12:00 AM
Right after the proof of the lemma

Well, it's just continuity. Determinant can't change sign because it can't go through 0.

@TedShifrin I know
But I'm asking if there is a more elementary proof

You can use the volume form, but I don't think that's necessary.

Anyway goodnight @Ted

Because do Carmo does not suppose one is familiar with differential forms.

12:01 AM
Gnight, @Ali. Be well!
Yes, one reason I don't like doCarmo — I love differential forms.

@Ted: Sorry, are you blaming the global political scene right now on the youth?

NOOOO, @MikeM. Au contraire. I'm blaming the parents for everything, not the youth.
Hang on, 0celo. I need to see how doCarmo discusses orientability.
@0celo: The same proof that you have an isometry should apply to determinant.

Ok. I was going to suggest you look at the age distribution of Brexit and US primary/general election. But I got you.

I'm not a dummy, Mike. At least I don't think I am.

@TedShifrin Hmm, I'm not sure what you mean, my isometry proof is that $\langle V,W\rangle$ is constant when $V,W$ are parallel
which follows from the defining equation of a metric connection

12:16 AM
Agreed.
What happens if you differentiate $\det(V_1,\dots,V_n)$ where the $V_i$ are parallel?

what is that determinant?

Oh, good point.
No, it's ok.

@TedShifrin I'm still not sure what that determinant is...
determinant of the components?

It makes sense to talk about the determinant of $n$ vectors in an $n$-dimensional vector space. You just pick any old basis ...

What's that mean? You can take the determinant of n vectors. There's only a slight problem getting a number as output of your manifold is not oriented.

12:22 AM
But it is oriented.

@TedShifrin but do Carmo talks about orientability in terms of the chart transitions

The question was to show that parallel translation is orientation-preserving.
You'll do arguments like this in G&P, 0celo. It's sort of a connectedness argument. It's locally fine, hence globally fine by connectedness.

Gotcha gotcha.

@TedShifrin What I don't get is the connection between your determinant and orientability

That's my favorite way to introduce special holonomy... a baby case of why restricting holonomy adds structure.
@0celo7 A positive oriented transformation is one with positive determinant.

12:28 AM
@MikeMiller I'm confused because do Carmo explains orientation in terms of chart transition functions, not in terms of orientation of the tangent spaces.
So without going through all of the proofs in Lee that connect the two, I don't see how one is supposed to know that "parallel transport preserves orientation" even makes sense.

do Carmo's definition immediately orients the tangent spaces, though.
Given a chart let $\{\partial/\partial x_i\}$ be an oriented basis. The transition function assumption says this is well-defined.

@MikeMiller Ah, because another basis $\{\partial/\partial y_i\}$ will be related by the transition function (chain rule) and this is orientation preserving in the vector space sense?

Right
I've never read do Carmo so I don't know if this is ever said explicitly.

I don't think so
Ok, so back to what Ted was saying...
@TedShifrin clearly something magical...
@TedShifrin It's probably zero
@TedShifrin IIRC that determinant will be a polynomial in $\langle V_i,V_j\rangle$

12:44 AM
I hope it's not zero! That would say parallel transport took a linearly independent set of vectors and made it linearly dependent.

No, the derivative is zero
Well, can't you just do the rest of my proof at this point
Simply argue it can never be zero so by continuity it has to stay positive
so the determinant of the parallel transport operator is positive

yo this cheddar does not smell right

So why does this not work to put an orientation on every manifold?

Why does what not work?
@MikeMiller Are you asking why we can't use parallel transport to orient manifolds?

12:52 AM
Pick an orientation on one tangent space and parallel transport it to every other tangent space. If that's well-defined, it gives you a consistent orientation on every tangent space. It seems you've proved that parallel transport preserves orientation.
Yeah.

Yes, that's a good question.
And I'm not sure of the error.

No error in what happened above, just a subtlety.

Which is?

You wouldn't rather look for it?

No, not really. I have a correct proof of my own already

12:56 AM
So why doesn't your proof orient jon-oeientable manifolds?

Because my proof uses the existence of a volume form.

Good, good. Ours here used well-defined mess of determinant.

Yes, I figured from an earlier comment of yours.
But I'm not sure why it's ill-defined on a nonorientable manifold.

Note that determinant is just the wedge product of n vectors, followed by the determinant isomorphism $\Lambda^n TM \to \Bbb R$.

I'm not sure you're supposed to know what a wedge product is when reading this book.

12:59 AM
I'm not talking to the generic reader of the book, I'm having s conversation with you. So I'm not worried about that.

Well I'm looking for a proof that the generic reader would understand...but go on

Take determinant with respect to one basis. Change bases (by a matrix A, say). Then the new determinant is the original times det A. Or maybe the inverse of that, I forget.
In particular if you can't say "Take determinant with respect to an orthonnormal basis" unless you can throw the adjective oriented in there - your determinant is not well-defined.

@MikeMiller Ah, I see.
But can we go back a bit -- I know I said the determinant is constant -- is that true?

1:22 AM
Yeah, it's always 1:

1:58 AM
@MikeM, @0celo, determinant is invariant under change of basis, because of $B=P^{-1}AP$ ...
@0celo7 You're thinking of the Gram determinant.

2:41 AM
@MikeMiller In a connected space, can we have disjoint open sets $U,V$ such that $\bar U$ and $V$ are not disjoint?

Am I missing something? It seems to me that $A\cap\left\{ \left(p-r,p+r\right)-\left\{ p \right\} \right\} \neq\emptyset$ is exactly what needs to be shown. And nothing more.

@RudytheReindeer Is that @ me?

It's to anyone here in the room. Not you in particular, don't worry about it.

@RudytheReindeer Well if you want to answer my question @ Mike, you're welcome to do so :)

Good question, let's see...
How about the finite complement topology on $\mathbb R$?

2:54 AM
Oh god what's that

No, not possible to find disjoint open subsets.
Don't worry I'm just thinking aloud.

@RudytheReindeer Basically, I'm trying to show that normality implies: given closed $A$, open $U\supset A$, then there is a closed $B$ such that $A\subset B\subset U$
And that question came up in my construction of $B$
I think I found a construction that does not depend on that, but my first try seemingly did

$$T = \{ \varnothing, \{a\}, \{b\}, \{a,b\}, \{b,c\}, \{a,b,c\} =X\}$$ with $U = \{a\}$ and $V=\{b,c\}$. Then the closure of $U$ is $\{b,c\}$ so they are not disjoint.
oops, I didn't check if this space is connected.... let's see.
It's not.
Sorry. We have to try again.

@Ted I'm confused. Why isn't $\Lambda^n TM$ trivial, then?

@0celo7 I'm starting to think that maybe the answer is no.
Oh, hello there mean square.
@0celo7 What prevents you from taking $A=B$?
Do you use $\subset$ to mean $\subsetneq$?

3:04 AM
@RudytheReindeer Yes, that's implied, sorry.

(I don't have a quibble with that, just need to be clear) Oh, okay.

And I also need $A\subsetneq B^\circ$ for my actual proof to work, but that might be immediate.
well it can clearly ever be equal because one is open
and the other is not
but anyway...
@RudytheReindeer Do you know a proof of that?
of the $A\subset B\subset U$ thing
It's definitely true in metric spaces

@0celo7 You implicitly require $U \cup V$ to be the whole space? That's impossible in a connected space. But if you don't require that you could to something like $U=[-1,0)$ and $V=({1\over 2}, 1]$ in $[-1,1]$ with the standard topology.

@RudytheReindeer No, I don't. I just said connected because the space I'm working with is connected.
And I didn't want some weird uncountable number of components disconnected thing :P

Ah not disjoint.
Sorry again ^.^;

3:07 AM
lol

@0celo7 Okay, I think the answer is no. $V$ is open so $V^c$ is a closed set containing $U$. Hence $\overline{U}$ is contained in $V^c$.

@RudytheReindeer Aha!
I'm dumb, I used that same method to construct my $B$, I think.
Thanks

@MikeM: Where do we get a nowhere-0 section in general?

3:24 AM
@Ted: The determinant defines a homomorphism $\Lambda^n TM \to \Bbb R$, if it's well-defined. A nowhere zero section would be the inverse image of 1, aka "the unique element with determinant 1."
Consider the silly case of the the standard basis in $\Bbb R^n$. Change basis to the same with first vector $e_1/2$. I think the determinant of the old atandard basis in this new basis is 2.
In fact if $A$ is the new basis and $V$ is the old vectors, I claim the new basis is $A^{-1}V$ or $VA^{-1}$ or something like that.

3:46 AM
We only get a real number if we normalize ... Thereby trivializing the bundle. Skew multilineairity isn't quite enough. @MikeM

How is that different than what I said?

@TedShifrin @MikeMiller I'm not sure what the argument is.

4:02 AM
math.stackexchange.com/questions/1839860/… Could you please review this question.
if you fimd ot to be clear I would please request that you vote to reopen it.

@TedShifrin Hey Ted!

Heya @Stan

How are you amigo? :)

But in @0celo's case, we have an orientable manifold, and he's got a parallel frame along the curve. So we just can use that frame field to normalize. I guess I didn't make that clear.
Doing OK, @Stan, and how're you?

Life is fantastic. Just keeps getting better and better. But my music needs work.

4:16 AM

One of my good friends has been teaching me theory
and pointing out how I don't really know any actual music theory, so I'm studying it hard. I'm sorta alternating between doing textbook exercises
and writing music to experiment
What's new with you?
You still tutoring?
Hahhahahaa
my sister is a teaching assistant
and she says she's tired of it already
can't handle 3-5 year olds all day

No, I'm on summer break.

That's nice :)
What did you think of Brexit?

I think the world's a mess.

Me too :/ But then again, it's been worse.
Democracy seems to have some issues though.
I'm not sure votes mean much
if everyone making them is poorly informed
myself included

4:21 AM
Yup, not sure it's so much better for us to be a democratic republic.

I'm just not sure how the system can be improved.

Just put Putin in charge of everything?

LOL
hahahahaha

@TedShifrin I agree, I was just fighting about you saying that determinant is basis-invariant because you told me I was wrong :)

@TedShifrin I'm sure that's how he sees it.

4:23 AM
Well, determinant is basis invariant for a linear map. I fibbed lied, yes.

Now that I agree with!
Hi @Sangchul.

I finally learned from @Balarka how to cross out :P

Hi everyone :)

How's summer treating you? Are you teaching?

I volunteered to give a talk to a bunch of REU kids in a few weeks (at CSUSB), @MikeM. Now I'm gonna panic thinking about preparing such a talk.

4:25 AM

Are you and @SangchulLee neighbors?
I think I'm gonna talk about connections in mathematics, @MikeM (fields crossing over, etc.).

We are, down in the dungeon. (Unless offices changed.)

That sounds cool!
I'd enjoy a talk like that.

Some of my favorite questions that have arisen going back to when I was a young'un.

What about connections coming from affine connections?

4:27 AM
@TedShifrin Back when people actually said "young'un" hahaha

Including the 4-skew line question, probably something about Morse theory, combinatorics and computing how many homogeneous polynomials of degree $k$ in $n$ variables there are, etc.
No, not those, Mike.
glares @Stan

Yes, that dungeon... :)

hides
lol

I didn't realize you all had a dungeon.

There's a mathchat dungeon?

4:28 AM
We're on the second floor. There are no windows. There are stocks.

Stocks are crashing, though.

Down on us?

Thanks to Brexit.

@MikeMiller that was my first thought

Well, that's what efficient markets are supposed to do.

4:30 AM
Yes, yes. I do think there's lots to say about connections. At least that's the easiest route between representation theory and differential geometry I know. :)

@0celo7 No problem, glad I could help!

@0celo7 yoooo watchu listening to these days?

@StanShunpike I'm feeling YG's new album, Riff Raff, TGOD Mafia
Oh lord DJ Khaled just dropped a single, brb

lmao his T-Mobile commercial is absolutely unintelligible

Dj Khaled is a living god meme
It's incredible

4:36 AM
scroll down a bit
and the video is there

Hmm, it's not very good.
(the single)

I don't get what his role is exactly
Is he literally just a marketer?

He's the producer

But like, does he have musical talent?
I can't tell.

No one knows lol

4:39 AM
yeah exactly

He's rich tho, must be doing something right

True that
I'm just trying to figure out what
lol

and he has guest producers a lot

yeah that's part of the marketing
integrating different people promotes his brand
@TedShifrin Do you know DJ Khaled?

London on tha Track, Mike Will, etc.
lol

4:40 AM
What's his best song in your opinion?

They Don't Love You No More, probably.
Although All I Do Is Win is a true classic.
@StanShunpike literally what

You said your sister is a TA?

Yeah, I like All I Do Is Win too

@StanShunpike Hmm, Khaled did produce Bugatti.

@TheGreatDuck to 3-5 year olds hahahahaha

4:43 AM
What is she teaching, math?

One of my friends aptly said
"That's not teaching, that's supervising"
She's just there to like help the classroom function

oh my god the Major Key cover :'D
can't wait for it

@StanShunpike here is an interesting game they could play
Count by a number that changes as you count
;)

@StanShunpike Snoop Dogg dropping an album in a few days

@TheGreatDuck example?
@0celo7 How long has that been in the works?
I can't remember his last dope one
@0celo7 dude that cover.....hahhahahahaha
there's just a fucking lion

4:46 AM
Count by a number that increases once every time you count

and some flowers

Hmm actually it's on Apple Music already

in terms we know: \sum x

@StanShunpike But yes, YG's new album is flames.

Children count by 2s or 3s to do things differently

4:48 AM
Except for 1 song which I deleted

but why not count by x? After all, it cannot be that hard to do for simple patterns

@TheGreatDuck oh that's cool. So 1, (1+2), 1 + 2 + 3.....so that would be 1,3,6,10,etc
like that?
@0celo7 what's the best song from the album?

Exactly!

@TheGreatDuck that's a really cool idea. i will suggest it :)

:)
I figured it couldnt hurt
when you really think anout it
simple series are not high level concepts

4:51 AM
@StanShunpike Why You Always Hatin'

multiplication is just repeated addition of fixed values
implying an unfixed value shouldnt be that complex
(Other than notation i suppose)

@StanShunpike And I'm still deciphering 2 Chainz's 2016 stuff

@TheGreatDuck that's awesome. I always find it fascinating to think about whats the best way to educate kids about math

I want to go back and listen to mid-2000 Jay Z, I've never really listened to it.
But I haven't been in the mood for it
And it's like 5 albums, so it's not a quick affair

@StanShunpike the best way I belive would be to explain all of what I refer to as the "atomic operations" as soon as possible

4:53 AM
@0celo7 I remember taking a music class this past quarter. everyone in the class kept saying "oh mozart is my fav, oh beethoven is my fav." and finally, one of my homies in the back was just like "yeah, i listen to 2 Chainz"

that is to say

@StanShunpike 2 Chainz is one of my top 5 rappers
He just doesn't care lol

addition, multiplication, power raising, exponentiation, logarithm, summation, large product operator, and floor.
Everything else is pretty much defined in terms of them
at least at my level (calculus 4)

the heck is calculus 4

Differential equations

4:55 AM
@TheGreatDuck Yeah! I felt amazed when I started studying abstract algebra. I felt like those kinds of basic operations should be explained more deeply early on
@0celo7 who are ur top 5?

It is calculus semester 4

@StanShunpike 1) Rick Ross. 2)-5) Lil Wayne, 2 Chainz, Kanye, Ludacris.
Last 4 in no particular order.
Maybe Jay Z over Kanye...
Every time I listen to Watch the Throne I move one up to my top 5 :)

Ouch, Eminem didn't get a mention

I have all of his albums

My top 6 musicians? I have no idea.

4:57 AM
But a top 5 is not a lot of dudes :P

whoever they have making video game music i gues
:p

@TheGreatDuck Pokemon music has been straight fire lately.

I havent played the recent games
ever play undertale?

@StanShunpike Em, Meek Mill, Jay Z, Future are good

I never actually played it
i just watched a long play on youtube

4:59 AM
I don't get Kendrick Lamar

@0celo7 He's been suuuuuuuuuuuuuuuuuuuuper successful tho
what don't u like about him?

I like his hits like King Kunta and Swimming Pools

That music is epic

@StanShunpike I don't dislike him.
I just don't get his lyrics.
And I can't sit through one of his albums
I have all of them...
Apple Music is amazing
@StanShunpike The Eminem Show is damn good.

IT IS. i like that one a lot

5:02 AM
Till I Collapse is probably a top 5 song.

Stan: pythagorean relation is easy to show too

Hmm, maybe Em does need to be on that list...

ever see the visual proof?

@TheGreatDuck Hasn't everyone?
It shows up on reddit every now and then

@0celo7 i did not until wikipedia one day. It just appeared one day in a book and I accepted it as true.

5:04 AM
@StanShunpike Throwing out 5 of my favorite songs: Till I Collapse (Em), Renegade (Jay Z, Em), John (Wayne, Ross), Birthday Song (2 Chainz, Kanye), Black Skinhead (Kanye)

@StanShunpike I forgot one atomic operation: the limit

what's an atomic operation?

@0celo7 they are not a defined term. I just accept them as a set of operations upon real numbers that define all other operations.

Uhhh, limits are totally defined...

@0celo7 i said "atomic operation" is not a defined term.
Of course limit is defined

5:06 AM
Then I don't know what you're doing

...

@0celo7 Till I collapse is classic
he's so classic em in that song
angry aggressive
in ur face

@StanShunpike No Love (Em, Wayne), My Way (Fetty Wap), Ridin (Chamillionaire), I Mean It (G Eazy), Heart of the City (Jay Z)
Actually, all of the Blueprint 1 is amazing

@0celo7 I was telling @StanShunpike that my approach towards teaching would be to teach as many of the atomic operations as possible. And then I rattled off a bunch of different operations. The idea is that while some of the, might define each other, they can be used to define all other functions... Or at least, all functions that everyday ordinary people learn.

@TheGreatDuck I got what you meant. I think that's exactly the right philosophy.

5:10 AM
@StanShunpike Oh of course, one cannot forget, Move by Luda

@0celo7 No love is dope

Well, Move female dog
This chat is rated G

@StanShunpike thanks. I admit they might not be "atomic" per se, but it is probably the best mentality for a simple analysis. If one needs to add another one it could be modulo which is the remainder when a is divided by b.

@StanShunpike If you want to feel like a thug, listen to Box Chevy.

@TheGreatDuck Building in an understanding of basic operations is crucial to learning how to think mathematically
@TheGreatDuck Sometimes when i've been tutoring, I get the impression the people I am teaching simply lack a basic sense of these operations and as a result their equation solving becomes mechanical and they don't really realize these operations are....exactly that operations. it is just sort of a mindless solving process they go through

5:13 AM
Well I have to sleep, cheerio.

@StanShunpike exactly! Plus, if one can express all other operations in terms of lesser "atom-like" operations, then the other operations become meaningless. To use a computer science term... They are "sugar syntax".

@TheGreatDuck I sometimes feel like CS and Math should be taught side by side from an early age
CS, math, and stat
all useful and important to being a scientist

@StanShunpike it is sometimes annoying when my fellow classmates in calc one continually tried to cancel the word cosine in fractional functions.

@0celo7 later
@TheGreatDuck hahahahaha that's the WORST lol
truly not understanding mathematics

@StanShunpike i hate statistics and it relies on integration heavily, but cs definitely needs to be taught more. It's concepts are useful beyond programming.
I never even considered that statements in math returned true or false until programming
And even then it wasnt until i actually posted a q&a question implementing an equality conditional
someone mentioned the iverson bracket "sugar syntax".
needless to say, it was quite... Intriguing.
The tv show i am watching right now is so weird
It is an episode of star trek where they land on a modern day rome...
O.o
i feel like its the beginning of a bad bar joke
"three gladiators and captain kirk walk into a bar..."
XD