@0celo7 lick it

No wonder you guys lost the World Cup :P

@Sᴋᴜʟʟᴘᴇᴛʀᴏʟ Fight me

::runs::

@Sᴋᴜʟʟᴘᴇᴛʀᴏʟ Brazilian vs. Oaklander

It's pretty even

@0celo7 He so much as come near my water supplies he'll get sick enough to die

Your guys Olympics isn't gonna be much fun without the Russians.

Where are the Russians?

In Russia?

They got banned.

Doping and all that.

@dmckee you think you're so smart

@dmckee doesn't everyone?

Or rather the IOC decided their anti-doping agency wasn't even trying.

@0celo7 Well, yeah. But they have to be pretty careful about it because the anti-doping agencies know a thing or two.

@dmckee I gotta ask, was that a typo or did you not know which one to use?

@0celo7 Smart or knot I have a terrible thyme with homophones.

lol

My brain knows, but my fingers just do their own thing.

Don't you mean there?

Well, I lost a lot of work today and probably a lot of respect from my prof.

And Pokemon GO doesn't even work.

@0celo7 I lost a day once trying to get Doom to display through aalib in the linux console.

Did it work?

That's foolishness, but I had discovered the mplayer would run video that way and it is just so goofy I wanted to make everything do it.

@0celo7 Not doom. But I got a bunch of less interesting games to do it.

@dmckee I lost about a week once trying to homebrew my PSP

I wanted Charmander but I have fat fingers.

wtf you have to walk to play this??

I don't have time for that

oh my god this is so cool

@MAFIA36790 I take it back

haha my building is considered a monument because it's so old

@MAFIA36790 Ok, I'm over it

it's a cool gimmick

Interesting crowd in the math chat @0celo7

It's a gym but I'm not lvl 5 yet

I can't find enough things around me

I don't want to go wandering off in the middle of the night

@EmilioPisanty ah, good catch - I changed the accepted answer accordingly. (I do wish you hadn't capitalized "standard model" in your edit.)

@Danu I learned to do proofs during the PhD

it does not take so much time

\o @yuggib

Who will you be cheering for on Sunday?

@DavidZ a shameless bit of self publicising by @EmilioPisanty :-) David, I must admit I generally capitalise Standard Model. Is the convention not to do this?

I'm not sure. Some people do (capitalize it), but I don't.

I'm not sure if there is an officially accepted convention.

Anything used as a title is capitalized, no?

Only proper nouns (at least that's the general rule)

I figure it's called the standard model because it's a model that we have adopted as standard; that's not a proper noun.

Big Bang or big bang - either way I get critical comments from someone :-)

Big Bang has more bang :-)

Nah, Big Bang has more Bang, while big bang has more bang

@Sᴋᴜʟʟᴘᴇᴛʀᴏʟ moderately for france

since it was my home for two years

Hmm...

Should be a great match.

@JohnRennie Yeah, that it was. But also I genuinely think it should be the accepted answer, which is why I took to pinging @DavidZ.

@EmilioPisanty well it has my upvote :-)

@DavidZ Sorry about the sm to SM thing. I've never seen it in lower case that I can recall, figured it was a slip on your part.

Perhaps relevant

@EmilioPisanty no, that was very deliberate. I (like to think I) very rarely make typos of that nature so if you see something like that, I probably meant to do it. Not worth changing back in an edit, though.

@Sᴋᴜʟʟᴘᴇᴛʀᴏʟ hm, well... I wasn't aware of that, though it doesn't exactly change my mind.

@DavidZ Yeah, that's the reasoning I normally use for the capitalization. And a CERN style guide is as official as it'll get.

But to each their own style.

Well, I don't like it, but I suppose we should stick to the standard

Oh no

I have to do the worst thing

BASIC ALGEBRA

dun dun duuun

@Slereah basic algebra is silly; abstract algebra is fearsome

how do you do strikethrough text in the chat?

(or everywhere else using ASCII)?

T̶h̶u̶s̶l̶y̶

like @BalarkaSen did

e.g. here

adamvarga.com/strike

---strikeout---

thanks :)

If you click "help" near the lower right of the window, it shows you the formatting commands

good to know, thanks

Why do you find basic algebra "silly" compared to abstract algebra? @yuggib

for basic algebra I mean operations in natural/rational/real numbers

Yes, so do I :-)

for abstract algebra I mean the theory of magmas, monoids, rings, fields, groups, algebras, ...

So, what makes one silly and the other not?

by silly I mean rather easy to tackle

he's just an algebra snob

Ok, @yuggib that makes sense.

@Slereah algebra snob algebra purist (?)

algebra purist bourbakist

@yuggib I am allegedly bourbaki style when it comes to geometry

I prefer Gromov style.

@0celo7 only because of your radicalism

radicalism?

I'm not even a mathematician

extremism?

I know what the word means

I don't see how it applies to me

that you are only interested in (differential) geometry

but I would not say that you have a bourbaki approach on math

This is true

you should however

as everyone

;-P

I am only interested in another field insofar as it helps my pursuit of geometry

I hate Bourbaki's writings.

So do everyone I know in real life.

@BalarkaSen why?

@BalarkaSen You're like 14, how do you know people who know Bourbaki

@BalarkaSen you don't know many people of the french school right? ;-P

Being unrealistically rigorous is not the point of mathematics. Gathering insight is.

@BalarkaSen bourbaki is not unrealistically rigorous

@yuggib My German PDE prof hates Bourbaki too

@yuggib You're mistaken.

@BalarkaSen glad to disagree

@0celo7 germans make natural numbers start with one...how would they know? ;-P

What is unrealistically rigorous?

But each to his own. This is a pointless discussion.

Russel?

@BalarkaSen you started such a discussion

@0celo7 as in bertrand russel?

I made a statement.

@BalarkaSen the statement "you're mistaken" is a stupid statement

oooooooo

fight fight fight

and seems a lot like arguing to me

if you say so

@BalarkaSen Is it normal where you come from to just say "no, you're wrong" and that's the end of the discussion?

@0celo7 if it is, I am glad to live in another country

but I doubt so

@0celo7 Nope, but I stop having a discussion when it comes to just personal opinions from both sides.

@BalarkaSen you must be fun at parties

for some reason I was just reminded of this weird house I saw in Germany

it had a toilet in the middle of the room some reason

up against a pole

was there a sign "in case of emergency"?

no, they wanted to build it against the wall

but for some reason couldn't

Also, no, I don't know a lot of people of the French school - most of whom I know are from the concrete geometry school. I also think the old French-style mathematics is quickly disappearing of late.

Concrete geometry school?

Geometry/topology. Concrete stuff, not abstract functor business.

@BalarkaSen I've been in France (working in a mathematics department) for two years, and I have to say that old French-style mathematics seems pretty well and healthy to me

What is French style math?

Perhaps just in France. :)

@BalarkaSen Ah, I think that's what my adviser would be

But I don't know: I heard this from people a lot.

He has no knowledge of nor desire to know about algebra

Heh, I recognize that attitude. I don't despise algebra though, nor category theory.

Weil, Grothendieck, Schwartz, Dieudonné, Lions, ...

Good and powerful stuff when used appropriately.

@yuggib But Grothendieck's stuff, however categorically oriented, are very geometric in my opinion!

Which is surprising, but yeah.

@BalarkaSen He's a geometric analyst

@BalarkaSen so? still is pretty much the symbol of "old French-style mathematics" as you call it

He knows plenty of algebraic topology (I think) but he prefers the "by hands" proofs in GP

No, I mean, always plugging in functors whenever necessary. The kind of thing of which the offshoot is nlab.

I told him someone (@ACuriousMind) told me one can prove $\pi_1(S^n)=1$ via Seifert-van Kampen and he just scoffed

Algebraic topology to him is Bott-Tu

But I didn't mean the work of the individuals in Bourbaki anyway - just their style of writing.

@BalarkaSen I think it is possible to find plenty of works with the same style of writing today in all "pure" mathematics, and in some applied mathematics as well

@0celo7 Um. S-vKT is probably the easiest way to do that, although not my favorite. I perturb a loop to miss a point and then stereographically project through that point.

@BalarkaSen There's a proof via Sard which makes it trivial.

"in all pure mathematics" don't you think that's a bit of a stretch?

Surely you mean some, not all.

Can someone explain what the hell Bourbaki style writing is

@0celo7 Ah, but the point is a loop on S^n, a representative of an element of \pi_1(S^n), need not be a smooth map S^1 --> S^n

@BalarkaSen I am saying that it is possible to find in all areas of pure mathematics some works with that style

It can be badly badly badly space-filling.

@BalarkaSen We only care about the $C^\infty$ case, really

$\pi_1$ is not defined in the $C^\infty$ category. But sure, then it's trivial, and not interesting.

Well, not trivial, I meant obvious.

Same way smooth Jordan is obvious.

Obvious?

Visually obvious.

The proof is many pages in GP

I see

Yes, I didn't mean proof :)

I don't find the continuous case very interesting though

The nonobviousity of Jordan curve theorem comes from things like Osgood curves, which are horribly not smooth.

@yuggib Ah, fair enough.

That's true, I suppose.

@0celo7 read the first chapter of a bourbaki book of your choice and you'll understand

I've realized that people have no clue what math beyond calculus is

@yuggib link to your fav?

please not set theory

I have shitty internet, have patience

springer.com/jp/book/9783540205852

I use chapter 9 reasonably often

I think they're useful as a reference, but not as a textbook.

@BalarkaSen I think they were not conceived as a normal textbook

and I mean, after graduation who cares about textbooks? ;-P

What

I almost never read undergraduate texts (i.e. textbooks) after graduation

@yuggib textbooks are only for undergraduates?

@0celo7 I assumed that was the meaning of "textbook" intended by @BalarkaSen

GTM books are for undergrads?

@yuggib Now that I look back, I guess "you're mistaken" sounded a bit aggressive, I meant to add a "I think" in front, sorry about that. But I stick to the fact that discussing about this is pointless, as it is primarily opinion-based.

@0celo7 they're called graduate texts

I'd rather prefer a more mathematical or physical topic of discussion, rather than pedagogy and books and stuff :)

@BalarkaSen I agree that it is not worth discussing about opinions; and no problem about the other thing, but yeah it was a bit too assertive ;-P

Ok, let's convince 0celo7 that $\Bbb R^n$ is second countable

I know it has a basis of balls with rational radii

I'm still not convinced that I can take the centers to be rational

rationals are dense in reals

Take $\Bbb Q^n \subset \Bbb R^n$. Consider balls of radius $r$ around each pt in $\Bbb Q^n$ so they'll cover $\Bbb R^n$.

@yuggib I know

@BalarkaSen It's this part I don't get, rigorously

@0celo7 and for any two reals, there's always a rational between them

@yuggib I know

@0celo7 I mean you're look at each point in $\Bbb Q^n$ and collection of all rational-radius balls around that point.

That's countable. Taking union over $\Bbb Q^n$ is countable. So that's a countable collection of open sets.

@BalarkaSen That's not the proof I had in mind

@0celo7 which of the two parts of the definition of a base do not convince you when taking the one above?

I just need to show that any ball with rational radius and irrational center can be filled by balls of rational radius and rational centers

@yuggib I am convinced

I'm just not sure what the completely rigorous proof is

@yuggib I can't download that book

something wrong with my Springer access, again

I'll have to email the librarian, again

@0celo7 that's a pity

take the subset of the topology sets formed by the balls with rational radii and centres, and verify that they constitute a base

You cover your open set with just balls at rational points.

@yuggib is that now what I just said???

Then you enlarge it slightly to get the radius to be rational.

What part of "I don't know how to rigorously prove that" is so hard

Can be done without breaking anything because density of Q in R.

@BalarkaSen that's not rigorous either

I'm sure you can make it rigorous if you try hard :)

@yuggib oh

for some reason it was giving me springer.jp

@BalarkaSen multiple days later and I do not have the proof

@0celo7 yeah I was in japan you know

this seems like a regular math book to me.

I cannot contradict you there

Is there a Japanese equivalent to Bourbaki? @yuggib

I am not the one who said they are "unreasonably rigorous" ;-P

I know that given any point in my ball, there is a rational ball arbitrarily close to it

@skillpatrol who knows...probably

So any point is contained in a rational ball

QED

But I want a proof that would make Bourbaki proud

Good luck.

Ok, let's teach 0celo7 category theory

you have to prove two things: $\bigcup_{c,r}B_r(c)=\mathbb{R}^n$, where $c$ is a rational centre and $r$ a rational radius

Is $\Gamma$ (section operator) a functor on the category of vector bundles in general or only over the vector bundles of some specific manifold $M$

and that follows from the fact that $\bigcup_{r}B_r(c)=\mathbb{R}^n$ for any fixed $c$

@yuggib wait wait why $=\Bbb R^n$

A definition of a basis is that any open set is a union of elements of the basis

Not the whole space

I can easily show that $B_{irrational}(c)$ is the union of some countable collection $B_{rational}(c)$

Simply take an increasing rational sequence $r_i\to r$ (the irrational), then $B_r(c)=\bigcup_i B_{r_i}(c)$.

In a similar vein I would like to show that $B_{rational}(irrational)$ is the union of $B_{rational}(rational)$

look at here: en.wikipedia.org/wiki/Base_(topology)

$U$ be your open set. Take each rational point $p$ in $U$, take a ball $B_p$ of radius $r_p$ around that point contained in $U$ - can be done by openness of $U$. So $U = \cup B_p$. Because $U$ is open, for any ball $B_p$ at $p$ of radius arbitrarily close to $r_p$, it's contained in $U$. So just pick a ball $B_p'$ of radius sufficiently close to $r_p$ which is rational, which can be done by density of Q in R. $U$ is then again union of these, so you're done.

That is all.

Bourbaki (& Landau) are the greatest books ever written, if you don't like them but you don't understand why every single section is where it is and why then give them time ;)

@BalarkaSen Once again, it's that "sufficiently close" that I'm unhappy with.

@yuggib What are you referring to? You need to prove your open sets in the base covers every open set, not just the whole space.

That it covers the whole space is just the special case of it covering every open subset.

Most of Bourbaki's proofs are very short and involve little work, just wave your hands man ;)

then you have simply to show that it is the same as the usual topology

@BalarkaSen if you show that the set of rational balls with rational points is a base for $\mathbb{R}^n$, then it induces a topology on it

probably it is the same amount of effort as in the other way

Still not sure what you're trying to imply, but sure.

@BalarkaSen Why is $U=\cup B_p$ in the first line?

@0celo7 You're taking balls around each point in $U$ contained in $U$. Union of those is all of $U$.

Because nothing else is there, and clearly all of $U$ is there.

Sorry, I mean, rational points. This is a consequence of density.

@BalarkaSen I know it is, but that's not a rigorous proof :(

That's easily rigorified.

Ok, that's what I've been asking for for 20 minutes

I don't know how to do it

@BalarkaSen Let $(B_n)_{n\in\mathbb{N}}$ be the collection of rational balls with rational points. 1) prove it is a base for $\mathbb{R}^n$ (it suffices to show that it covers $\mathbb{R}^n$ and the other property on intersections). Then there is a unique topology on $\mathbb{R}^n$ that is generated by $(B_n)_{n\in\mathbb{N}}$. 2) prove that it is the usual topology

So, you took balls of radius $r_i$ around each rational $x_i$ in $U$, yeah?

@BalarkaSen I am already rigorously convinced I can take $r_i$ to be rational as well

let's just assume $r_i$ are rational

@BalarkaSen how do you like print GP

@yuggib you're welcome to rigorfy Balarka's statement too

Ok I feel like both of you are ignoring me

I already have most of the proof, I'm literally missing one step

I cannot do this by myself

I've tried and tried and tried

I do not know how to make it rigorous

I am explaining to you how to do it

No, just that one step

I do not want to do it your way, I need help with that one step

You are not telling me how to do that one step

clarify better that one step, as it is written it seems wrong

Sorry, I left for a while. Where was I?

How can it be wrong? $B_{rational}(rational)$ forms a basis and $B_{rational}(irrational)$ an open set

so it is clearly the union of $B_{rational}(rational)$

define your f*****ing objects properly instead of complaining

what?