Mathematics - 2019-06-20 (page 1 of 3)

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.

Ted Shifrin

howdy, a @Balarka

Balarka Sen

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

Ted Shifrin

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 $t$ ...

Balarka Sen

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 $d/dt (\Phi^X_{\sqrt{t}} \circ \Phi^Y_{\sqrt{t}} \circ \Phi^X_{-\sqrt{t}} \circ \Phi^Y_{-\sqrt{t}}) = [X, Y]$

anakhro

@BalarkaSen fave diff. form?

Ted Shifrin

12:52 AM

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

Balarka Sen

Ah OK

Ted Shifrin

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

Balarka Sen

@anakhro Uhh, the linking form

Ted Shifrin

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

Balarka Sen

Oh that's a good one

Ted Shifrin

12:53 AM

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

anakhro

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

Ted Shifrin

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 $e_2$.

Balarka Sen

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

Ted Shifrin

@Balarka: Do I even bother responding to this?

Balarka Sen

Heh probably not

Ted Shifrin

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.

anakhro

$\cos(r)\,dz + r\,\sin(r)\,d\theta$ is kind of cute on $\mathbb R^3$.

@TedShifrin count me as one of those inordinate number of people.

Balarka Sen

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

anakhro

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

Ted Shifrin

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

Balarka Sen

Good comment

Ted Shifrin

1:00 AM

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

Ryan Unger

big do carmo

isn't it thinner than the baby one actually

Ted Shifrin

yes, @Ryan, way thinner.

Ryan Unger

@BalarkaSen @TedShifrin hi

Ted Shifrin

hiya

Joe Shmo

hey all

Ted Shifrin

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?

Joe Shmo

1:02 AM

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

Ryan Unger

Joe Shmo

Ted Shifrin

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

Ryan Unger

Do we know each other

Ted Shifrin

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

Balarka Sen

@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

Ted Shifrin

1:04 AM

Or maybe my brain truly is decaying.

Oh no. It's a MikeMiller.

Hi @Mike!

heya @RyanUnger

hi

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

Érico Melo Silva

@MikeMiller wtf lol

Ted Shifrin

1:05 AM

Didn't you visit the west coast? Or am I addled?

That's what I was thinking, Eric.

Mike Miller

@BalarkaSen just thinking about this

Joe Shmo

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

Ted Shifrin

Ohhh, got it. Somehow I thought you were visiting schools.

Weird pair to choose.

Joe Shmo

well i wouldn't be going to both

jk

Ted Shifrin

Duh.

CalTech may be a bit too small/elitist.

Ryan Unger

1:07 AM

Lu Wang is going to CalTech

Balarka Sen

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

Joe Shmo

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

Balarka Sen

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

Ryan Unger

I haven't seen the famed Princeton elitism yet

Ted Shifrin

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

Joe Shmo

1:08 AM

ryan are you at princeton?

Mike Miller

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

Ryan Unger

yus

here for a summer school atm

Joe Shmo

woah! congrats

Balarka Sen

@MikeMiller I have a better question for you

Ryan Unger

thanks

Mike Miller

1:08 AM

It's a local computation, of course, that your two descriptions agree

So you can do everything in euclidean space

Érico Melo Silva

@RyanUnger I didn’t either when I visited

Balarka Sen

That's true.

Ted Shifrin

I figured you would have done/assigned/graded that standard exercise by now, @MikeM :P

@Ryan: I have little patience for people like that.

Ryan Unger

deleted because it can identify easily

Ted Shifrin

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

Mike Miller

1:10 AM

I don't remember these things if I did

Joe Shmo

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

Érico Melo Silva

Tbh I would’ve gone elsewhere had my circumstances been like barely different

Joe Shmo

@TedShifrin you got into a doctorate program in princeton?

Ted Shifrin

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

Yes, @JoeShmo.

Ryan Unger

Princeton is the best place in the world for general relativity. But Berkeley has people too

His interests are very specific

Ted Shifrin

1:11 AM

The people I knew at Berkeley who knew/did GR are long gone.

Joe Shmo

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

Ryan Unger

@TedShifrin were you there during the RK Sachs days

Balarka Sen

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 $\Bbb R^m$.

Ted Shifrin

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

yes, @Ryan.

Mike Miller

That's not a question

Ryan Unger

1:12 AM

his book is a little nutty haha

Ted Shifrin

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

Balarka Sen

$\mathcal{M}_n^\infty$ 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.

Ryan Unger

he gives the penrose and hawking singularity theorems as an exercise

Balarka Sen

@MikeMiller There you go

Ted Shifrin

With hints? :)

Ryan Unger

1:13 AM

the hint is to read hawking's book

Ted Shifrin

LOL, fair. :P

So he obviously doesn't intend a reader to do the exercise at that point.

Ryan Unger

hah, no

Mike Miller

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

Ted Shifrin

I knew him, not his book.

Ryan Unger

it's called "general relativity for mathematicians"

it's like GTM 5 or something crazy

Ted Shifrin

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.

Ryan Unger

yeah

Joe Shmo

@TedShifrin you are an embarrassment of riches

Ted Shifrin

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

Ryan Unger

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

Ted Shifrin

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

Ryan Unger

1:15 AM

oof I would not call the book good exposition

Ted Shifrin

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

Ryan Unger

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

Ted Shifrin

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.

Balarka Sen

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

Ted Shifrin

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.

Ryan Unger

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?

Mike Miller

I can't see anything yet

Ryan Unger

Differential forms do show up at some point sure

Érico Melo Silva

When I spoke to people there during open houses I honestly didn’t hear anything bad about the quals

Ted Shifrin

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

Érico Melo Silva

people complained a lot about how much teaching there was tho

Ted Shifrin

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.

humn

Anyone here at the moment think about randomly orthogonal bases?

Érico Melo Silva

@TedShifrin it’s bad, folks

Ted Shifrin

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

anakhro

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

Ted Shifrin

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.

Ryan Unger

1:22 AM

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

anakhro

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

But I like that it makes a pizza.

Ted Shifrin

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

Balarka Sen

It's uh swirly isn't it

humn

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

Érico Melo Silva

U gotta love to see the hollowing out of higher education

Balarka Sen

1:23 AM

swirls inwards, not quite radially

Ted Shifrin

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

anakhro

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

Ted Shifrin

Yeah, I get that.

Ryan Unger

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

Ted Shifrin

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

I'm glad to hear that.

anakhro

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.

Mike Miller

@BalarkaSen I don't see it, sorry.

Ted Shifrin

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

anakhro

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

Ted Shifrin

I really was a complex geometer ...

Ryan Unger

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

Ted Shifrin

1:25 AM

Simple person, complex geometer.

anakhro

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

Érico Melo Silva

@TedShifrin I like geometry and I’m afraid of everything

Ted Shifrin

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.

Ryan Unger

but a good part of the book uses the standard notation or fixes a specific set of coordinates

nature of the beast I think

Ted Shifrin

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

Balarka Sen

1:27 AM

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

anakhro

or pizza, si

Ted Shifrin

I wish I had done more physics in college altogether.

Balarka Sen

ya pizza

Daminark

Pun potential is excellent in relativity I imagine

anakhro

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

Ryan Unger

1:27 AM

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

Ted Shifrin

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

humn

I once made a function called "unfurl."

Ryan Unger

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

Ted Shifrin

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.

anakhro

@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.

Tim Campion

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

Ted Shifrin

But, still, "on the edge" is misleading. If you wiggle, it stays tight.

Wow, putting bounty on a 2012 question.

humn

Never too late!

Balarka Sen

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

humn

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

Ted Shifrin

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

Tim Campion

1:30 AM

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

anakhro

@TedShifrin that's more a property of contact structures, not tight structures.

That they are stable.

humn

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

Ted Shifrin

OK, so in what sense the edge?

anakhro

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

Daminark

Tight compact structures skateboard without safety gear

Ryan Unger

1:32 AM

arxiv.org/pdf/math/0411109.pdf

Balarka Sen

@Daminark lol

Ryan Unger

@TedShifrin can you do this with frames

Daminark

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

Balarka Sen

That is art

anakhro

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.

humn

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?

anakhro

Not formal in any sense, that is. :P

Ted Shifrin

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.

Daminark

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

Ted Shifrin

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.

anakhro

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.

humn

1:39 AM

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

Ted Shifrin

I have no idea what you're talking about.

humn

Bitwise XOR.

Ted Shifrin

You're not clarifying.

I know what exclusive or is.

Daminark

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

humn

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

Ted Shifrin

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.

Balarka Sen

Enjoy

Ted Shifrin

Thanks, Balarka :)

Balarka Sen

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

anakhro

I cooked dinner and missed Mike

Never again will I cook dinner.

Balarka Sen

LOL

Joe Shmo

1:42 AM

dinner.. mike.. what to do..

humn

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

Ted Shifrin

LOL, joe shmo.

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

humn

anakhro

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

humn

@anakhro , thank you!

I can only imagine.

anakhro

1:49 AM

@humn no thanks needed, apart from your first born.

humn

!

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.

Balarka Sen

lmao

Mike Miller

@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 $BS_\infty$

humn

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

Mike Miller

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

humn

1:55 AM

Care to expand?

Balarka Sen

@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

Mike Miller

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

humn

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

Mike Miller

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

humn

!

Mike Miller

1:57 AM

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

anakhro

m i k e

Balarka Sen

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

humn

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

anakhro

@humn what sort of math do you like?

Joe Shmo

@TedShifrin i need to email you ask about grad schools

Mike Miller

2:02 AM

@BalarkaSen I looked up the original paper here: books.google.com/…

Seems like there's some abstraction which seems unavoidable

humn

@anakhro , thank you for asking:

3

A: MathJax: ones and zeros

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...

anakhro

I don't know what I am looking at.

humn

It was clear once. The system keeps changing.

anakhro

fight the system, d00d

Balarka Sen

@MikeMiller Looks scary!

humn

2:07 AM

But, yeah, i can take any bull by the horns and wrestle it down.

And have taught others the same.

Mike Miller

@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.

anakhro

Tell me a mathfact you enjoy, @humn

Balarka Sen

Yeah I'll have a look. Thanks

humn

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

anakhro

I said mathfact, not humnfact.

humn

2:09 AM

One thing leads to another.

anakhro

Lead me to the mathfact

humn

Numbers close to 1 behave as interestingly as infinitesimals.

More interestingly.

Trigonometry close to 1 is geometric.

(That was a cheap call.)

anakhro

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

humn

\true!

Two places where numbers behave: near 0 and near 1.

anakhro

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

Balarka Sen

2:12 AM

what is this conversation

anakhro

@BalarkaSen have you ever read Alice in Wonderland?

Balarka Sen

Yes

anakhro

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.

humn

. . . Annotated by Martin Gardner?

Balarka Sen

This is more like the Jabberwocky poem from the mirror land

I always liked Through The Looking Glass more than Wonderland

humn

2:15 AM

!

You know the actual titles, @BalarkaSen.

anakhro

@BalarkaSen are you vegetarian

just curious, while I still remember I wanted to ask you.

humn

After i was a mathematician i became a publisher.

anakhro

@humn what was your Ph.D. in?

Balarka Sen

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

Cheaper

humn

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

anakhro

2:19 AM

Can you help me finish my thesis then

pls

humn

Y? Tempted.

anakhro

Why not?

@BalarkaSen neat. I like the idea that veggies can be cheap.

The prices fluctuate in Canada too much to be reliable.

Balarka Sen

Ah

humn

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.

Were i kidding about this there'd be bugs on my smiling teeth.

anakhro

@humn It's in LaTeX

humn

2:30 AM

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

anakhro

Will this write my thesis for me? @humn

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.

anakhro

@humn contact topology.

3-dimensional

humn

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

anakhro

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

humn

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.

anakhro

humn

!!!!!!!!!!

anakhro

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

humn

It is . . . beautiful!

anakhro

thanks d00d

Balarka Sen

2:44 AM

what do you use to make these with

humn

Takes one to know one.

dont tell me tikz

tikz

nooo

This one was not tikz

humn

2:47 AM

!!!!!!!!!!!!! (wearing out my shift key)

Balarka Sen

looks great

what was this made with

anakhro

That was in inkscape.

But with the pdf export + tex file.

Balarka Sen

Aha

anakhro

For the nice text

humn

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

anakhro

2:49 AM

Yeah, pictures make math fun sometimes.

They can also be so misleading.

humn

(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.

anakhro

Heh. No.

I just can appreciate the need for rigour when pictures don't suffice.

Want to see something sort of cool, @humn?

humn

Whew.

@anakhro , please

anakhro

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

humn

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

anakhro

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.

humn

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

anakhro

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

humn

Just what i was getting at.

anakhro

humn

We speak the same language.

anakhro

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.

humn

anakhro

Nice! we will make something that sort of looks like that.

By perturbing the surface.

humn

Again, we speak the same language.

Transcript for

$Mathematics$ Mathematics