@Knight No. He was at NeXT for most of the time I worked at Apple. When he came back to Apple, there were massive layoffs and the project I was working on was slashed.

I didn't know this bit of your history, @robjohn.

@robjohn What sort of work did you do at Apple?

@Ted: I have a naive question. When do complex things admit holomorphic tubular neighborhoods? That is to say, a tubular neighborhood along with a holomorphic retraction to the base

Hi, a @Balarka.

This is probably going to be tough for projective analytic sets. What about affine sets?

@TedShifrin I wrote a program for teaching Logic at UCLA, and although the professor in charge of the project said that the way the program presented derivation proofs was a huge improvement (and similar comments on the symbolization of English sentences into symbolic logic, and other facets of the program), when he got an award from UCLA for the program, he did not share the spotlight.

Oh hi

That's a good question, a @Balarka. So the tubular neighborhood theorem works in the holomorphic category, doesn't it?

@TobiasKildetoft I wrote graphics software for the QuickDraw GX project.

That sucks, @robjohn. I think you told me about that endeavor once.

Oh wow.

Neat

You should forever be our Mac expert, @robjohn :) I've merely been dabbling with it since 1989.

@TedShifrin That's better than what happened to a friend when I was at Princeton. His advisor said that one of my friend's papers was garbage, and then quoted it, word for word, without attribution, in one of his papers.

Oh geez. That's immoral.

My only analog of that is I once told Phil Griffiths (who was de facto half my thesis adviser) some easy thing I'd figured out on the way to something more interesting. A year later he asked how it went. It ended up in a paper of someone else he was advising (or maybe a postdoc, I forget), no attribution to me.

@TedShifrin I don't believe so. In the smooth proof you flow along some vector fields and some such.

That's why I was thinking maybe affine is easier, having Stein stuff and all

No, @Balarka, you certainly don't need that (and flow of vector fields is fine, anyhow). It's just inverse function theorem.

@TedShifrin yeah, that happens, and it is immoral, but academics are not always models of ethical behavior.

I'm trying to think of we use partitions of unity, but I don't think so.

@BalarkaSen Elias Stein?

What is the statement you want? Given any complex submanifold, there's a neighborhood which is biholomorphic to an nbhd of the 0-section of a complex bundle on it?

Hah @robjohn, no. Different Steins

@BalarkaSen okay, just checking

I was thinking of the (holomorphic) normal bundle, yes.

@TedShifrin But this cannot possibly be right. In particular this implies you have found some nbhd which admits holomorphic retraction to the submanifold -- take some torus in $\Bbb P^2$, then arbitrarily close to it you can find torii which are not biholomorphic to the original torus

I'm wondering if Chern class (positive or negative) might be relevant. But let's think about $\Bbb P^1\subset\Bbb P^2$.

@TedShifrin What changed? I have the third edition, always wanted to know what are the changes.

If you have a holomorphic retraction that'd be a contradiction

Don't mix topology and complex, @Balarka. You mean a smooth elliptic curve?

@robjohn I am glad I never experienced that sort of behavior while I was in academia (at least not that I noticed)

@TedShifrin I just looked up my friend's advisor, and he died in 2007. I guess I can let it go now ;-)

Yes @Ted

Sorry for calling them torus.

That's confusing on my part

@Billy: I had notes on limits (giving less crazy $\delta$-$\epsilon$ proofs) and a bunch of stuff on Taylor polynomials and applications which he incorporated. So some problems got added, as well.

I won't ask who it was, @robjohn. I might have known this person.

@TedShifrin There was even a line of housewares that were influenced by him and carried his name. I won't go more than that.

@robjohn I like how the wikipedia page for QuickDraw GX attributes you as the resident mathematician

Hmm, I'm not sure I know about that, @robjohn.

@TedShifrin Do you still have the notes and/or could share them with me? I'll eventually buy the fourth edition but at the moment it's kinda impossible because the dollar is too expensive 1USD ~ 5.36BRL and we earn much less money on average.

I can send you the pdfs of the short notes I handed out in class. Spivak embellished some on his own. There were also some other exercises added/changed, which I cannot remember at all. But email me and I'll send you the two pdfs, sure.

Nothing hugely novel, but things I thought were important to do.

@TedShifrin Ok. Thank you. Gonna e-mail now.

@Balarka: To be honest, I've never thought about the point you raise. But does the fact that $\Bbb P^1\subset\Bbb P^2$ has self-intersection $1$ tell you we can't do it there? I'm being addled.

But $\Bbb P^2 - p$ where $p \notin \Bbb P^1$ is a tubular nbhd of $\Bbb P^1$ which admits a holomorphic retraction to $\Bbb P^1$, not?

That's what I had been thinking.

So you're making it seem like it's a deformation theory question.

I think it works out in this case

Yeah I guess it must be

That's why I don't like projective varieties.

Affine should be easier

So I have to rethink the usual proof I used to give of the tubular nbhd theorem.

I think the proof in the compact case for the tubular neighborhood theorem should go over just fine. You just use the holomorphic version of the inverse function theorem. Riddle me that, Batman.

Uh oh I don't want to be the Batman

Hm

What is the map from a nbhd of the $0$-section of the normal bundle $N_Y X$ to $Y$? Given complex compact submanifold $X \subset Y$.

Oh, you're right, I was just thinking about Euclidean space. Um, so I need it for $Y\subset\Bbb P^N$ to start with.

Alright. But is the map still clear?

I can certainly do it locally by the affine case you're thinking about. But how to patch. Hmm.

Yeah, maybe that's where partition of unity enters and messes up our lives

I don't think I've ever thought about this, shockingly.

Probably because it's a dopey question lol

@Billy: Two emails sent.

No, it's not a dopey question at all.

@TedShifrin Thank you very much!

There are all these different notions of positivity for vector bundles, and I keep thinking it's germane to that, @Balarka. But I'm way rusty.

Hmm looks relevant to me

I might browse through some of Griffiths's articles that I used to know ....

@Balarka: Relevant ... this also mentions one of the Griffiths papers I was thinking of.

What are we talking about

Good question. Is there a holomorphic tubular neighborhood theorem?

Interestingly, @Balarka, that Overflow question has an answer which says that $\Bbb P^1$'s in a surface with negative self-intersection have holomorphic nbhds.

Makes sense because you can blowdown

Ah, there is not. I wanted this a while back and found that there is almost never a holomorphic tubular nbhd.

Blowdown, take a ball nbhd of the blown down point, take preimage under blowdown map

@MikeMiller What about the affine case

Projective is going to be terrible as we have convinced ourselves

Maybe affine has Stein stuff

I think still then but I'm not sure. You're thinking Stein manifolds I guess

ha

Lol

Caught in act

Yeah, @Balarka, Griffiths's papers "Extension Problems in Complex Analysis, I, II" seem highly relevant.

Thanks, let me have a look

I read through these in grad school, but I'll have to look more closely after lunch. Good thing I kept Griffiths's Selected Works when I gave everything away.

Hi, @MikeM.

Hiya

Ok maybe you can tell me what to look at once you have taken a look already

So I'm getting hundreds of emails because UGA is about to open up to in-person classes (thanks to a wonderful Tromp-loving governor) and the faculty are freaking out, @MikeM.

OK, @Balarka. I'll do that later today.

Thanks

@TedShifrin But why are you getting emails about that?

What I know is true is that totally real submanifolds are more flexible than complex ones; if L is a totally real submanifold of a complex manifold M then it always admits Stein neighborhoods

Unsurprising/uninspiring because being totally real is an open condition

@Tobias: I still use my uga mail account. Anything that goes to "all" comes to me.

ahh

@Balarka: That surprises me on an intuitive level. It feels closed.

I could be wrong lol, is it open or closed

Um

LOL

Don't listen to me

closes ears

yeah its open

I mean perturbing a totally real subspace a little bit still gives a totally real subspace

satwcomic.com/masquerade expresses the situation here pretty neatly

@Tobias: Just read your earlier exchange that was starred. I have corrected so many people on MSE who think that you solve $A=0$, $B=0$ by setting $A=B$. Even people who've studied linear algebra.

@Balarka: If I'm supposed to open my ears, why?

@TobiasKildetoft Lmfao

@TedShifrin And if the person had just done an example, they should have immediately (hopefully) recognized that they would get the equation of a plane again

My common phrase is "throwing away information."

I also commented on an answer a few weeks ago where the answerer had done the elimination of a variable and said that the resulting projection was the intersection. Got upset at me for saying this was incorrect. "Oh, anyone would know what I mean."

I'm thinking my teaching days are getting close to the end. :D

So why is $V=\bar V$ an open condition, @Balarka? I'm confuzled.

For half-dimensional subspaces that's the same as saying $V$ and $JV$ transversely intersect, yeah?

If you wiggle $V$ you still retain transversality

Is it?

Total reality of $V$ is saying $V \cap JV = \{0\}$

@TedShifrin No shock. In addition to GA's ... general temperament, let's say ... I think you'll see on the balance sheets that they're not doing very healthy. Probably desperate to stay open for fear of lost tuition.

College football of all things? :(

For less-than-half dimensional subspaces $V$, you can find a maximal totally real subspace $W$ containing $V$ which is half-dimensional, and then argue.

I don't mean specifically UGA about the balance sheets though.

So my definition is wrong, @Balarka.

Ah yeah complex subspaces also satisfy your condition of course

Yeah, having the wrong definition explains a lot of stoopidity. Thanks.

I think I still remember calculus, though :P

All of math should be written in a tautological language so that one doesn't have to remember anything

I hate remembering formulas

Should learn to think like Meeshaa

LOL

I can't be Yau

I never wanted to be.

yeah math is nerd stuff anyway

i wonder what meeshaa is writing about nowadays

last math paper: arxiv.org/pdf/1908.10612.pdf

i dont understand anything but i can tell that its written by a maniac

"What exactly happens in the limit when $\varepsilon \to 0$ to the Dirac operator used in the proof of $\color{blue}{[X \to \bigcirc]}$?"

Nuts

He doesn't even use \bigcircle, he uses some custom command for ellipse

What's that notation at page 31 man

two blue wheels

why is this file the same paper twice in a row

What

OH LMAO

its 163 pages, but twice

Beautiful

I laughed for like 10 minutes straight

Nobody bothered to proof read his submission

that final paragraph on p156

beautiful

Oh nooo

HAHAHA

why does he switch fonts like every two paragraphs

@Thorgott lmao what

"inductive DECENT method"?????

Lol

Thor: He just likes a lot of italics.

Oh do you guys know how the arxiv work? To you submit a .tex and it gets compiled on their servers or do you submit both pdf and .tex?

Because I found a paper that only had the .tex once file and I didn't know that's possible

I don't even mean the italics, but he uses different straight fonts too

Lol

So much \mathsf

Feels like the nlab

it is too beautiful, I'm crying

some of the equations in this look like they're straight out of one of these math paper generators

hYbRiDizE

The "[≷ ∓mean]-Condition."

yeah those are amazing

he refers to one equation by an image of a pair of scissors LMAO

I can't even copypaste this symbol, how did he do it

@AlessandroCodenotti You just submit the .tex files

I forgot which files they need for bibtex

@Thorgott thanks for reminding me of the math paper generator

Definition 2.2. A Riemannian matrix $\ell$ is reversible if the Riemann hypothesis holds.

lmao

my favorite type of matrix

My Notendurchschnitt went up to 1,9 this semester hahaha

just checked my Notenspiegel

hilarious

Hello everyone

@TobiasKildetoft i see, thanks

@TedShifrin I did some more investigation and evidently the line of housewares was not his advisor, but someone with a very close name. I did however find a very scathing article about this guy from somewhere else.

@Alessandro Take $X_n = \Bbb R^2 - B_{1/n}(\frac{1}{n} \Bbb Z^2)$

If $n \to \infty$, does this converge to $\Bbb R^2$ in the ultralimit?

Can you adjust the radius $1/n$ of the ball to make it converge to $\Bbb R^2$?

How are you @TedShifrin

How are you @TedShifrin

It converges to R^2 in the pointed GH sense, so it should also converge as a ultralimit, no?

Yeah it just converges to R^2

I mean it'd be a pretty garbage convergence otherwise

Right I was just getting puzzled by something; of course it converges to R^2 in the GH sense

Easy to prove

The stuff that is "obviously converging to a clear limit" should also converge in the ultralimit sense

anyone know how one might try to analytically continue the following function $\varphi(z)=\sum \exp(-n^z)?$

I'm really confused

Can anyone help?

I'm deriving Divergence in Cylindrical Coordinates

Find Unit Vector Transformations ✔

Take Dot Product ✔

Take Partial Derivatives ✘

That's what I'm stuck on

If anyone can help, that would be great

Here are the PDs I'm trying to understand:

@AlessandroCodenotti Any ideas?

@BalarkaSen If anyone could help, that would be great

Hello everyone. Anyone here familiar with Denis Diderot Université Paris VII and its math faculty?

@DarkRunner looks to be $\vec a=(\overbrace{\delta\cos(\phi)}^{\rho\text{-component}},\overbrace{-\delta\sin(\phi)}^{\phi\text{-component}},\overbrace{\,\,\quad\gamma\quad\,\,\vphantom{\phi}}^{z\text{-component}})$

where $\gamma$ and $\delta$ are constants

I'm trying to follow this proof for "If f(z) satisfies CR at $z_0$, then so must f(z)^n for any positive integer n"

But I don't quite understand how the partial derivative of $f(z)^n$ w.r.t $x$ turns out to be $f(z)^{n-1}$ multiplied by the partial derivative of $f(z)$ w.r.t x

I know nothing else about $f(z)$.

@Threnody: It's wrong. There should be a factor of $n$. This is just the power rule (chain rule).

@TedShifrin I'm not too strong with the properties of partials... but it should be $n.f(z)^{n-1}\frac{\partial f(z)}{\partial x}$ right?

Nothing to do with partials. It's just the usual chain rule for a single variable. But if you're doing complex analysis, you do need to know your multivariable calculus.

Am I right in saying that since partial derivatives are normal derivatives where the variables not concerned are taken constant, then all the algebraic properties of derivatives apply to partials as well?

Yes. Of course, the multivariable chain rule is a different thing.

I'm not sure I understand.. this proof is not using the 'full' derivative... isn't that where the chain rule usually applies?

Yes, but lots of people write it in terms of the partial derivatives, of course.

I'm just saying that your statement applies here but not in the general, more interesting setting of the multivariable chain rule.

Hey Ted!

Hey Demonark!

How's everything going?

Still bumbling along, and you?

Pretty much same. Turns out I'll be hanging around at home this coming semester, so saving some money on rent

Yeah, education is going to be non-traditional for quite some time.

I guess there are no TAs for calculus lectures? How are they working that?

Discussion sections are mixed online and in person

So if a TA prefers to teach online they can

Oh, so they aren't making all classes remote, which means undergrads at least have to risk their lives on campus?

UGA seems to be way behind ... the faculty are revolting now, but it's intended that most classes will be in person — until tragedy strikes and they're up the creek.

Not all are remote, but even in person classes can be taken online somehow

So nobody is required to come to campus

I mean, they have to fit something like 1/6 of the expected lecture size in the room even if students go.

I don't know how much of an option professors (who tbh are prob at highest risk) are given regarding teaching in person vs online, the grad classes here are all online anyway as well as the huge lectures

I'm sure glad I'm a fossil.

So the optimistic assessment is that anyone who doesn't want to do things in person doesn't have to

Hopefully that's the case here. Sucks about UGA though

I would miss my 10 hours of crowded office hours. Don't know how to do testing without cheating, etc. shrug

(As a result of this though, TA assignments are still not yet decided)

I bet.

Yeah testing is gonna be tough for sure. Last semester there was def some grade inflation in my calc class on the second midterm, so on the final things were made tougher

There were 4 versions of the test, and people didn't have as long of a window

I think we're partnering up with some online proctoring thing for next fall?

It's hard to write exams where you aren't looking at and grading work ... and where Wolfram Alpha or whatever can't get the answer for you.

Hiya @Amin

Hey, what's up Edward?

Not muuuch, how's it going?

Long time no see

Demonark, did you see that Hippa returned briefly a while ago?

But yeah the online proctoring program I imagine would video people, supposedly an AI detects suspicious behavior (and it can tell if you open a new tab or program, so that's flagged), presumably students would have to make very subtle movements to check on a cell phone or smth

Ted: No I didn't actually! It's been a little while since I was last here

Hey Mike what's up?

Yeah, I suggested to a friend of mine that he make his students take exams on Zoom and watch like a hawk.

Our exams all take place in person, but all lectures are online

strange

Edward: things have been doing alright! Making a bit of progress on math, also been trying to lose some weight over the summer

@Amin oh nice, what kind of math are you up to? And nice, that's always a nice idea :) I'm doing the same,

maybe I need some m

some better vocabulary, "nice" seems to be my go to word

Been reading about basics of automorphic forms!

NICE I'm taking a seminar on the Langlands correspondence for GL_2 this semester

Seems like just yesterday that little Demonark was doing Spivak. :D

Hahaha, yeah :)

local Langlands anyway

Edward: eyyyy nice! My tentative advisor doesn't quite work in Langlands, he's more the analytic side of automorphic forms

I see :)

But I'd like to learn some stuff about Langlands nonetheless!

I'll relay information to you if you're around and I survive

Sounds good!

Will also be talking about elliptic curves over global fields for another seminar, you did your undergrad research project on elliptic curves right?

Prob a stretch to call it undergrad research project, like the thing I wrote up was just expository, but yeah!

It was specifically on the $\mathbb{Q}(i)$ analogue of Kronecker-Weber

Eventually I would like to learn more about elliptic curves, but at the moment I've got quite a long list of things to do lol

that's fair haha

Where are you right now?

In what sense? My physical location? rofl

Yeah

In Heidelberg :)

Ah nice! Also hey Alessandro whaddup?

Aye, there's a crazy amount of number theory going on here

That's fun for sure! Here we've got 4 people (as of this fall) in number theory which is a good time.

Mainly in analytic number theory?

Nah my advisor is the only analytic number theorist, and he's more automorphic formsy rather than prime gaps anyway

One guy does... I think Shimura varieites?

Oh yeah there's such a guy here too

lol

And we've got two arithmetic geometers

One just starting this fall, he's actually a mix of the more arithmetic stats and rational points on varieties + Shimura stuff

Cool :) There's an "Iwasawa theorist" here, who I'd like to write my masters thesis with, and a bunch of arithmetic geometers and Langlands people

That's dope!

What's Iwasawa theory about?

Urr about growth of class numbers in infinite towers of number fields

Sounds pretty nifty :0

What's Lukas been up to btw?

We haven't seen him in months

Yeah, he's MIA. I just hope he's OK.

Hopefully yeah

He's changed his profile pic on WhatsApp a couple of times

I guess that's something

Eyy Akiva

The gang is all coming in

Either the Riemann hypothesis implies the Goldbach conjecture, or the Goldbach conjecture implies the Riemann hypothesis

Heyo

'cause $(p\Rightarrow q)\lor(q\Rightarrow p)$ is a tautology

That is true actually :0

It's the Boolean version of "$x\le y$ or $y\le x$"

If one of them's false I can say it implies the other; if they're both true they imply each other

amazing

Can someone explain to me how $Qc\zeta_5 + \zeta_4) = Q(\sqrt{5})$? It's said I can solve for this in Dummit and Foote. I just know that $x = \zeta_5 + \zeta_4$ satisfies $x^ 2 + x - 1$ and that both fields are degree $2$ over $Q$, but I don't see how that shows $x = \sqrt{5}$

Am I misreading this, or is $\zeta_5+\zeta_4=\frac{1\pm\sqrt5}2$?

one or 'tother

Then $\mathbb{Q}\!\left(\frac{1+\sqrt5}2\right)=\mathbb{Q}\!\left(\frac{1-\sqrt5}2\right)=\mathbb{Q}\!\left(\sqrt5\right)$

i meant

$\zeta_5 + \zeta_5^4$

$x = \zeta_5 + \zeta_5^4$

Okay, I was going with the root of $x^2+x-1$.

@Hawk $\Delta(x^2+x-1) = 5$

oh right the $1/2$ is rational

so it gets absorbed