Yeah. Set difference A\B is defined to contain every element of A which isn’t also in B @ShaVuklia

Hi @Ted

Heya @Balarka @Semiclassic. Did I miss anything fun for the last 9 days?

Not particularly. Are you safely back home?

Hey can I ask a personal quesiton?

even with typos :P

Yup, safely back home. And some of the students I ran into at UGA who're taking the course out of my book said they knew I was on here ... but I didn't try to find out how :P

You just did, @SumNeuron.

Hah

I like math, but I am just bad at it. Even though I can solve some PDEs by hand, I feel like I have no understanding of the underlying mathematics

So, Balarka, you survived all your exams. What's next? Oh, I asked Mike Usher about Eliashberg's stuff and he had no insight; he said he had wrestled with that book as a grad student but doesn't know any better source.

What resources would y'all recommend to get that understanding

So what is your background in math, @SumNeuro?

I took several courses in college, Calc3, Combinatorics, PDEs, etc

Prob / Stats

Yep, exams went reasonably well. I think my math exam was the best I have had in years. All thanks to you because geometry of lines and planes in R^3 is amazingly easy when thinking in terms of vectors.

So you don't know any real analysis (proofs for calculus in one and higher dimensions).

Yup, @Balarka, that's really the right way to do that stuff.

Learning linear algebra payed off, I didn't have to study for them at all

Right

See? :P

So you only have to get the answer, not show work according to what they want?

Several of the math courses were proof based, but I did them off intuition. I have no formal background in proofs.

Oh, no, I do. We are allowed to use vector geometry to solve these things.

OK, @SumNeuron. I recommend you look at Fritz John's introductory book on PDE. I just recommended that to someone else in here a week ago.

Let me get you a link.

Oh, that's excellent, @Balarka.

@Semiclassical oh that's weird then, because spivak seems to be doing something different?

i'll post a screen shot

@TedShifrin It's not morally speaking hard to translate from unit direction vector to direction cosines lingo, either way.

But yeah

We don't know if $D_1^o\cup\dots\cup D_k^o\cup U_{k+2}\cup\dots\cup U_n$ is contained in $A$, no?

@SumNeuron: This is it, although I don't know what changes were made between first and fourth editions. But it's a good, well-motivated book.

@TedShifrin will that provide the entire ground work, or just PDE related

it's supposed to be just a cover, I would think

Just PDE, @SumNeuron. But that's what you asked about. I can't give you one source for all of mathematics. :P

I'm not following. My point is that the set difference A\B is well-defined regardless off whether $B\subset A$

@Sha: You should read my multivariable book instead of that book by Spivak.

lol

@TedShifrin true, but it would be nice though :P thanks for the input

It's his worst book by far.

oh, @ted. I had a question related to my stuff from last week, which Balarka was helping with

@Sha: If you are nervous about open sets that cover $A$ but go outside $A$, then intersect everything with $A$ to start with. (That's called working in the subspace topology.) But I don't see why you're having issues.

@TedShifrin Thanks for passing on my textbook recommendation question to Mike Usher. Really appreciate that.

I think I finally understand PVAL's construction, anyhow

Let $f(x,y,z)=4xyz-(x+y+z-1)^2$, and let $K=\{(x,y,z)\in[0,1]^3:f(x,y,z)>0\}$.

It's really about changing norms as you isotope the loop

@Balarka: If it makes us feel better, Mike agreed that it's very hard and unintuitive, and he had no reply when I reiterated my complaint about turning nonzero integrals into zero integrals with a small perturbation. :P

That renders the uniformity argument invalid

@TedShifrin Heh.

Makes me want to understand this better :) It'd be nice have on my imaginary CV

Balarka, when I have time and mental fortitude, I'm gonna look at that book a little bit.

Thanks!

I hope @Danu comes back. Now that Adeek is learning complex geometry, Danu can help teach him :P

Daminark is supposed to learn some Riemann surface theory with me after his exams are over, I suspect

It's cool that everyone is slowly getting free

I want you guys to learn about linear systems :)

I'm wanting to prove that there's no way to represent $K$ as a system of polynomial inequalities of degree at most 2.

Ah, dynamical things?

But the analysis part would be good for you, Balarka.

Hello everyone!

@Ted how was Georgia?

Hi Demonark. It was fun, bits were depressing, but lots of good cooking and it was good to catch up with old friends and former students. I also met the group of 1st and 2nd years who are suffering with my book ... they were an energetic bunch ... they had their last hour exam yesterday so they were studying. :P

(I also discovered an identity for $f(x,y,z)$ which I can check algebraically but which I have no intuition for. So that's neat.)

@Semiclassic: I think that sort of thing is very hard and subtle. I don't think about inequalities, but even with equalities it's hard. It turns out lots of things can be cut out by quadratics (but more than the codimension of the algebraic variety).

Yeah.

@Balarka: No, linear systems in algebraic geometry are totally different. I can explain a little or you can google.

Some interesting facts about that $f(x,y,z)$:

@TedShifrin I'd be all ears if you tell me about it a little (unless you're tired from the trip, of course)

@TedShifrin What do you recommend as a starting point for complex alg. geometry? I have half a mind to go through some of Claire Voisin's book on Hodge Theory this winter break. We'll see if that comes to fruition, of course.

1) It's got an S3 symmetry in its arguments, and it's also symmetric with respect to the rotation (x,y,z)->(1-x,1-y,z). That gives rise to an S4 group action on the points (1,0,0), (0,1,0), (0,0,1), (1,1,1) on f(x,y,z)=0

Also, nice to have you back :-)

@Antonios: I hear great things about Voisin, but I don't know it personally.

2) It contains the line (x,y,z)=(t,t,0), and from the S4 symmetry we get 5 other such lines.

I learned mostly from Griffiths/Harris. Despite its errors, it provides a good intuition for things. My main complaint (despite their not fixing the errors I pointed out and others did) is that there are no exercises. I can send you my typewritten exercises from when I taught a grad course on that many, many moons ago, if you want.

Hey @mago

Long time

Wow, @Semiclassic, you've turned into an algebraist too! :D

That would be a nice thing to have @TedShifrin

So it's cut out by a cubic but contains 6 lines.

lol

If you have a chance, my email is [email protected].

Is the probability of 100 heads in a row appearing infinitely often = 1?

@Semiclassic: Beautiful classic fact. The generic smooth cubic surface in $\Bbb CP^3$ contains precisely 27 lines.

I've heard of that

@Ted this is what happens when you're out of town. If this keeps up we'll all become finite group theorists. I know Balarka would love that

though this case isn't homogeneous (obviously)

@Semiclassical You can homogenize it, prolly

well, sure

You make it homogeneous when you put it in projective space, Semiclassic. The issue is how singular is it at infinity ...

hmm

hey @BalarkaSen

Oh, @Antonios, I was gonna tell you to email me. OK, hang on.

$F(x,y,z,w)=4xyzw-w^2(x+y+z-w)^2=0$, I think

Shouldn't you have updated to an NYU email by now? :P

@orbit-stabilizer The proof will show you why

Probably, but I have separate usages for my emails

NOOOO, @Semiclassic. Don't mess up the cubic term.

So it'd be a quartic surface in CP^3

You made everything into a quartic unnecessarily.

blah, you're right

That happens occasionally.

The idea is that you take any open cover of $F$, and then you can extend that to an open cover of $K$ be also looking at $X\setminus F$

$F(x,y,z,w)=4xyz-w(x+y+z-w)^2$

so cubic

There you go.

@Daminark thanks. I get hung up on the first thing I don't understand and I don't proceed further. I need to fix that.

degree never changes when you proj'vize

Look at Miles Reid's little Undergrad Alg Geo text. He talks about the lines on the cubic.

Hi chat

But for that to be an open cover, you need $X\setminus F$ to be open, so $F$ has to be closed in all of $X$

Oh, there's demonic Alessandro :P

hi @AlessandroCodenotti

Yeah I have found it helpful to suspend disbelief sometimes until the end of a proof

@Daminark pls no finite groups

Yo @Alessandro!

@TedShifrin Ted :D

Finally Ted is back :D

Heya @Kasmir. I presume that was your email I got (but I didn't remember your name).

@BalarkaSen i rechecked my geometrical physics book, finding it say diffeomorphism of a manifold form a group but not in the sense of a Lie group because it's infinite dimensional.

Yes.

Kek @Balarka, with time you'll come to live and breathe it

@TedShifrin haha yes dont say it here >< i want to stay as kasmir here =p

@TedShifrin was worried about ya :D

But finite groups are so finite

It's an infinite dimensional object. I think it's a Frechet manifold or something

Now all is good with chat

@Daminark This is me everytime I hear finite group theory: youtube.com/watch?v=uF7TsMDMyRM

I was on "holiday" cooking Thanksgiving dinner for 10 people and visiting various old friends, @Kasmir.

@TastyRomeo and that's great because then you can do counting arguments!

@TedShifrin hmm, at the point $(1,1,1,1)$ both $F=0$ and $\nabla F=0$

That's the boring part! :(

fixed

Finite groups are all Abelian, right?

Nope

Lol counting arguments with the Sylow theorems to prove that stuff isn't simple is like, my favorite part

Alessandro knows what's up

@BalarkaSen so Lie groups are required to be finite dimensional? in the chapter of Lie group of that book, it never says that.

@TedShifrin Haha nice =p I assume you are a good Cook? =p otherwise they wont chose you to be the Cook between your friends :D

I heard all finite groups are abelian and simple

@Antonios-AlexandrosRobotis He knows, he's showing his disdain for nonabelian finite groups by asking a question

@TedShifrin So while this is a cubic, it's not a smooth one

@CaptainBohemian Well, manifolds are often required to be finite dimensional

@BalarkaSen phew

Manifolds are finite dimensional, usually, yes

@Kasmir: I am actually, in total immodesty, a very good cook. I taught my sous-chef a lot and he enjoyed it

They are locally R^n

@TastyRomeo Corollary: all numbers are prime

@Antonios: Sent.

@TobiasKildetoft so diffeomorphism can't be pictured as a manifold?

i just cook at codechef, (wonder if anyone understands)

@Daminark except 0 and 1 :^)

@TedShifrin Thanks so much!

You have stuff like Banach or Hilbert manifolds which are locally modeled on infinite dimensional spaces

@Semiclassic: Right.

But I can live with 1 being a prime

@BalarkaSen Phew.

@TastyRomeo Ah, I thought we were talking about finite groups, not finite fields

@TedShifrin Well that is good to hear =p I did my exam today btw , Went super good against all odds ><

@Semiclassic: I've forgotten what's true about lines on a singular cubic.

@Tasty yeah yeah

@AlessandroCodenotti Same thing barring the extra operation :D

Well, that's good, @Kasmir. You do work hard, which is a good thing!

:P

@CaptainBohemian Not sure what you mean by "pictured" here

Multiplication is just repeated addition anyway so it's not like fields have anything "more" than groups, right?

haha well i need to work hard and be more productive ><

the activity in this chat just got bass boosted

thanks Ted

But really I'm all good with non-abelian and non-simple groups, they're the fun stuff, it's just that when you see a cyclic group of prime order it's like a cute little kitten that makes you happy. Though there are so many times that I've tried to be like "Oh yeah these are totally isomorphic because they're the same size! Sick that was fun" and then whoops the groups aren't assumed to be finite

so Kasmir outta learn how to work hard

Not sure what you mean, @Balarka.

Suppose that $G$ is a free group on the $n$ generators $u_1,...,u_n$, and let $U_1,...,U_n$ be unitary operators (not necessarily matrices). Will $u_i \mapsto U_i$ always extend a unitary representation of $G$?

so Kasmir outta learn how to work hard

@TobiasKildetoft formed

@Kasmir: I think you need to spend more time mastering definitions before you go too much further. That seems to be your biggest weakness.

heya @Tobias

@Daminark What does it mean when Rudin says: "If $F^c$ is adjoined to $\{V_{\alpha}\}$"?

@Kasmir: I think you need to spend more time mastering definitions before you go too much further. That seems to be your biggest weakness.

@BalarkaSen up to 11

@TedShifrin Hi

Not sure what you mean, @Balarka.

@orbit: Add that open set to your open covering.

I suspected Ted was off somewhere for the holiday. Welcome back.

@orbit $V_{\alpha}$ is some open cover, just add $F^c$

And the sniping resumes...

Thanks, @Kevin. Yup, I was, among other things, an indentured slave.

@CaptainBohemian That also does not make sense to me

Ah, thanks.

Hey @Tobias!

@TedShifrin In general your presence increases the activity on this chat, is all I meant :) It was a shit joke on the "bass boosting effect" only memestars of this chat would get

@Daminark Hi

Anyway , was fun talking to ya Ted

I figured it was your meme personality, Balarka.

I figured it was your meme personality, Balarka.

Are other people having trouble with the chatroom denying comments and, upon retrying, getting double entries?

Yep.

@TedShifrin So true >< also i never use notes in one Place, like each day , different stock of empty pappers and the ones I wrote I dont look at them >< so thats few things I have to fix =p

@TedShifrin yes

@Kasmir: Keep an organized binder or notebook for each class.

I just had that earlier

I can't see any reason why it won't always be a unitary representation.

I can't see any reason why it won't always be a unitary representation.

@TedShifrin So true >< also i never use notes in one Place, like each day , different stock of empty pappers and the ones I wrote I dont look at them >< so thats few things I have to fix =p

yeah, I was

And Kasmir.

make that "yes I am"

anyone else feels that the chat is acting odd?

Apparently :P.

I see many messages twice of them

oh boy

Yeah

It just sent me an error message

Something's wrong.

@Kasmir same

it's a lag in ACK i guess ...

I think someone was spamming spam flags

Maybe we should all leave in a huff.

@Daminark Dami :D

Maybe we should all leave in a huff.

i was seeing 10 of them piling up

some metapost was opened in this regard.

@Semiclassical Oh huh

@Semiclassical Oh huh

If we all flag everything, will they act on this? (not serious)

Is this the end of the world?

This is apocalypse y'all

Maybe North Korea is blowing us all up.

I have a Galois theory question but I'm not sure whether this chat is about maths or memes

@TedShifrin Ted !you said and kasmir ... then you did not say more :D

am still waiting =P

Trump started WW3 on Twitter again

Why would SE do this to our chat when we would never call SE badly moderated? Oh well, we try so hard to be shog9's friend... maybe someday that will happen!

I meant that you were having the double comment problem, Kasmir.

Oh haha

@TedShifrin hmm, $F(x,y,z,w)=F(w-x,w-y,z,w)$

and it's also symmetric in its first three arguments.

All righty then ! was fun talking to you Ted :D

now kasmir will go sleep =P good night yall

There are all sorts of experts on invariant theory stuff, @Semiclassic. I'm not one (other than a little bit of classic stuff).

Night, @Kasmir.

I guess nobody got my joke...

Sad!

@Balarka I did

don't be a sad! baby!

called it

@Narcissusjewel O hell yea

flag story still hitting again ?

Now Balarka is sounding like Trump, too. Oy vey.

Haha

@BalarkaSen well memed

Antonios, I'm just curious. Is your family of Greek origin?

Yeah, I'm Greek

So if we have $F\subseteq K\subseteq E$ a tower of extensions we call $K$ stable if $\sigma(K)\subseteq(K)$ for all $\sigma\in\text{Gal}(E/F)$

Sort of a triple-Greek name, so I figured :)

Did demonic Alessandro ever say hi to me? (I might have missed it.)

It's quite a mouthful. I usually go by Alekos, but my name is displayed as it is, so Antonios is just fine.