That's very bad approach... I can say that a cylinder size of earth and some charged object ,size of a bull dog and and then calculate V(r) since cylinder is effectively infinite .....lol

Are all manifolds first-countable?

On the widest definition of manifold

@Slereah Manifolds are locally R^n

R^n is first countable

Ah yes

I wondered because some aren't second countable

Right the point is second countability is a global criterion

First countability is a local one

Hi @PrathyushPoduval

Heyya

The Jimmy Carr videos were really good

Haha, right?

Hey @BalarkaSen

I think I'll remember one or two of his phrases for future use :P

Do you want a PRIVATE COMMUNICATION biblio reference for this :

I'm putting it here

samuel-lereah.com/articles/Mathematics/…

He roasts people in frying pans, and sticks the frying pan inside the boiler and roasts the frying pan too

100 marks for that

@Slereah You can use it without acknowledgement

How rude

I'll do it

Nah, just copyleft

I saw the heckling video compilation, it thought me some new things

Well I mean, it's a math thing

It's not copyrighted

@BalarkaSen You an open source guy?

But acknowledgement isn't the same thing as copyright

@Slereah Fine, acknowledge me then. "private communications with a memelord"

@Slereah At what level are you writing your book?(Basic, Undergrad, grad etc)

I don't write at levels

I just write

He levels at writing

good enougg

probably not a good level for people who don't know GR

@Slereah Write an intro blogpost on GR for interested differential geometry grad students

I would read it

I did plan on writing a thing about what a spacetime is

Geometrically speaking

Aha

A Zee is quite a good book

The many structures of the spacetime

That sounds exactly the kind of thing I'd like to hear about

@Slereah you gonna put c=1 in your book?

@PrathyushP Are you following the idubbbz drama

$c = 1$ except for the experimental sections

In which all units are put in full

@BalarkaSen Idubbbz yeah, but what drama?

@Prathyush Man what rock to do you live under

idubbbz put up a content cop on Jake Paul two days ago

@BalarkaSen earth

(it's not actually on Jake Paul, it's on RiceGum)

@BalarkaSen Ah okay, I'll watch that now

Englad is mah city

@Slereah Why do authors do that? Is it like a trend?

It simplifies equations

@BalarkaSen fyi, I also forgot to follow Idubbz, which is why I didn't know 'bout it

And you don't need it for mathematical properties

@Prathyush kk, I am not subscribed to idubbbz really

But it's a big thing, it went viral on the internet after a few hours since it dropped

@Slereah I used to always check units everywhere while reading relativity and things became very confusing, took some time getting used to

@BalarkaSen See anything shocking in that page btw?

@BalarkaSen Ah okay

or is it all good

@BalarkaSen I would caution against identifying the corner of the internet you frequent with "the internet" :P

looool

What is the "internet"

I spend a lot of time on the internet and I rarely have any idea what you're talking about, for instance :P

yeah you spend time on the normie corner of the internet

the actual internet is the meme culture

@Slereah In the page you linked me?

@BalarkaSen

Da

this is what you were talking 'bout?

yes

Ok, checking

It's fuckin 30 min long

Thats true

I'm not sure the limit proof is quite good yet

anyways I got nothin else to do today

Since it doesn't use the fact that it's first countable or anything

@BalarkaSen Frankly, I find this attitude in itself off-putting.

@BalarkaSen I watched only them which are less than 15min

Oh wait

I didn't even finish that proof yet

Darn the luck!

@ACuriousMind Surely you realize I'm joking?

@BalarkaSen What's the joke? The things you say are entirely consistent with you actually believing that.

Who would unironically believe internet is equal to meme

you

That's a silly thing to say, which is the joke :P

I do spend time on the meme corner, but that's because I am a procrastinator with no life

(Hm, on the other hand, it's better than saying internet is equal to 4chan)

@BalarkaSen How do you know there aren't people who actually believe that? (Of course not in the sense that these is nothing else on the internet, but in the sense that the other things are somehow lesser)

@BalarkaSen Yeah, that's true

People believe all sorts of outlandish things, so the default assumption when someone states something is to believe them, not assume it's a joke.

For example, I believe the earth is flat and moon landing is a hoax

What outlandish things do you believe @ACuriousMind

@BalarkaSen Don't you find school boring?

@ACuriousMind Good guess? People involved in the meme culture have a heightened sense of humor, so usually they can tell the difference between what are worthless jokes that are only their for cheap entertainment and the vast wide world which is known as "The Internet"

In fact that's part of the meme behind saying "memes are the internet"

@BalarkaSen See, there's that off-putting arrogance again - "heightened sense of humor". That's not a joke, that's you literally saying those not participating in your subculture are deficient in the humor department.

But you are, @ACuriousMind

Get on his level

@Slereah Well, if I'd consider my own beliefs outlandish, I wouldn't believe them, would I? ;P

I only believe outlandish things

Damn the Man and whatnot

Whats up with your profile picture @ACuriousMind?

>you're

@PrathyushPoduval I've changed it, as I'm wont to do regularly

@Slereah Get them mixed up all the time

Who is she?

@ACuriousMind You seriously think saying a certain group of people have heightened sense of humor is belittling the others not involved in the group? So saying to someone that he doesn't appreciate 100 levels of irony and saying they don't find disgusting filth as funny is putting them beneath the others?

I am sorry, I don't see your logic at all

huhuhuh

I mean who in the whole universe would find "you don't get 4chan/b/ humor" as off-putting

@ACuriousMind Saying someone has something is different from saying everyone else doesn't have that thing

@BalarkaSen Not finding something funny is different from having a heightened or lowered sense of humor. People can find very different things funny, that's okay. Saying that your sense of humor is higher because you find certain things funny is the problematic phrasing here.

Ah, I see what you are annoyed about. But you get what I mean, right? I didn't mean to say meme humor is in any sense the greatest level of humor on earth or anything

Just that it's a specific form of humor that people outside of the meme culture would find hard time appreciating

Eg, saying "internet is meme" is such an example. You didn't get that I was actually making a joke when I said that

@ACuriousMind even better

@Slereah I have been thinking about non-Hausdorff spaces a bit these days

Lots of time to waste, too?

Well, not in the context of non-Hausdorfficity, but a lot of sheaves are non-Hausdorff

True

Although I don't know much about it

I'm not much into sheafs

@Xasel That's a standard approach, so you'd better get used to it. And while there is often some relatively sloppy language, it is important to think carefully about just why it is justified (which it is) and what the limits of that justification (and therefore of the formalism and the language that encodes it) are.

Gotcha. It's fun actually

Isn't it like

Foliations of manifolds by solutions of PDEs

Or something

The identification of "long" as a shorthand for "long enough to be effectively infinite" is pretty much the only thing that "long" means in a technical setting.

@Slereah Oh, those are leaf spaces

Ah

Now that you say it, they are kinda like sheaves actually

Which ones are the sheaves

sheaves and leaves

Oh wait

Is it the...

Category theory thing

Yeah but I like concrete examples

About maps between open sets and... something

You know what germs of functions are?

Or is that pre-sheaves

Yes

So say $M$ is a smooth manifold

$M$ is a smooth manifold

lol

For each point $p \in M$ there is an associated "space of germs"

I s'ppose

I'm guessing related to the space of smooth functions on the manifold

in some bundle bullshit

Yeah. The space of germs at $p$ is defined as follows; look at all the smooth functions defined around some (unspecified) neighborhood of $p$

And define the following equivalence relation

If $f$ and $g$ are smooth functions on $U$ and $V$ respectively where $U$ and $V$ are neighborhoods of $p$, say $f \sim g$ if $f = g$ on some open subset $W \subset U \cap V$ containing $p$

Is there a constructive definition of germs?

Like are germs the sequence of all derivatives of the function at that point

Aha.

@BalarkaSen grandayy came up with a idubbzz JP video now i think :P

got to watch it

@Prathyush I'll watch it, thanks!

grandayy was tweeting a lot when the drama happened

You use twitter?

Also are germs associated with a unique analytical function?

Nah just look at the meme gods once in a while

alright

This guy insults people very calmly

in a controlled, destructive way

thats idubbbz

yeah

@Slereah Well, k-jets are actually a quotient of the space of germs

But some of the videos are too long

and k-jets, as you might know, contain the information of the function's Taylor series upto order k

@Slereah Not necessarily

You can take, say, the germ of "$f(x) = e^{-1/x^2}$ if $x > 0$ and $f(x) = 0$ if $x \leq 0$" at the origin

@BalarkaSen that example is just too common

That's not a germ of any analytic function whatsoever

@Prathyush It's a standard example of a smooth but not analytic function

But if a germ has an analytical function associated to it, is it unique?

Ya. Identity theorem.

Ah yes

In fact if you have a complex manifold and you do the same construction I did

but look at holomorphic germs instead

Then a holomorphic function on the manifold is determined by it's germ

IIRC you can define vectors as germs

Or somesuch

Straumann does it

you can define tangent vectors of a manifold as derivations on the germs

I think

Well, actually, only for $C^\infty$ manifolds

for $C^k$ something bad happens

@0ßelö7 taught me that once

oh no

not the bad things

he's an analyst, there is no good analysis in the world

only bad analysis

@Slereah So to each point on $M$, you could associated the space of germs, right?

That actually gives you a bundle

$E \to M$ where each fiber is the space of germs

I'll believe you on it

if you take a local section of this bundle over $U$, that's the same as choosing a bunch of germs at every point on $U$ smoothly

but that's the same thing as a smooth function on $U$

so local sections of this ""bundle"" are precisely the smooth functions

That's where the sheafy thing is happening; to each $U$ you associated $C^\infty(U)$

Isn't that just the line bundle

The sad thing is that $E$ is badly nonHausdorff

Oh wait

Line bundle isn't necessarily smooth

So... Subset of the line bundle?

@Slereah Well the fibers are not necessarily lines, or even vector spaces for that matter

It's not a vector bundle

But is the bundle isomorphic to a subset of the line bundle?

Of which line bundle?

Real line bundle on the manifold

@BalarkaSen Going through grandayy's tweet was a wonderful decision :P

there are lots of real line bundles on a manifold

@PrathyushPoduval I loved the "this guy needs to work on his quick mafs"

@Slereah But no, it's not a subset of any line bundle on the manifold in general

The reason is $\mathcal{O}_p$ is a non-Hausdorff space

True

Eg, germ of the function $f$ I wrote there, and germ of the zero function, are arbitrarily close

That's why Taylor expansion fails, eg

renyi.hu/conferences/lrb17/slides/Luc.pdf

Hey

That's pretty recent

Two months ago

Oh no Poland

the Polish people do all the crazy topology

They need a lot of toplogy to figure out their borders

and it's by a lady

pretty rare in GR

oh Hawkings-Ellis talk about nonHausdorff chaps in their book?

non-Hausdorff manifolds pop up in some maximal extensions of spacetimes

@BalarkaSen you're god damn right

like the maximal extension of the Taub NUT spacetime

But about Lipschitz ones for sure

what's a maximal extension

@BalarkaSen is tom Dieck really algebraic? It looks like it

Uh, yeah

Not many non-algebraic books on algebraic topology out there

@BalarkaSen Take a spacetime $M$, an extension is a spacetime $M'$ such that there's an isometric injection of $M$ into $M'$

Or something

It is maximal if there's no more extensions beyond that

@Slereah Ah.

(basically when there's no more singularities)

@BalarkaSen basically an lub of extensions of spacetimes

Got it

The same thing works for Riemannian manifolds. Do Carmo mentions it

something something completion

yeah

Except it gets weird with spacetimes

I think you can even get weird non-hausdorff extensions with Rindler wedges?

@BalarkaSen but it's got scary stuff like hocolims

He defines everything right?

You just need to pick up the terminology real quick

Hey @0ßelö7

"Schwarzschild's paper is strange to modern eyes, however, in that, when he considers positively curved space, he only discusses $RP^3$, which he calls \the simplest of the spaces with spherical trigonometry." In fact, he explicitly rejects $S^3 $as a physically acceptable model for spatial geometry, on the grounds that the light emitted from a point in $S^3$ would collect again at the antipode, and 'one would not consider such complicated (sic) assumptions unless it were really necessary'."

Scharzschild wrote about spacetime topology in 1908

@Slereah huh

Well

Space topology

He didn't write about spacetime yet

Message lagged

@BalarkaSen So apparently the Yamabe error is that he messed up the order of Lp embeddings. The large ones embed in the small ones. He did the opposite

Oh, 1900 even

Who made that error?

Uber das zulässige Krümmungsmass des Raumes

Hm

There doesn't seem to be an english version

@BalarkaSen Yamabe in the original paper

Ahh

Schwarzschild died pretty young

42

on the battlefield

"Der Vierdimensionale Raum"

from some disease

that's what people did in 1917

And good thing he died, damn kraut!

Trying to steal my french soil

amazon.com/SECOND-ORDER-PARABOLIC-DIFFERENTIAL-EQUATIONS/dp/…

what kind of cover is that