".

But say, DF level theory of groups, rings, fields, and modules, for sure

I like algebra in context, for example understanding groups while doing basic algebraic topology
Otherwise it's too sterile

5:05 AM
Though here they go pretty hard on the stuff, everyone does a year of graduate algebra which is a quarter of representation theory, a quarter of commutative algebra and algebraic geometry, and a quarter which is more generic topics. Usually Kato teaches it and does more algebraic geometry, plus algebraic number theory

Trying to figure out how I draw an operation table for cartesian product. Internet has nearly 0 resources on this.

Given how many people do algebra for its own sake I think that's your extremely specialized idea of what it means to be sterile @Zee

well, probably start by deciding what you'll label the elements as.

@Semi backtracking slightly, "general" in what context, for everyone? Stem?

I was thinking an undergrad math major in general.

5:07 AM
@Daminark maybe am being close minded, but frankly I look at algebra the same way algebraists look at logic

Or as anyone in math views Bessel functions :)

@Semiclassical don't even know how to do that much :(

Well, how many elements are in Z2^2?

But yeah I mean, I'd say it's generally good to say that everyone ought have a minimum of knowledge across various areas. It shouldn't be like, a 4 year ordeal that takes everyone to research level

...4?
:|

5:09 AM
Right.

@Avery @bwDraco well, I looked. Turns out @TheGreatDuck and @Dodsy have been trading flags here for a couple of days. Latest volley was courtesy of Dodsy, but TheGreatDuck still leads by a fair margin for raw numbers pointless flags.

So:

Both of 'em should find a better hobby.

x 1 2
1
2

5:10 AM
...

???

Actually I think undergrad should be about pushing you to get to the research level

wat

...

oy vey

5:10 AM
****

___

@SimplyBeautifulArt I think we need to have a chat on discord.

@Shog9 yes, I may have stooped a little low today.
However, he got me banned yesterday

Ah, that makes sense @Semi. Usually I think of the "general math topics" as the basics of real analysis, complex analysis, point-set topology, and algebra

Right.

5:11 AM
Probably should've been more mature.

What about algebraic and differential topology?

Then after that it's more electives

...I definitely have not been making any flags to target anyone and I have no recollection of flagging dodsy or anyone else much lately. As I said, I've been largely outside of the site recently.

@Shog9 At least we know now that it was in fact him who has been flagging relentlessly here.

5:12 AM
@Dodsy do you have discord. I think it would be best if we chat somewhere private for a bit, please.

@Zee They're important to pick up by the time you're reasonably well into your graduate career, that's for sure

How many flags have I done in total? 5. Who have I flagged? The Great Duck, that's it. I was banned for using the word "titty".

wat

@Daminark I think they are foundational

I don't even mind cursing. I probably curse the most here.
Mostly I just don't like it when people discuss breaking the law.
:/

5:14 AM
I mean for sure, it's a question of how much comes in at the undergrad level and how much comes later

The facts are out, Duck.
I knew it was you doing this shit.

I have something with the word 'm*therfucker' on the starboard

@Shog9 Sorry, was playing Shovel Knight.

and I told you not to lie to me.

I didn't even censor the right half of the word

5:15 AM
Well, this makes a lot of sense.

@Dodsy yeah, and it is actually quite disturbing. Also, why were you asking about indiana a while back?

Because I knew it was you duck.

that doesn't even make any logical sense
what does flagging have to do with my location?

So... now that we've gotten to the bottom of this, hopefully, the flags will stop.

So the great duck has been relentlessly flagging here for how long?
and it took me finally flagging the fuck out of him to actually get it out in the woodworks
So, I don't even care if what I did was immature. What he's been doing to me and balarka for some time has been immature.

5:18 AM
@Zee I'd say perhaps the way my school does it is reasonably good. The first year everyone does the core courses, which are a year of analysis (real, functional, complex), a year of algebra (rep theory, comm alg + alg geo, topics), and a year of geometry/topology (atop, difftop, diffgeo)

It makes absolutely no sense at all for people to be flagging over a personal issue.

@Dodsy You misunderstand completely. Dodsy. If you would just talk somewhere private I think I might be to explain, but I'm not comfortable giving out such information here...

@bwDraco and yet, that's exactly what duck has been doing.

@bwDraco I actually like Balarka Sen a lot. So... that's absurd.

And we've asked and asked for assistance, nobody has gotten to the bottom of it.

5:19 AM
Mystery solved.

Duck, everytime you get into an argument with someone, you flag them.

no I do not
that's ridiculous

I debated with you last night about existentialism and 5 minutes later I was banned for half an hour.

and I enjoyed that conversation
besides

It is beyond me that people would flag each other over such silly reasons as a personal dispute.

5:20 AM
titty is offensive
but I didn't flag it

How is titty offensive?
I used to go to nude drawing sessions.
Is that offensive?
I don't even care if this tarnishes my image, at least I dragged you through the mud with me, Duck.

I'm being honest. I find that to be pushing the boundaries of what is considered allowed, but I did not find it bad enough to choose to flag it. Plus, I barely even talk to Balarka and yet they've been getting tons of flags. Why would I flag someone for no reason for not even no reason.
@Dodsy I don't flag people for personal reasons. At least get that through your skull. and I have definitely not been flagging as much as you claim. I only flag things that are really bad.
and that hasn't happened in a while
besides, like i said: I cannot approve flags
so who the heck is going around approving nonsense flags, hm?

here you go, @Dodsy and @TheGreatDuck. I'd like to see the math chatroom chat about something other than this dispute (unless there are mods who can weigh in).

Nah, I'm done, thanks Anon.

@anon thanks, I was just about to discuss that.

5:25 AM
I'd recommend that we don't have this fight here, migrate to another room if you want to keep going. For factual claims, proof can now be requested, and the discussion on intention and what's offensive can go on forever. I will suggest that we return to math (and also will not continue participating in this conversation)

@Dodsy and just to clarify. I really don't want to get into a large discussion. It's kind of embarrassing but... you're gravely misinterpreting what I'm saying and I suggest that if you have some other form of external communication (such as email) that we have a little chat. There's been some stuff going on in my online life I haven't told a lot of people. Simple Art knows.
@SimplyBeautifulArt I messaged you on discord. You'll know what this is about.

That TGD has been flagging things has been public suspicion for a long while, and that's what has been clarified by community mods. I don't think there's any point to beating a dead horse to further levels of death.

"Further levels of death"
inducts on levels

like what, hell?

Maybe your soul is dead and has a soul which goes to its own afterlife?

5:31 AM
Anyone notice that @tedshifrin talks in the third person a lot?

I used to a few months ago
got rid of that bad habit
hopefully that bad habit doesn't return
@Dodsy and I should be very clear dodsy that i am not at all mad at you. In fact, I thought we were having a nice discussion the other night. I was very positive about it. But finding out these flags have been going on just makes my blood boil. If what that mod said is true I will most certainly be dealing with some shit.
>:(

@Daminark Death by induction.

@Dodsy alrighty. and stop bringing me back into chat with stuff. I'm technically chat banned.
you want me to be perma-banned for evading a ban?
I was told to stay out.
@Dodsy and if you have an email, I would please ask that you email me at [email protected]. I'll explain why I have been flagging everyone.
@BalarkaSen Just to clarify. The flags were primarily just legitimate flags at first. I guess when people starting making fun of the whole thing (see I just assumed they'd get deleted) I guess I just figured marking a few once in a blue moon to make people laugh wouldn't be a big deal. I guess after a while it just got out of hand. I honestly never had anything against you. It was legitimately just you happening to say something profane at the wrong time and the mods being way to trigger happy.
but still use the email. I think there's more things that need to be said of off stack exchange. Too private for this site. If I were to say them, we'd probably all get banned.
and I'm not joking
@Dodsy one thing I should heavily emphasize though Dodsy is that I am far more self-destructive than rude towards other people and if I somehow could with a dupe account, I'd probably go through my whole chat history and flag any bad language I see... just to shit on myself more. You don't realize that many of my favorite posts on here are gone because I either reported them or cast close and delete posts on them just to punish myself for spending time on here.

5:55 AM
i am not asking for any justification of the flags. i am not sure what you gives you the idea that i want to continue this discussion any further

@BalarkaSen you're friend with simple art right?

i'm probably also just going to put you on ignore for the walls of text i don't want to see on the chat space... sigh

wait
are you on the googology server of his by any chance?