Mathematics - 2016-08-27 (page 1 of 3)

He guys, Ive got some proof here I dont understand the beginning of it. It says that a finite division ring R has |Z|^n elements. Where Z is the center of R. We dont know that R is commutative ( because that is tonbe proven)

So I guess the argument is that R is a vector space over the center?

How can I see that?

user227867

1:39 AM

@user1618033 I am trying to improve my singing. Until then, I won't put up more. Hehe. Don't forget your destiny as a mathematician!

user227867

@0celo7 Physics can be rigorous, but many physics professors are not. Some math departments in the world teach lots of physics, like Cambridge.

0celo7

@JasperLoy I like mathematical physics

lorentzian geometry especially

user227867

I like you. =)

0celo7

thanks

user227867

Anyway, I gave up learning physics because there was too much math.

0celo7

1:44 AM

huh?

user227867

I mean too much pure math.

0celo7

I gave up on physics because (a) QFT is BS (b) there are no good string theory books (c) physicists can't explain group theory

user227867

So instead of focusing on physics, I wanted to do pure math.

0celo7

I cannot put my frustration with physics group theory into words

it boils my blood

user227867

But maybe I will do some physics in my next life.

Sir Cumference

1:45 AM

Uh...er, why is QFT BS?

0celo7

@SirCumference No one knows why it works

interacting QFT cannot be rigorously defined

user227867

I will leave you two to discuss QFT, poof.

0celo7

lots of problems

Sir Cumference

@0celo7 So? Doesn't it still seem to work?

0celo7

@SirCumference that's too pragmatic for me

Sir Cumference

1:46 AM

Oh

0celo7

and for most mathematicians

and renormalization is hell

Sir Cumference

Welp, you broke it to me. Physics doesn't explain truth, it just models the world

But even modeling the world seems amazing. Being able to predict how the Universe will act is something in itself

0celo7

nothing explains the truth

when you truly realize that you will spend a few hours on the floor in the fetal position

Sir Cumference

Well...I still am confident that one day, we will find a theory of everything

0celo7

uh, I think it's been proven that's not possible

Sir Cumference

1:48 AM

Why?

Because GR and QFT are incompatible?

0celo7

there will always be undefined variables

no

because you have to point to things in nature and say "nothing more fundamental than that"

and that's a nonanswer

Sir Cumference

Welp, at the very least, we can come closer and closer and closer to the truth

Coming closer and closer is worth something, isn't it?

0celo7

yeah, maybe

Sir Cumference

It's the best we got

user227867

Speaking of LaTeX, all mathematicians must read 'More math into LaTeX' by Gratzer. The fifth edition was published this year. Read it for proper texing.

0celo7

1:54 AM

proper?

user227867

Well, there are good and bad ways to typeset everything.

user227867

This is now the definitive book for LaTeX. There is no competition.

user227867

It is only 600 pages long. =)

user227867

A beginner can read the first chapter and begin texing. It is only 30 pages long. =)

0celo7

I don't know what tex is

Sir Cumference

2:24 AM

@0celo7 Abbreviation of Texas

0celo7

that's TX.

Sir Cumference

Close

O'''''''''''''''''''-

@0celo7 @SirCumference and more specifically: Texarkana, Texas

Sir Cumference

Yeah sure, that

O'''''''''''''''''''-

"Display name may only be changed once every 30 days; you may change again on Sep 18 at 7:20"

Oh well, I'm stuck with this.

user58865

2:32 AM

Nice name, @O'''''''''''''''''''- :)

Anybody here study Algebra?

If so, have you ever studied from Artin's book?

Would you recommend it? :)

Semiclassical

I have no idea what this question is asking: math.stackexchange.com/q/1904953/137524

0celo7

was half expecting the question I just asked :)

1

Q: Every Riemannian metric is conformally related to a complete metric

If $(M,g)$ is a Riemannian manifold, there is a metric $\tilde g=hg$ on $M$ which is complete, where $h$ is a positive smooth function. I've been given the hint to let $f:M\to \Bbb R$ be a smooth exhaustion function, which means that $f^{-1}(-\infty,c]$ is compact for all $c\in\Bbb R$. Then one s...

Semiclassical

hah. that one I'm at the least confident you understand what you're asking :P

the one I linked...not so much.

0celo7

i say the surface of a three ball so that any topologists reading dont think im talking about a glome if i say 3 sphere and geometrists dont think im talking about a circle if i say 2 sphere — dave scherer 2 mins ago

what the hell is a glome

Semiclassical

typo of globe, maybe?

0celo7

2:43 AM

oh lol

Semiclassical

i had to stare at it myself

I just don't even know where to start with that question.

0celo7

@BalarkaSen Your tubular neighborhood trick doesn't exactly work for the smoothing. You need to make the first derivatives close, not the curves themselves.

Semiclassical

At least that question did get me to look up what the name of a 24-sided polyhedra is. (icositetrahedron, apparently)

arctic tern

@Semiclassical presumably talking about the permutohedron

Semiclassical

hmm, that'd work

come to think of it, i assumed he meant 24 faces

arctic tern

3:00 AM

either that or the 24-cell

which I guess is what you said

so probably that actually

0celo7

...what is that good for

Semiclassical

good question. i doubt the person asking it knows either

arctic tern

classifying higher-dimensional analogues of platonic solids.

0celo7

eh...why?

Semiclassical

because they're there, i suppose

arctic tern

3:02 AM

why did they classify platonic solids? they're heavenly, or so I'm told.

Semiclassical

@arctictern Could be that what they're talking about is something related to this bit of what you linked: en.wikipedia.org/wiki/24-cell

0celo7

> Show that any closed subset of a compact topological space
is compact.

Hmm

I thought that was only in Hausdorff spaces

Semiclassical

what an "open column of points" is compared with a "closed row of points", though...

(when I say it's possibly 'what they're talking about' I am being extremely generous, of course)

arctic tern

@Semiclassical oh, I guess you weren't talking about the 24-cell before I chimed in. I didn't recognize the difference between icositetetrahedron and icosatetrahedroid with a glance-over.

0celo7

good god, there's a billion typos on my analysis problem set

arctic tern

3:05 AM

that's quite a large number

Semiclassical

@0celo7 huh: "In mathematics, a 3-sphere (also called a glome) is a higher-dimensional analogue of a sphere."

from Wikipedia's page on the 3-sphere

0celo7

WHAT

2

arctic tern

seconding 0celo7

0celo7

what's their reference

arctic tern

checked the edit history just to make sure it wasn't OP putting that in there...

0celo7

3:06 AM

3-sphere

In mathematics, a 3-sphere (also called a glome) is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space. Analogous to how an ordinary sphere (or 2-sphere) is a two-dimensional surface that forms the boundary of a ball in three dimensions, a 3-sphere is an object with three dimensions that forms the boundary of a ball in four dimensions. A 3-sphere is an example of a 3-manifold. == Definition == In coordinates, a 3-sphere with center (C0, C1, C2, C3) and radius r is the set of all points (x0, x1, x2,...

bruh

Semiclassical

it also shows up at mathworld, though without citation: mathworld.wolfram.com/Glome.html

0celo7

mathworld.wolfram.com/Glome.html

WHAT

wait wait wait

please just hold on

since when do topologists and geometers disagree on which balls are which

Semiclassical

you haven't run into that before?

0celo7

that guy spewed a bunch of nonsense

why is this coming true

my world is crumbling

Semiclassical

my figuring is that they read about something that was true, didn't understand it, and just aped it without knowing enough terminology to actually get it right

0celo7

3:08 AM

@Semiclassical no, what is the issue?

$S^n=\{x\in\Bbb R^{n+1}\mid ||x||=1\}$. No debate.

arctic tern

"Glome: A gamer's term glome is what you get after playing the Nintendo Gamecube for too long.This condition is especially prone to happen if one game is played for a long period of time and all data is lost. Damien got glome after playing The Legend of Zelda,The WindWaker for 43 hours straight then losing all save data because of a power outage." - urbandictionary

0celo7

@arctictern urban dictionary is bad for you

Semiclassical

evidently topologists prefer to label by the dimension of the surface itself, whereas geometers prefer to label it by the dimension of the space it's embedded in

0celo7

...literally no geometer.

maybe in 1850

Semiclassical

second paragraph: mathworld.wolfram.com/Hypersphere.html

and the quoting of Coxeter 1973 therein

0celo7

3:10 AM

That's BS.

I own ::counts::

Semiclassical

I wouldn't be surprised if that's something that's largely disappeared, to be sure

0celo7

like 20 geometry books (wow)

6 topology books

let's see...

well, let's go for my oldest geometry book

it's the geometers who supposedly name it wrong?

Kobayashi-Nomizu is pretty old.

Semiclassical

that's what shows up in the Glome entry on Mathworld, btw. "A glome is a 4-sphere (in the geometer's sense of the word) $x^2+y^2+z^2+w^2=r^2$ (as opposed to the usual 3-sphere)."

0celo7

does KN even define the sphere lol

I suppose they don't.

They do not, afaik

who needs examples lol

do Carmo has the correct definition, and it's from 1979

Semiclassical

The more I think about it, the more I find myself wondering about that "geometers v. topologists" notion. the fact that Coxeter has it that way makes it seem like it's something that's lasted far longer than I'd have expected

0celo7

3:17 AM

noooo I got too many books off of the shelf and the remaining ones fell over

@Semiclassical Kosinski Differential Manifolds has it with the weird definition.

1993.

Semiclassical

huh, i find that one surprising

0celo7

Oh, no, it's a typo.

Semiclassical

ah

0celo7

On page 16 he writes $S^n\subset\Bbb R^{n+1}$.

Interesting book btw

has some nice folklore theorems in it

Semiclassical

there's some discussion on the Talk page of 3-sphere re: geometers, btw

[en.wikipedia.org/wiki/Talk%3A3-sphere](link)

with a number of people saying "Mathworld's claim is BS"

0celo7

3:25 AM

dead link.

Semiclassical

woops

arctic tern

remove last ]

url-haxoring

Semiclassical

well, i failed at link posting

there's also a bit about the glome reference, namely that it's also BS

0celo7

> Geometry uses a different notation - for example, see H. S. M. Coxeter, the 'greatest geometer' of the 20th century

Coxeter is hardly the greatest geometer

pls

Chern? Grothendieck? Milnor?

Hamilton?

Lots of better ones

Freaking Whitney

Einstein :P

that thread is cancer @Semiclassical

Semiclassical

pretty sure Hamilton can't be counted as 20th century

0celo7

3:32 AM

@Semiclassical Richard Hamilton.

Semiclassical

ah.

this is, i'll be honest, a claim I can't judge

0celo7

I honestly don't know what coxeter did

maybe he proved the inverse function theorem, let's see

Semiclassical

the AMS had some essays when he died: ams.org/notices/200310/fea-coxeter.pdf

those aren't exclusively about his work, but they do talk about it

0celo7

ugh, that's weird geometry.

projective geometry and stuff

Semiclassical

the one by Peter McMullen in particular talks about his work on polytopes, and his central role in discovering their connection to group theory

0celo7

3:35 AM

not something I particularly care about, tbh

Semiclassical

okay?

0celo7

but "the greatest" is certainly wrong.

oh, also Weil

Semiclassical

I'm not a geometer, so I"m not qualified to judge

0celo7

but geometry is really broad

@Semiclassical who is the greatest physicist of the 20th

inb4 Einstein

Semiclassical

too late :/

hard to pick anyone other than him, really

0celo7

3:37 AM

Einstein was also the greatest geometer. Without him Riemannian geometry would not really be a thing

Semiclassical

ehh, i'm dubious about that

0celo7

and Riemannian geometry is what so much of geometry relies on for various reasons

Semiclassical

to what extend did he develop Riemannian geometry versus apply it?

0celo7

He popularized it

Semiclassical

my sense was that it was far more the latter than the former

0celo7

3:38 AM

it was pretty obscure before GR.

Semiclassical

that's a very specific interpretation of 'greatest'

0celo7

perhaps

O'''''''''''''''''''-

As far as I know Coxeter developed an area of geometry quite different from the other people you have listed. I think in that context it is fair to call him "the greatest".

Semiclassical

The word 'geometry' is problematic

O'''''''''''''''''''-

I guess it may be better to say "one of the greatest" as to not spawn one of these debates, which is super subjective anyway.

Semiclassical

3:42 AM

Woudl someone working on algebraic geometry call themself a geometer?

His work seems more in the realm of classical geometry, though his use of group theory in that context is definitely more modern

@0celo7 moving on to something where I think we're more likely to be in agreement: thoughts on the comment here?math.stackexchange.com/questions/552424/…

user227867

4:00 AM

The best is not to call someone an algebraist or analyst or topologist or geometer or logician or combinatorialist or number theorist but simply a mathematician.

2

0celo7

4:19 AM

@Semiclassical back

@Semiclassical Uh, not sure

what do they mean by rapidity

geometer always sounds so arcane to me

also not very likely to elicit respect from high schoolers :P

@Semiclassical Yeah, not sure what we're supposed to agree on. Rapidity means something different to me but I've studied a lot of relativity.

Semiclassical

4:37 AM

My point is that I've never heard of speed being referred to as rapidity, whereas I am familiar with the relativistic meaning