The h Bar

John Rennie

3:00 PM

I think the way you've seen Le Chatelier's principle worded is very confusing

user228700

Yeah :'-(

John Rennie

Phrases like oppose the stress are largely meaningless.

user228700

@JohnRennie Yeah, before I kill my brain, I'll kill those people who write our textbooks.

John Rennie

That's why if you did chemistry at university Le Chatelier's principle would never even be mentioned.

user228700

@JohnRennie I see :-(

John Rennie

3:02 PM

Any system always tries to reduce it's potential energy

That should be obvious.

user228700

@JohnRennie Yeah.

John Rennie

In this case reduce potential energy basically means reducing the Gibbs free energy

user228700

@JohnRennie OK. So that's why it has to be negative and all, for the process to be spontaneous. Alright...

John Rennie

You need to be careful about the sign of enthalpies and free energies in chemistry. But yes, a negative $\Delta G$ means the system is reducing its Gibbs free energy i.e. its potential energy.

user228700

Anyhoo, thanks loads sir. I'm ever so sorry for troubling u like this every day. I really hope u don't mind. I'm gonna go have my dinner now.

John Rennie

3:06 PM

I have to dash off now as well (to install a PC) so that's good timing. See you tomorrow.

user228700

Bye :-)

Balarka Sen

@Loong That happened to me once.

0celo7

3:26 PM

@JohnRennie what principle is that?

user218912

@0celo7 question 4, if I have the same lagrangian density term as yesterday but with 1 up 1 down i.e. $\partial_\alpha A^\alpha \partial_\beta A^\beta$ how does the metric look?

user218912

the same?

user218912

i did the same metric as the one with both lowered and i get the wrong answer 0.o

ACuriousMind

What do you mean, "with both lowered"?

Write out the formulae you're talking about instead of using those confusing words

user218912

$\partial_\alpha A_\beta\partial_\alpha A_\beta$

ACuriousMind

3:33 PM

@IceLord That doesn't make any sense.

If more than one index with the same name is in the same position, something is wrong.

Unless you are using a strange variant of the summation convention

user218912

so I know the metric for that term is
$\eta^{\alpha\epsilon}\eta^{\beta\delta}\partial_\alpha A_\beta\partial^\alpha A^\beta$

user218912

sorry i fixed it

user218912

my bad @ACuriousMind

0celo7

@ACuriousMind lel, applying for a job

user218912

@ACuriousMind that one is wrong too

0celo7

3:35 PM

Bavarian is a separate language from German

user218912

what i mean is

ACuriousMind

@0celo7 lol

@IceLord I know.

0celo7

@IceLord You are honestly terrible at index notation.

ACuriousMind

brb, need food

0celo7

I'm sorry, someone had to say it.

I don't even know how to help you.

user218912

3:36 PM

$\partial_\alpha A_\beta\partial^\alpha A^\beta$ and $\eta^{\alpha\epsilon}\eta^{\beta\delta}\partial_\alpha A_\beta\partial_\epsilon A_\delta$

0celo7

You do the same thing wrong every time.

user218912

is that right?

0celo7

Are those supposed to be equal?

user218912

yes

0celo7

I'm pretty sure I spoon-fed that to you last night.

So it's probably correct.

user218912

3:37 PM

now how about for this $\partial_\alpha A^\alpha \partial_\beta A^\beta$

user218912

how does that look like with all lowered using the metrics

user218912

i get the answer wrong so i assume my metrics/indices are wrong

0celo7

I'm not telling you.

This is basic undergraduate material

user218912

:(

0celo7

I don't care about the book, you're not learning when I tell you things.

user218912

3:41 PM

where can I learn it then?

Balarka Sen

"you're not learning when I tell you things" Then we should stop spoon-feeding you math stuff too.

4

0celo7

@BalarkaSen You're hardly spoon feeding me

user218912

in my defence I did figure out a lot on my own as well

Balarka Sen

I remember doing that, and you saying that you like to be spoon-fed.

0celo7

I do like being spoon fed

But you don't spoon feed me

vzn

3:47 PM

17 hours ago, by 0celo7

@ACuriousMind So these students sit in the back of the room, don't take notes, and always ask pedantic questions

user218912

I like to be spoonfed too. because I learn a lot from it. I don't use my thinking skills in the process but I do gain knowledge which I can use next time.

vzn

↑ ofc 0celo7 doesnt sit in the back of the chat room always asking pedantic questions...

Balarka Sen

@0celo7 Why talk against spoon-feeding when you like to be spoon-fed? Stick to your opinions.

vzn

@Balarka hi, are you still interested in number theory? heard about recent pythagorean triples breakthru?

0celo7

@BalarkaSen I like being spoon fed but I don't like it.

user218912

3:49 PM

@0celo7 o. k.

Balarka Sen

lol, that doesn't even make sense

@vzn Nope, what's that?

vzn

@0celo7 sounds like 0celo7 following schroedinger

0celo7

Balarka Sen

OK, we should stop dogpiling him.

vzn

in theory salon, Sep 16 at 16:21, by vzn

math celebration 2016: pythagorean triples empirical attack conquest / Tmachine blog

are you in college right now? india? whats your major?

Balarka Sen

3:51 PM

Nah

vzn

@BalarkaSen catfight! :P

0celo7

@IceLord I will tell you if what you write is correct

user218912

ok

user218912

firstly there are 2 metric tensors right?

user218912

is it

0celo7

3:56 PM

2 metric tensors?

vzn

@IceLord hi, studying physics? what country are you in?

0celo7

@vzn North side.

user218912

$\eta^{\alpha\epsilon}\eta^{\beta\delta} \partial_\alpha A_\epsilon \partial_\beta A_\delta$? @0celo7

vzn

@0celo7 ?

0celo7

@IceLord YES

user218912

3:57 PM

woo

user218912

::pats self on back::

user218912

@vzn 0celo7 answered, do you need clarification?

vzn

@IceLord "north side"?

0celo7

@vzn Standard term for Canadia.

cf. East Coast n**gga repping North Side* - The Weeknd.

Hmm, how to censor that word

Don't want to get banned

ACuriousMind

How about...not typing it? :P

vzn

3:59 PM

@0celo7 lol, have/ recently bought a weeknd song for rockband ps4. "I cant feel my face when Im with you" ... had to look up those lyrics to figure em out... amusing video too... come to think of it reminds me of chat room(s) :P

0celo7

@ACuriousMind The quote doesn't make sense w/o it.

@vzn Yes that song is pretty good.

ACuriousMind

@0celo7 If you can't find a quote with "North Side" without that word in it, that'd indicate it's not a standard term

0celo7

I like "Low Life" by Future & The Weeknd.

vzn

@0celo7 seen the video? check it out

0celo7

@ACuriousMind It's the first song that comes to mind.

@vzn I think I have, but I don't remember it.

@ACuriousMind It is in rap songs :P

vzn

4:01 PM

@0celo7 he sings for an audience with many varied reactions and at the end catches fire.

user218912

@0celo7 but when I take $\frac{\partial}{\partial(\partial_\mu A_\alpha)}$ of it I get 3 $\alpha$s.

user218912

what should I do?

ACuriousMind

@vzn 0celo7 might ask pedantic questions, but he certainly doesn't sit at the "back" of the room, usually.

0celo7

@ACuriousMind I do sit in the back of the room in probability

vzn

@ACuriousMind front, back, who can tell? think hes easily at least as annoying as those he criticizes, in much the same way... & big lol at him complaining about the disruption :P

0celo7

4:02 PM

I have a suspicion the prof smells.

In topology I sit in the middle

Analysis at the front

ACuriousMind

@IceLord Think about what type of index that index in the derivative is supposed to be - bound (i.e. summed) or free?

0celo7

QM at the front but off to the side

user218912

@ACuriousMind free right?

0celo7

engineering, in the back

programming, in the front because the TA whispers

E&M way in the back

ACuriousMind

@IceLord Yeah, so why would you choose $\alpha$ to denote it if $\alpha$ already appears in your formula?

0celo7

4:04 PM

@ACuriousMind I sit pretty much evenly distributed between the back and the front.

@vzn I'm not annoying, unlike you. And you're the disruption.

ACuriousMind

@0celo7 I meant that you don't sit in the back of the chat room, given your lively participation in the chat

0celo7

@ACuriousMind oh

vzn

@0celo7 hey disruption can be benefiicial aka disruptive technology :P

ACuriousMind

But nice to now know where to locate you in your classes, anyway :D

user218912

@ACuriousMind if I use $\omega$ for it I get $2\partial_\mu A_\omega$

user218912

4:06 PM

is that right?

0celo7

@IceLord $\omega$ is a bad index

user218912

okay it's the first thing that came to my mind

ACuriousMind

One usually just goes further down the alphabet, i.e. $\mu,\nu,\rho,\sigma,\tau$

Or $\alpha,\beta,\gamma,\delta$

0celo7

$\mu\nu\rho\sigma\tau\alpha\beta\gamma\delta$ after that give up

vzn

@0celo7 now reminds me of that internet character the annoying thing... will start mentally picturing you in that role, a perfect fit :P

user218912

4:08 PM

@0celo7 @ACuriousMind okay I see but am I right?

0celo7

@vzn the orange?

ACuriousMind

@IceLord With what? You have a rather bad habit of always just posting minimal information and scrolling up through all the unrelated conversation to find the original problem is rather infeasible

2

user218912

$2\partial_\mu A_\omega$ from $\frac{\partial}{\partial(\partial_\mu A_\omega)}$ of $\eta^{\alpha\epsilon}\eta^{\beta\delta} \partial_\alpha A_\epsilon \partial_\beta A_\delta$

user218912

@ACuriousMind

ACuriousMind

You can tell yourself that that cannot be right because the index positions of your result are wrong.

0celo7

4:13 PM

maybe he doesn't understand index notation...

user218912

@ACuriousMind D:

user218912

what is the right answer?

vzn

@0celo7 orange?

user218912

@0celo7 i'm figuring it out.

0celo7

@vzn the annoying orange

ACuriousMind

4:14 PM

Also, please try to write actual equations when possible. It took me a bit to realize that with "$X$ from $\partial$ of $Y$" you meant $\partial(Y) = X$.

0celo7

@IceLord I'm curious. What is there to figure out?

Index notation seemed so natural when I learned it

I never understood the "fear of indices" that's apparently a thing.

ACuriousMind

@IceLord Why are you asking for the right answer? Why do you not rather ask what the correct index positions would be?

user218912

@ACuriousMind same thing?

user218912

because the right answer includes the right index positios

4:16 PM

No, I'm not telling you the answer. I will ask you however whether the index in $\frac{\partial}{\partial x^\mu}$ is upper or lower.

0celo7

@ACuriousMind Good question

user218912

@ACuriousMind it's lower.

ACuriousMind

(the standard abbreviation for that operator tells you already, but it's necessary to realize the general principle at work here)

vzn

↑ apparently crazy frog is aka the annoying thing... luv that robot

0celo7

@IceLord Prove it.

It will do you good.

user218912

4:17 PM

that's just how its defined.

0celo7

Huh?

ACuriousMind

@IceLord Aha. First, why? Secind, so what position do $\alpha$ and $\omega$ have to have in your final result, regardless of whether they are on a $\partial$ or on a $A$?

0celo7

I hate physicists

user218912

@ACuriousMind top position for both?

user218912

it's because i used the wrong kronecker delta

ACuriousMind

4:18 PM

@0celo7 Don't go all formal, we're doing stuff as you initially learned it when doing GR here

user218912

the kronecker delta is supposed to have mixed indices

0celo7

@ACuriousMind When you go formal, things get confusing.

$\partial_\mu$ is a vector but has a lower index.

ACuriousMind

@IceLord Right. But first I want you to tell me why the index on $\frac{\partial}{\partial x^\mu}$ is lower

0celo7

Actually, in the current context I don't think your question makes sense @ACuriousMind .

You need to ask why the index on $\frac{\partial f}{\partial x^\mu}$ is lower.

this is actually very confusing @ACuriousMind

ACuriousMind

No, you just have to look how the thing transforms under a coordinate change.

user218912

4:22 PM

i knew that

user218912

brb btw class

0celo7

@ACuriousMind But saying that $\partial_\mu$ has a lower index makes one think that it's a covector $\partial$, but that's wrong.

In fact each $\partial_\mu$ is a vector.

It's $\partial_\mu f$ that is the covector.

ACuriousMind

What's probably confusing you is that $\partial_\mu$ is not a component of a covector, but a basis vector.

But I didn't ask whether it's a vector or covector, I asked whether the index is upper or lower

And that's not a confusing question

0celo7

Yes it is!

Upper and lower is defined on the components of tensors.

$\partial_\mu$ is not the set of components of a tensor.

The question does not make sense.

@ACuriousMind I'm not being overly pedantic, this confused me quite a bit when I first learned some geometry.

vzn

↑ think of the robot as like a room mod :P

0celo7

4:35 PM

@ACuriousMind I take the silence as implied disagreement and an invitation to a duel.

DanielSank

@ACuriousMind I've been called?

Slereah

Hey hey

DanielSank

@ACuriousMind That's not the point.

vzn

@DanielSank hi whats new did ya go on a trip?

@Slereah its a ghost! o_O

Slereah

Oh no

Best gauge it away

DanielSank

4:48 PM

The point is that your comment poked a particular (bad) logical node in the question, but failed to address the most important part of the question and the alleged duplicate is not a duplicate, but could the thought to be one by making the same "mistake" made in your comment.

Slereah

What are we talking about

vzn

@Slereah are you still working on that stats/ datamining job? (dont) wanna talk about it?

Slereah

nah

Let's talk about non-archimedean Hilbert spaces instead

Trying to find a Hilbert space with $^\rho \Bbb C$ as a field

vzn

if a vacancy opens up at the hilbert hotel, is it called a hilbert space? :P

0celo7

@Slereah I found a book on convex non-Archimedian analysis.

ACuriousMind

4:54 PM

@0celo7 Take the silence as me being otherwise occupied

DanielSank

@ACuriousMind How dare you?

Slereah

Found a nice paper on how to embed distributions in smooth functions on $^\rho \Bbb C $

0celo7

@ACuriousMind it better be a woman

DanielSank

5:11 PM

@vzn Columbus Ohio.

Emilio Pisanty

i am voting to close the question. — Lamichhane88 3 hours ago

↑ Poster's rep: 131

Obliv

@emilio he probably thinks that the VTC system is just saying "i'm voting to close this question" lol

0celo7

Hehe

John Rennie

@0celo7 Le Chatelier's principle. Victorian era empirically based law from a time when physicists didn't know any better. All mention of it should be removed from textbooks!

0celo7

Pretty sure I learned that in chemistry

Can't tell you what it is though

John Rennie

5:23 PM

@0celo7 you don't want to know. Trust me on this one :-)

Emilio Pisanty

@JohnRennie what's so bad about it?

It sounds pretty reasonable to me

John Rennie

Aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaarrrrrrrrrrrrrrrrghhhhhhhhhhhhhhhhhhhhhhhhhh

Emilio Pisanty

@JohnRennie This is $$\frac{|\text{me being contentious}⟩ + |\text{me being genuinely curious about why you hate it so much}⟩}{\sqrt{2}}$$

John Rennie

Le Chatelier's principle dates from a time when thermodynamics wasn't understood. These days we understand that any system will adopt the configuration with the lowest Gibbs free energy.

Emilio Pisanty

post-select your pick.

John Rennie

5:28 PM

And we put this on a quantitative basis because we know how to calculate Gibbs free energy (in principle at least).

Slereah

gib free energy plz

Emilio Pisanty

> (in principle at least)

famous last words

John Rennie

Le Chatelier's principle is just a vaguely worded statement of how the free energy is probably minimised, sometimes.

Emilio Pisanty

@JohnRennie but is it wrong?

i.e. does it fail?

John Rennie

It is unreliable. It sometimes works and sometimes doesn't. To be fair it works most of the time.

@Slereah I don't get it ...

Slereah

5:32 PM

and you never will

John Rennie

@Slereah German for give?

Slereah

English slang for give, actually

John Rennie

@Slereah you wacky youngsters

Slereah

I turned 30 last week

I am ooold

Obliv

@slereah you're so young wtf

I thought you were like late 30's tbh

Slereah

5:35 PM

Why, how old are you

Obliv

well I turned 19 last week but I know people in their early 40's who are more immature/fun than you

wait

that wasn't supposed to be an insult or anything.I just mean you aren't as old as you think

Slereah

Youuu whippersnapper

Obliv

D:

-1

Q: How life began?

How life began? First theory: Atoms duplicate themselves Second theory: Life came from below, archaea. Third Theory: Life came from Mars by a meteorite. What do you think? and why?

What do you think is meant by: life came from below?

user116211

@Obliv good question; let me downvote it first followed by vc....

0celo7

I'm 18...

user116211

5:46 PM

@0celo7 Dude, I'm elder than you ;))

user116211

@Sanya: done.

Sanya

@MAFIA36790 thanks

Sanya

6:23 PM

@MAFIA36790 hope I could answer some of your questions, but I'll be very happy if you ask me about the things that remain unclear

user116211

@Sanya reading

user116211

6:34 PM

@Sanya I'm not familiar with the Polar Decomposition Theorem; I have looked up the wikipedia but the contents are still not read by me. Apart from that, I get the other things; I'm accepting it.

Slereah

Oh I saw the pinned post

Elect me mod

user116211

@Slereah 0celo wants Lumo to be the mod.

user116211

He would ban this chat first.

Slereah

Among other things

user116211

@Slereah that's for sure.

user116211

6:39 PM

Well, speaking of Lumo; lemme sneak into his blog...

Sanya

@MAFIA36790 let's say F = \partial_j u_i, then we can decompose F according to F = VR with V being positive semidefinite and R being a rotation (orthogonal, but as det F > 0, it needs to be a rotation) and thus antisymmetric. If we linearise the product VR for small distortions, we get F = 1 + E + \omega (which we can calculate because the proof of the polar decomposition theorem is actually constructive) with E being the infinitesimal strain tensor coming from V

and \omega the infinitesimal rotation tensor coming from R. They are exactly the symmetric and skew-symmetric parts of the gradient tensor field of the distortion field.

user116211

You should use chatjax @sanya.

Sanya

sorry ... I should have used correct TeX, you're right

personally, I'm happy with the source tbh

user116211

Media downgrade Roger Penrose to an invisible appendix of crank Lee Smolin

user116211

Lumo's latest post ^^

user116211

6:44 PM

One can print this page too (why?).

Slereah

To read on the can

0celo7

@Slereah I need Steenrod.

I'm not kidding.

I will do whatever you want.

Slereah

0celo7

How much?

I have $

Slereah

I mean buy your own

You mooch

0celo7

6:52 PM

how?

Slereah

0celo7

That doesn't help

Slereah

amazon.com/Topology-Fibre-Bundles-PMS-14/dp/0691005486/…

0celo7

I don't know how to buy things.

@Slereah I don't have an Amazon account

Slereah

put money in the computer

0celo7

6:54 PM

Cash?

Sanya

a bookstore? @0celo7

0celo7

@Sanya The campus bookstore is too expensive

Sanya

then amazon it is - I'm happy to be living in a communist country where book prices are fixed

0celo7

I don't know how to get an Amazon account

How does the Internet work

Sanya

sometimes, you really drive me insane

2

0celo7

6:56 PM

I don't even know you...

Who starred that? Not nice.

You could get your account deleted

user116211

That's why we want Lumo to be the next mod.

0celo7

Lump would be a great mod.

user116211

@Sanya flagged as offensive...

bolbteppa

That Lumo article is so good: "Nor is Mr. Penrose’s critique of string theory as clearly argued as Lee Smolin’s 2006 “The Trouble With Physics.” In part because of the persuasiveness of Mr. Smolin’s arguments" oh man...

user116211

@Sanya You are from Cuba then.

user116211

7:09 PM

Speaking of fixed price, though my country is not run by the commies, we have Maximum Retail Price - the max. price anyone can sell the product at.

0celo7

We didn't we invade Cuba?

user116211

@0celo7 Those Castros....

user116211

JFK is infamous for the Bay of the Pigs invasion.

0celo7

That was a shit show

We should have annexed it.

bolbteppa

Is this a joke: $u(t,x) = 2a^2 \text{sech}^2[a(x - 4a^2t + d)] = 2 \frac{\partial ^2}{\partial x^2} \ln [ \text{cosh}(a(x - 4a^2 t + d))] = 2 \frac{\partial ^2}{\partial x^2} (1 + e^{2a(x - 4a^2 t + d)})$?

user116211

7:12 PM

There was that big Soviet guy too.

user116211

@bolbteppa complete the parenthesis...

user218912

my condensed matter prof is a little bit lazy so he assigns all the problem set problems from the book.

user218912

and the book has a solutions manual.

user218912

so...

user218912

I definitely know what I'm gonna be doing. :)

bolbteppa

7:15 PM

That describes a 1-soliton, what is a 2-soliton, or an n-soliton

Oh the first equality isn't that bad actually

0celo7

@bolbteppa no.

Look at Lp growth estimations on Riemannian manifolds

Slereah

@bolbteppa Backlund transform

0celo7

The equations are pretty terrifying.

I would try to understand it but I can't solve basic equations...

Slereah

well maybe learn math before learning solitons

user218912

@0celo7 yes you can.

user218912

7:18 PM

I can't though.

0celo7

I need to relearn high school algebra

Slereah

Solitons is basically all math

0celo7

What's the point of it all?

user218912

@0celo7 teach me high school algebra and it will come to you.

0celo7

Quantum cohomology bundles?

@IceLord what do they even learn there

I need to relearn logarithms

I never understood what they are

Slereah

7:20 PM

it is like a xylophone

Logs with rhythm

2

user218912

@0celo7 nothing.

bolbteppa

You can derive that $\text{sech}$ solution for KdV easily enough, then you're supposed to notice a countably infinite number of 'eigenspeeds', something to do with phaseshifts, but the $c$ in the solution indicates an uncountably infinite number of solutions to KdV :\ Then there's that weird derivative trick allowing you to get 2-solitons or n-solitons nicely

Backlund seems to come up later naturally

user218912

@0celo7 from morning's discussion was my answer right but with down indices instead of top?

user218912

or was it wrong wrong.

bolbteppa

omfg youtube.com/watch?v=i7ORX97drdg

No wonder people like solitons

user218912

7:26 PM

@ACuriousMind do you know?

bolbteppa

Oh my god, what the heck youtube.com/watch?v=X4tYrnbsWes

This stuff can actually be calculated with this nonsense, amazing

ACuriousMind

@IceLord Once again, I have no idea what you're talking about.

user218912

forget about it...

bolbteppa

From now on I'm modelling those classical mechanics car around a loop/ferris wheel problems as a 2-soliton problem

peterh

@JohnRennie You should be one of the next mods. It doesn't matter how long you wait before nomination, but please nominate yourself.

bolbteppa

7:38 PM

lie.math.brocku.ca/~sanco/solitons/kdv_solitons.php

dmckee

7:54 PM

I'd like to do a one or two hour familiarization lecture on general relativity for my modern physics class, but I don't have a really great resource.

We're talking about college juniors who still don't have the full panalpy of math skills nor the confidence to tackle hard problem without a lot of hand-holding, so I'm just looking for a collection of heuristic description of prominent effects rather than a proper treatment.

I have such a treatment of the gravitational red/blue shift and consequently of gravitational time dilation.

But I'm less happy with my existing resource on lensing, frame dragging, horizons and other subjects under the general heading.

Any one have ideas?

user116211

@dmckee Anthony Zee?

user116211

Well I have no idea.

user116211

But @dmckee, check this: einsteinandtheevidence.wordpress.com/recommended-references

dmckee

@MAFIA36790 I've seen 0celo7's blog. THat's not what I asked for.

user116211

@dmckee hmm.

dmckee

8:00 PM

Some math should probably be involved, but mostly words and pictures. We don't have time to dig into the math and these students are still boggling at special relativity.

0celo7

@IceLord how the hell am I supposed to remember that

ACuriousMind

^my sentiment precisely

vzn

@bolbteppa did someone say solitons? =D

ACuriousMind

8:33 PM

@DanielSank Yes. That's something I already conceded: I misread the question, and VTC as unclear. OP left a comment that clarified what the question was, and when I read that, I edited the question to be clearer (at least to me), and reopened it. What I don't get is what overarching pattern you are apparently seeing here or why that meta post was necessary and what the aim of that meta post is.

I agree I made a mistake, but you're apparently seeing some systemic fault here and I can't really understand what that is.

0celo7

@ACuriousMind Do the individual Betti numbers mean anything?

ACuriousMind

What do you mean by "mean"?

They are homotopy invariants

And since they are the dimensions of the homologies, they count the number of independent k-dimensional "holes" in the space, if you accept the premise that homology detects "holes"

0celo7

@ACuriousMind I don't. What's a "hole"

ACuriousMind

It doesn't have a rigorous definition

But in all reasonable examples of homologically non-trivial spaces, you see that there's usually some part of the space "missing", which, if it were there, would allow you to contract the homologically non-trivial cycle.

Like the punctured plane - the missing point prevents a curve around the origin to be a boundary, so the first Betti number being 1 counts exactly that point. This extends to a plane with n distinct punctures having Betti number n. It also works in higher dimensions, but unsurprisingly our capability to understand that ends at 2D holes in 3D :P

0celo7

8:55 PM

@ACuriousMind Ok, one of the first triumphs of geometric analysis was proving that if the Ricci curvature is nonzero, the first Betti number is at most $\mathrm{dim}(M)$. I'm trying to figure out why anyone gives a shit

ACuriousMind

@0celo7 Because it's a homotopy invariant, and bounding those for things with a given property effectively means bounding the "number" of things with that property. Furthermore, it can be computed purely topological, but this means that you might be able to prove Ricci-flatness purely topologically: By showing the Betti number is greater than the dimension.

0celo7

No, that implies the curvature is always negative somewhere, not that it's flat

ACuriousMind

Oops, misread what you said, of course

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