12:39 AM
@user681391 Right. Such a thing is known as a symplectic form; a closed 2-form $\omega$ with $\omega^{\wedge n}$ being a nonzero volume form.

howdy, a @Balarka

Hi @Ted!
@user681391 I like the following interpretation of this fact: If $X$ and $Y$ are two arbitrary vector fields, then flowing along $X$ for time $\sqrt{t}$, then along $Y$ for time $\sqrt{t}$, then along $X$ again for time $-\sqrt{t}$ (backwards) and then along $Y$ again for time $-\sqrt{t}$ (backwards), you get a "square of flowlines" which doesn't close up - in fact, what closes it up is the flowline of $[X, Y]$ for time $t$.
Think of "evaluating $\omega$ on each side of this square" - that is (roughly speaking) what $X\omega(Y) - Y\omega(X) - \omega([X, Y])$ is, a signed sum of those.
So $d\omega$ evaluated on the (truncated) square spanned by $X$ and $Y$ is $\omega$ evaluated on the boundary of the (truncated) square.
This is an infinitisimal Stokes theorem

I'm not sure I've worked out the details of that, @Balarka. I'm not sure I see it ...
I mean, totally intuitively, yeah.
Not sure how the $\sqrt t$ stuff works out versus $t$ ...

I don't have anything beyond intuition to offer, I haven't worked the details out either. (Let me draw a picture elsewhere though)
@TedShifrin Oh that's because $d/dt (\Phi^X_{\sqrt{t}} \circ \Phi^Y_{\sqrt{t}} \circ \Phi^X_{-\sqrt{t}} \circ \Phi^Y_{-\sqrt{t}}) = [X, Y]$

@BalarkaSen fave diff. form?

12:52 AM
I understand that, of course, @Balarka. I don't see how it works out in this particular computation.

Ah OK

I guess I usually do it with $t$ and $t^2$, but no big deal.

@anakhro: My favorite, of course, is $d\omega_{12} = -K\,\omega_1\wedge\omega_2$. :P

Oh that's a good one

12:53 AM
So, if you force me to write one differential form, it'll be the difference :P

What is the notation $\omega_{12}$?

The linking form is just the pullback by the chord map of the area $2$-form on the sphere :)
Connection form on a surface.
It measures how fast $e_1$ twists toward $e_2$.

$\nabla e_1 \cdot e_2$, rate of change of twisting of $e_1$ in the $e_2$ direction

@Balarka: Do I even bother responding to this?

Heh probably not

12:57 AM
There seem to be an inordinate number of people trying to do grad diff geo without knowing super simple basic things ... I told one person doing big doCarmo to start by learning basics about manifolds first.
I think I was more patient 5 years ago.

$\cos(r)\,dz + r\,\sin(r)\,d\theta$ is kind of cute on $\mathbb R^3$.
@TedShifrin count me as one of those inordinate number of people.

I mean I suppose I myself have been a victim of learning too fast

ya balarka doesn't even know how to count to 3

I did put something on it, Balarka, not too rude.

Good comment

1:00 AM
Yuck, @anakhro. Why should I like that $1$-form? It is well-defined, I grant you, but ...

big do carmo
isn't it thinner than the baby one actually

yes, @Ryan, way thinner.

@BalarkaSen @TedShifrin hi

hiya

hey all

1:00 AM
oh wow, it's a JoeShmo.

why wow?

It's been a while!

how're things!!

Hi @Ryan!

Still alive and mostly kicking. How're you?
Did you ever tell us what you're doing next year?

1:02 AM
im still trying to wrap up masters. im doing it part-time. thinking about Phd. probably going to need advice

Joe Shmo

Oh, somehow I thought you'd done some apps elsewhere.

Do we know each other

Didn't you tell me you'd visited some school(s) on the west coast?

@user681391 Picture. Difference of $\omega$ evaluated on the two flowlines of $X$ should be thought of as $Y\omega(X)$, difference of $\omega$ evaluated on the two flowlines of $Y$ should be thought of as $X \omega(Y)$ and $\omega$ evaluated on the truncation edge is $\omega([X, Y])$. Modulo signs, if you add them up, you get the formula for $d\omega(X, Y)$
Of course, difference really means difference quotient

1:04 AM
Or maybe my brain truly is decaying.
Oh no. It's a MikeMiller.

Hi @Mike!

heya @RyanUnger

i haven't been applying, although since ive been hanging out here long enough, i see how you would think that im a doctroate student at this point

@MikeMiller wtf lol

1:05 AM
Didn't you visit the west coast? Or am I addled?
That's what I was thinking, Eric.

i did indeed visit the west coast. i visited family, and drove down by route 1
i am thinking now about applying to ucsb and caltech, though

Ohhh, got it. Somehow I thought you were visiting schools.
Weird pair to choose.

well i wouldn't be going to both
jk

Duh.
CalTech may be a bit too small/elitist.

1:07 AM
Lu Wang is going to CalTech

I should say it means $\Phi^X_{\sqrt{t}} \circ \Phi^Y_{\sqrt{t}} \circ \Phi^X_{-\sqrt{t}} \circ \Phi^Y_{-\sqrt{t}} = \Phi^{[X, Y]}_t$ upto first order

how come? i mean i guess i see the elitist part of it
and yeah the department seemed on the smaller end from browsing online

There are second order badness but I was giving the infinitisimal picture so it doesn't quite matter

I haven't seen the famed Princeton elitism yet

Huge schools for grad school can be a problem, but tiny schools can be a different sort of problem.

1:08 AM
ryan are you at princeton?

It's too hard for me to do these formulas in my head.

yus
here for a summer school atm

woah! congrats

@MikeMiller I have a better question for you

thanks

1:08 AM
It's a local computation, of course, that your two descriptions agree
So you can do everything in euclidean space

@RyanUnger I didnâ€™t either when I visited

That's true.

I figured you would have done/assigned/graded that standard exercise by now, @MikeM :P
@Ryan: I have little patience for people like that.

deleted because it can identify easily

I got into Princeton, among other places, and was much happier to go to Berkeley.

1:10 AM
I don't remember these things if I did

the issue for me is that i dont have any research under my belt, since i don't go to school full-time and i dont have time for research

Tbh I wouldâ€™ve gone elsewhere had my circumstances been like barely different

@TedShifrin you got into a doctorate program in princeton?

No way I wanted the pressure cooker of needing to finish in 3 years. (I realize Princeton has mollified that slightly.)
Yes, @JoeShmo.

Princeton is the best place in the world for general relativity. But Berkeley has people too
His interests are very specific

1:11 AM
The people I knew at Berkeley who knew/did GR are long gone.

i dont know whats more impressive. the fact that you got in, or the fact that you turned it down

@TedShifrin were you there during the RK Sachs days

Better question: Suppose I take the space of $n$ distinct points in $\Bbb R^m$, which is a subspace of $(\Bbb R^m)^n$ (the diagonals thrown out). $S_n$ acts on this by permutation, and let the quotient be $\mathcal{M}_n^m$, the configuration space of $n$ points in $\Bbb R^m$.

I see ... that's a fair issue, @Ryan. But being successful/slightly happy in grad school is far more complicated a question.
yes, @Ryan.

That's not a question

1:12 AM
his book is a little nutty haha

And I got to work with Chern and Griffiths, so I won big time.

$\mathcal{M}_n^\infty$ is a model for $BS_n$. Apparently there should be a natural map $\mathcal{M}_n^\infty \to \Omega^n S^n$ but can't quite figure out what it is.

he gives the penrose and hawking singularity theorems as an exercise

@MikeMiller There you go

With hints? :)

1:13 AM
the hint is to read hawking's book

LOL, fair. :P
So he obviously doesn't intend a reader to do the exercise at that point.

hah, no

@Balarka It suffices to construct a map $\Sigma^n \mathcal M_n^\infty \to S^n$, at least.

I knew him, not his book.

it's called "general relativity for mathematicians"
it's like GTM 5 or something crazy

1:14 AM
@JoeShmo: Harvard turned me down for college and for grad school, but ultimately offered me a job, which I turned down :P ... I had already committed to a postdoc at MIT.
right, @Ryan, I remember. With H. Wu.

yeah

@TedShifrin you are an embarrassment of riches

Wu was a fantastic expositor/teacher, and a damn good geometer.

they have this whole formal theory for observers that I haven't seen in any newer books

Well, @JoeShmo, I never amounted to that much ... so it's OK.

1:15 AM
oof I would not call the book good exposition

Well, I honestly don't know who brought what to the book.

they do everthing without coordinates which for GR really is a mess
@TedShifrin what do you make of the stories that Berkeley purposefully fails students on the quals

Yeah, I don't like physicists' love affair with indices and total lack of intrinsic meaning, but I like computations ... I far prefer Cartan's moving frames to all the $\nabla$ stuff.

@MikeMiller Right. I was wondering if it's like the scanning map but got confused in trying to write it down.

I am not up on the modern stories. When I was there, we had oral qualifying exams on 1st year stuff, which they got rid of because so many students froze. Then they switched to the prelim + advanced oral. I honestly know nothing about how that has gone.

1:18 AM
Ok maybe this is a stupid question but GR has actually very little to do with forms for the most part. The objects involved are symmetric tensors. How would moving frames make anything easier?

I can't see anything yet

Differential forms do show up at some point sure

When I spoke to people there during open houses I honestly didnâ€™t hear anything bad about the quals

That doesn't stop you from working with orthonormal coframes ... you still use (non-alternating) tensors in that framework (pun intended).

people complained a lot about how much teaching there was tho

1:19 AM
I'm just saying the structure equations to compute curvature make everything else look horrible.
Eric ... all the CA schools are completely overrun with undergrads. UCSD has done something like triple the math majors/courses in the last few years. It's insane. No more faculty, either.
I think Mike escaped most of the undergraduate service teaching insanity at UCLA, but endured some.

Anyone here at the moment think about randomly orthogonal bases?

Why "ly"? Only once in a while are they orthogonal?

@TedShifrin it's kernel is the standard overtwisted contact structure.

No argument from me, @Eric. I chose not to volunteer to be an adjunct prof at UCSD. I'm pretty sure they would happily have hired me.

1:22 AM
@TedShifrin Oh sure. When actually computing the Riemann tensor, GR chads use frames

I guess that makes it less special than $\ker(dz - y\,dx)$.
But I like that it makes a pizza.

@anakhro: Oh, really? With the $\sin(r)$ term in there? I had no idea.

It's uh swirly isn't it

They're as independent as trig functions. You're tight, @TedS, "random" says it all.

U gotta love to see the hollowing out of higher education

1:23 AM

I've actually never seen that before, @anakhro. Interesting.

@TedShifrin as opposed to things like $\ker(dz + r^2\,d\theta)$, it twists a lot.

Yeah, I get that.

@TedShifrin there's a GR book by Norbert Straumann (a physicist) that uses moving frames about 50% of the time

Interesting, @Ryan. Most geometers and physicists seem sadly afraid of differential forms and moving frames.

1:24 AM
And I guess you know the tight/overtwisted dichotomy in contact geometry but I will say it anyways: the overtwisted and standard structures are not contactomorphic.

@BalarkaSen I don't see it, sorry.

@anakhro: I know very little contact geometry, in fact.

That is, no diffeomorphism which pulls back one contact form to the other.

I really was a complex geometer ...

@TedShifrin moving frames are the path to sign and p! errors

1:25 AM
Simple person, complex geometer.

@TedShifrin it's pretty cute. Did you do anything with Kahler manifolds?

@TedShifrin I like geometry and Iâ€™m afraid of everything

My research was stuff in $\Bbb CP^n$, so sure.
@RyanUnger Not moving frames. Just the definitions of wedge and, ultimately, whether you do left- or right-frame bundles. Meh.

but a good part of the book uses the standard notation or fixes a specific set of coordinates
nature of the beast I think

I really never learned any relativity, although I wouldn't have minded learning it. Oh well.

1:27 AM
@anakhro Overtwisted means that there's a disk on which the singular foliation is that swirly thing right

or pizza, si

I wish I had done more physics in college altogether.

ya pizza

Pun potential is excellent in relativity I imagine

Like the disk of radius $\pi$ in the standard one

1:27 AM
@TedShifrin there's no one book that captures everything mathematicians could find exciting about the field, IMO

Well, then, Demonark, you must go into that field.

I once made a function called "unfurl."

the field has a deep divide too so there's little hope in getting one book that talks about both sides

Mike Spivak started his physics writing decades ago with the ultimate goal to "understand" GR. I don't think he'll ever quite get there.

@TedShifrin the overtwisted contact structures are the easy contact structures because they, in a sense, surpass the contact condition with flying colours. The "tight" ones (non-overtwisted ones) live dangerously on the edge.

1:28 AM
If any moderators are in the house, I just flagged the bounty I set on this question to ask that it be cancelled so I can fix a link formatting typo in the bounty text and re-set the bounty. I forgot to mention that it was suggested on meta here that this is the appropriate way to handle such things

But, still, "on the edge" is misleading. If you wiggle, it stays tight.
Wow, putting bounty on a 2012 question.

Never too late!

I think there are h-principle results about approximating tight with overtwisted ones

I've set more bounties than rep. And still owe a few.

Oh, OK, @Balarka. But can you fall off the edge?

1:30 AM
It looks elementary enough to me -- if I weren't feeling lazy, I'd try to work it out myself

@TedShifrin that's more a property of contact structures, not tight structures.
That they are stable.

Remember the joke?: How many mathematicians does it take to fill an empty room?

OK, so in what sense the edge?

A smooth family of contact structures that agree off of a compact subset are indeed contact isotopic.

Tight compact structures skateboard without safety gear

@Daminark lol

@TedShifrin can you do this with frames

I actually have only heard of contact geometry from the "I am the very model of a Heegaard Floer homologist" song

That is art

But I mean that it is on the edge in the idea that contact structures are just twisty hyperplane fields that are too twisty to be integrable. Overtwisted structures are above and beyond twisty, while tight ones are less twisty.

1:35 AM
I did enter with a serious query. This place is fun! The serious query was: A random basis is as good as an orthogonal one. Agreed? Help?

Not formal in any sense, that is. :P

Nah, @humn. Only with orthogonal bases can you recover coordinates of a vector just with dot products.
@Ryan: I have no clue. This stuff is Greek to me.

If someone gives you a basis immediately use Gram-Schmidt tbh

That's not true, Demonark. You can't do that and stay in the holomorphic category, for example.
You just have to do matrix inversion.

If someone gives you a basis, immediately return it with the remark, "Gentle(wo)men don't use bases", as well as a card and flowers.

1:39 AM
@TedShifrin , i beg to disagree. exclusive or is the next best thing, and good enough.

I have no idea what you're talking about.

Bitwise XOR.

You're not clarifying.
I know what exclusive or is.

Hmm, what does that mean exactly? If you have a basis of holomorphic vector fields on a complex manifold or something?

I've been through this before and learned each time. Thank you @TedS

1:41 AM
Or for a complex (holomorphic) vector bundle, yes, Demonark.
OK, so humn is our newest troll.
Good thing I'm leaving momentarily to cook dinner.

Enjoy

Thanks, Balarka :)

I woke up early, I need to figure out what to do with my day

I cooked dinner and missed Mike
Never again will I cook dinner.

LOL

1:42 AM
dinner.. mike.. what to do..

@TedShifrin , i'll take that as a welcome. I have helped many engineers and scientists gain fame. In my own way.

LOL, joe shmo.
Mike said he had departed permanently. I'm glad to see he'll be back occasionally.

@humn I am independently responsible for your fame on this chat. Consider me your PR person.

@anakhro , thank you!
I can only imagine.

1:49 AM
@humn no thanks needed, apart from your first born.

!
Most of the time i feel like a mathematical maverick. I see art in functions and vice versa.
The rest of the time i get banned.

lmao

@BalarkaSen Are you positive you have the correct statement?
It seems more plausible to me to construct a map $\Sigma^m \mathcal M^m_m \to S^m$.
$\DeclareMathOperator{\colim}{colim}$ And it should still be true in the limit that $\colim \mathcal M^m_m = \colim_n \colim_m \mathcal M^m_n$, and in particular that should be a $BS_\infty$

Oh, @MikeM, do not \backslash unless you're ready to go full MathJax (just kidding).

But still, I feel like I want framed points. So I'm confused.

1:55 AM
Care to expand?

@MikeMiller I was thinking about that actually. Given a configuration of $m$ points in $\Bbb R^m$, I can build a map $S^m \to S^m$ which is identity on a neighborhood of those points and sends points away from them to infinity.
This should be the correct map $\mathcal{M}_m^m \to \Omega^m S^m$
I think that's what the scanning map is

The map you want to do uses a framing at those points

(I knew i was gonna learn something here. Thank you for making me feel stupid.)

Don't feel stupid because we're talking about things you don't know; knowledge and capacity to understand are different

!

1:57 AM
In any case, there's certainly nothing to be gained in feeling stupid (:

m i k e

The Barratt-Priddy-Quillen theorem says that $BS_n$ and $\Omega^n_0 S^n$ have the same homology in grade $\geq 2n$, so I thought there was a map $\mathcal{M}^\infty_n \to \Omega^n_0 S^n$ though.
@MikeMiller Ah

(I can make anyone feel stupid when it comes to (La)TeX/MathJax, so it's all fair.)

@humn what sort of math do you like?

2:02 AM
@BalarkaSen I looked up the original paper here: books.google.com/…
Seems like there's some abstraction which seems unavoidable

@anakhro , thank you for asking:
3

A $\texttt{\binary}$ Klein bottle These 72 replacement code tokens form 2 static macros and 1 dynamic macro, while blurring the distinction between inside and outside, reminiscent of a Klein bottle, with data that are not just produced by recursive calls but are explicitly those calls themselves...

I don't know what I am looking at.

It was clear once. The system keeps changing.

fight the system, d00d

@MikeMiller Looks scary!

2:07 AM
But, yeah, i can take any bull by the horns and wrestle it down.
And have taught others the same.

@BalarkaSen I didn't see anything that wasn't a little spooky. But you're not looking for a proof, just the map, so maybe read through for the map.

Tell me a mathfact you enjoy, @humn

Yeah I'll have a look. Thanks

@anakhro , sweetheart, i play with monotesimals (numbers close to 1).

I said mathfact, not humnfact.

2:09 AM

Numbers close to 1 behave as interestingly as infinitesimals.
More interestingly.
Trigonometry close to 1 is geometric.
(That was a cheap call.)

These are numbers of the form $1\pm\varepsilon$ for a positive infinitesimal $\varepsilon$?

\true!
Two places where numbers behave: near 0 and near 1.

Is this in the context of real numbers, or integers/rationals/something smaller?

2:12 AM
what is this conversation

@BalarkaSen have you ever read Alice in Wonderland?

Yes

Now then you might think I am going to say something about following a rabbit down a rabbit-hole.
But you'd be wrong
I am skipping to the caterpillar with the hookah pipe.

. . . Annotated by Martin Gardner?

This is more like the Jabberwocky poem from the mirror land
I always liked Through The Looking Glass more than Wonderland

2:15 AM
!
You know the actual titles, @BalarkaSen.

@BalarkaSen are you vegetarian
just curious, while I still remember I wanted to ask you.

After i was a mathematician i became a publisher.

@humn what was your Ph.D. in?

@anakhro I don't hold a moral stance either way but I eat vegetables when I don't have the cash
Cheaper

Never a phD, @anakhro, but thank you for mistaking. I've helped so many others gain theirs.

2:19 AM
Can you help me finish my thesis then
pls

Y? Tempted.

Why not?
@BalarkaSen neat. I like the idea that veggies can be cheap.
The prices fluctuate in Canada too much to be reliable.

It's probably already finished, @anakhro. Is it in LaTeX by any chance? Plain text is good enough.
You are typing with an expert.
Or maybe joking.
But i'd rather type about how hyperdimensional random bases are better than Fourier/Hilbert transforms.

@humn It's in LaTeX

2:30 AM
@anakhro , oh oh oh, if you dare, pick up: web.stanford.edu/group/cslipublications/cslipublications/…

Will this write my thesis for me? @humn

It will format it.
Oh wow, they've made it more difficult. Sorry.
@anakhro, what is your subject? An inquiring mind wants to know.
Again, i help careers when i can. When not, i don't undermine them.

@humn contact topology.
3-dimensional

@anakhro , that summons so much imagery. Good luck with the diagrams!

@humn yeah some of the diagrams have been fun to make

2:39 AM
I once helped someone publish a book on 3-D philosophy. Each dimension brings new connectibility and new challenge.
@anakhro , wanna show any of the diagrams here? One thing i've learned is that plagiarism is a slow game, so let it rip.

!!!!!!!!!!

fun to make sadly doesn't translate to exciting to look at.

It is . . . beautiful!

thanks d00d

2:44 AM
what do you use to make these with

Takes one to know one.

dont tell me tikz

tikz

nooo

This one was not tikz

2:47 AM
!!!!!!!!!!!!! (wearing out my shift key)

looks great

That was in inkscape.
But with the pdf export + tex file.

Aha

For the nice text

Apologies for being who i am, but mathematics never looks as wonderful as when it is visual.

2:49 AM
Yeah, pictures make math fun sometimes.
They can also be so misleading.

(Not one of mine, just one of my favorites)
Don't tell me, @anakhro. that you're from the French school of mathematics without diagrams.

Heh. No.
I just can appreciate the need for rigour when pictures don't suffice.
Want to see something sort of cool, @humn?

Whew.

That "plane field" (the last picture I posted)---this is an example of the standard contact structure on in R^3.

(At last looking it up. Thought i could fake it.)

2:53 AM
Basically a contact structure on R^3 is just a twisty plane field that you cannot "integrate" to a surface in any small neighbourhood.
All that you really need to know is that in no small little neighbourhood will there be a surface which is everywhere tangent to the plane field.
So this means that in general if you embed a surface in R^3, it will "foliate" the surface into lines where the tangent plane to the surface intersects the plane field.

Is your goal to summarisze it? Too many twists and turns stymie that.

And where they coincide (which is necessarily isolated) will be a singularity.

Just what i was getting at.

We speak the same language.

2:57 AM
So referencing that plane field above, if you have a plane given by the equation $z = \varepsilon y$ (a tilted plane in the y direction), it foliates like this.
But note that there are no singularities. Never does the plane field coincide with the surface.
Basically the lines asymptotically approach the $x$-axis.

Nice! we will make something that sort of looks like that.
By perturbing the surface.

Again, we speak the same language.