Mathematics - 2017-02-14 (page 5 of 6)

Kasmir Khaan

18:00

when i drew it the circle is on the top and ellipse is on the bottom

Ted Shifrin

(When $y\ge 0$, yes.)

DHMO

@Astyx math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/…

Lemma 1.2, pp.2-4

Kasmir Khaan

its a 3/4 of the top unit circle

from pi /4 to pi

Ted Shifrin

So travel along the line $y=x$ from the circle to the ellipse. That point on the ellipse does not have $t=\pi/4$.

Kasmir Khaan

but they both end at ( -1, 0 ) right?

Ted Shifrin

18:01

Yes.

Alessandro Codenotti

@Maks chances are someone else can answer your question if you just write it

Ted Shifrin

So if you set $\cos(t) = \sin(t)/\sqrt3$, this means that $\tan t = \sqrt 3$, so what is $t$?

Kasmir Khaan

why would i equate those?

I get pi /3 that way

oh

Ted Shifrin

So that's the starting point on the ellipse.

Kasmir Khaan

because they are on y=x

Ted Shifrin

18:03

But their solution is totally sloppy. You do need to adjust by the line integral along that little piece of $y=x$.

However, that line integral is $0$. Why?

Kasmir Khaan

you mean from (1/2 , 1/2 ) to ( 1/sqrt(2) , 1/sqrt(2) ) ?

Ted Shifrin

Right.

Kasmir Khaan

Well it is not clear to me why without doing that line integal

Ted Shifrin

Well, the vector field is orthogonal to that line segment along the line segment.

Kasmir Khaan

in the solution this were not even mentioned

Ted Shifrin

18:05

Yeah, so they're wrong.

Kasmir Khaan

ahh because of -y, x

Ted Shifrin

Right.

Kasmir Khaan

okay that is smart :D

but one thing pls !

what did you mean by not normal polar

and how do I do in those situations

Simply Beautiful Art

When someone isn't busy, does there exist f such that $g(x)=\int_a^x f(t)\ dt$ is defined for all $x\in\mathbb R$ but $g$ is not continuous? Any type of integral is fine.

Ted Shifrin

I will post a pic from my diff geo notes. Hold on

DHMO

18:07

@AlessandroCodenotti do you have other exercises like the one you just gave?

Kasmir Khaan

Thanks alot @TedShifrin You are THE best! :)

2

Ted Shifrin

Oh, I don't have it in there. Hold on.

Kasmir Khaan

Take your time ! :)

Alessandro Codenotti

that wasn't really an exercise, just an assertion in a thing I'm reading that was regarded as obvious but I wasn't seeing

Ted Shifrin

The ellipse is outside instead of inside, but you can see what I'm talking about.

On the right, the angle to the point on the ellipse is not $t$, but $t$ is the parameter you're using.

Kasmir Khaan

18:09

Okay I see why the line y=x there does not mean pi/4 now :D

it is different period kind of

Ted Shifrin

You're parametrizing the ellipse by stretching (in your case vertically, not horizontally) the parametrization of the circle. So angles are distorted.

Kasmir Khaan

Very sloppy correction of that exam indeed

Paul Plummer

@BalarkaSen I meant what book are you learning from, if any

Kasmir Khaan

Okay thanks again @TedShifrin :)

Ted Shifrin

Sure, @Kasmir.

Back later ...

Alessandro Codenotti

18:11

have a nice day @Ted

DHMO

@AlessandroCodenotti alright, thanks

Astyx

@DHMO thanks

JeSuis

nice day @TedShifrin

Mike Miller

18:28

@PaulP Candel and conlon

Balarka Sen

18:43

@MikeMiller Objections regarding what? I just had the wrong counterexample

by the way, regarding the problem you gave me, is the codimension 1 foliation supposed to be co-orientable?

Paul Plummer

Cool

Balarka Sen

It's a nice book but I wish there were more cool pictures

Daniel Guldberg Aaes

Hi. I'm trying to turn this cubis equation ( x^3-9x^2+(1/2)x-(9/2)=0) into a depressed equation whit this formula(http://www.texpaste.com/n/vcqsur2h) for the depressed equation x^3+px+q
But for some odd reason the two equations doesn't have the same roots?

Paul Plummer

Yah, I am wondering if there is some software, or software packages, where you could somewhat easily have some visualizations of foliations (at least for surfaces it would be nice)

DHMO

@DanielGuldbergAaes what are your roots?

Balarka Sen

18:48

I think I have a decent collection of examples of foliations. What I struggle with is converting a symbolical proof into a picture-proof (the latter is the only way I can make sense of things)

Daniel Guldberg Aaes

I only need one root btw. and the root or the solution to the equation should be x=9.

DHMO

@DanielGuldbergAaes and what did you get instead?

Balarka Sen

@PaulP There aren't many surfaces which admit foliations, right? I guess you mean singular foliations

Paul Plummer

Yah

Vrouvrou

@DHMO please how to prove that $\forall \varepsilon, ]-\varepsilon, \varepsilon[\cap \{\frac1n, n\mathbb{N}^*\}\neq \emptyset$ ?

Balarka Sen

18:50

What kind of pictures are you looking for?

DHMO

@Vrouvrou You might be looking for Archimedean property

Paul Plummer

I guess I am actually more interested in seeing some concrete visualizations of the dynamics and what is happening to curves @BalarkaSen

Daniel Guldberg Aaes

@DHMO 2 sec

PVAL-inactive

Are quotes a good way of showing that you are defining a term? What about using "so-called"?

DHMO

@PVAL-inactive An integer is said to be even if it is an integer multiple of 2.

Balarka Sen

18:52

@PVAL You can also italicize it, IMO

PVAL-inactive

Here's the passage

However, if we assume a 4-manifold admits a symplectic form and the boundary is a certain contact manifold (with a natural compatibility condition), classification problems suddenly become tractable. We will show how such classification results of Eliashberg, McDuff and Lisca in the case of lens spaces can be used to put topological restrictions (completely determining the Betti numbers and signature) on these symplectic fillings

I'd like it to be clear that the meaning of "symplectic fillings" corresponds to the hinted at definition in the first sentence.

so-called sounds like I am disapproving of the terminology

Daniel Guldberg Aaes

@DHMO here is the soulutions and a graph of bot equations owncloud.techknight.eu/index.php/s/iZpekAu1QVqmVWq

PVAL-inactive

and quotes don't really work.

Balarka Sen

how about: [...] on these, known as symplectic fillings

Paul Plummer

So if you have a foliation, invarient under a pseudo Anosov, that would correspond to a teichmuller geodesic, and I want to see what is happening to curves as you travel up the geodesic, which is streching the foliation. This could be fun to play around with, so you can see how complex these curves get

Mike Miller

18:56

@BalarkaSen Nah. In that case it just fibers over the circle.

PVAL-inactive

@PaulPlummer Well there are these things called measured laminations ya know...

DHMO

@DanielGuldbergAaes sorry, no idea

Mike Miller

@PaulPlummer I think you probably want to look at Danny Calegari's foliations book

@PVAL-inactive wouldn't have figured that to be your thing

Daniel Guldberg Aaes

@DMHO Honest anwser thoug ;) I just can't grasp that it isn't the right anwser.

Balarka Sen

@MikeMiller What if the same leaf hits the back the transversal once again?

PVAL-inactive

18:57

I directed the reading seminar here toward it so I could take a passive role.

But they do matter to me I think.

Balarka Sen

I guess by compactness of the leaves it would fiber over a covering space of circle so that's ok

Mike Miller

@BalarkaSen Then it's not co-orientable.

Paul Plummer

@PVAL-inactive @MikeMiller I was were talking about a software to see what is happening easily

PVAL-inactive

I'm pretty sure people like Wand use hard MCG stuff, and I definitely care about the stuff they do.

Paul Plummer

and have pictures/video

Mike Miller

18:58

Oh, nevermind, I see your point.

The transversal could cross the leaf infinitely many times if you want.

PVAL-inactive

I know Kaloti uses some like Thurston stuff specifically in his thesis.

Balarka Sen

Can you do that with compact leaves? I think you required that in your question

Mike Miller

I should know all this but I don't.

@BalarkaSen Let $\varphi$ be a diffeomorphism of the circle corresponding to (some smoothing near the end of) $x \mapsto x^2$ on $[0,1]$. Then there are no periodic points of this diffeomorphism, and only one fixed point. The induced foliation of the mapping torus has a transversal going through a non-fixed point comes back around to phi(x), then to phi(phi(x)), etc

PVAL-inactive

@Paul Somehow Thurston's proof of classification of PA autos is just seeing how it acts on a generalization of foliations called measured laminations

@Mike Is the passage I wrote clear?

Mike Miller

Yes

If you really want to make it more clear you could write something like "on these bounding manifolds, known as symplectic fillings".

But you've already put more effort into this sentence than most authors do their intros.

PVAL-inactive

19:02

its an abstract

for a talk

Mike Miller

Then you're totally fine.

Vrouvrou

@DHMO thank you, please is $\{\frac1n, n\in\mathbb{N}^*\}\cup \{0\}$ is compact in $(\mathbb{R},|.|)$ ?

DHMO

@Vrouvrou que veut dire "compact"?

PVAL-inactive

for the paper I feel fine talking about terms that are common in the literature. But I don't want to use foreign terminology to generic topologists/geometers when giving a talk directed at them.

Balarka Sen

@MikeMiller Ah fair. The transversal is free to be noncompact, even if the leaves are not

Thanks.

PVAL-inactive

19:04

uh

The transversal should be compact

At least in the interesting cases I know.

Vrouvrou

Compact space

In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other). Examples include a closed interval, a rectangle, or a finite set of points. This notion is defined for more general topological spaces than Euclidean space in various ways. One such generalization is that a space is sequentially compact if any infinite sequence of points sampled from the space must frequently...

Paul Plummer

@PVAL-inactive I am somewhat aware, Thurston introduced train-tracks to help understand these laminations, and somehow it is involved in the classification. But I am thinking about how it could help see what is going on when a pA acts on the curve graph (vertices homotopy classes of curves, edges if there are representative which are disjoint)

(I mention train tracks because I was told I should learn about train tracks soon)

DHMO

@Vrouvrou is your set closed and bounded?

Balarka Sen

@PVAL If you look at the mapping torus of a diffeomorphism S^1 --> S^1 and start at a point x which is not a periodic point then the transversal is not compact

Vrouvrou

is it bounded i think @DHMO

DHMO

19:06

@Vrouvrou prove that it is

Vrouvrou

$0<\frac1n\leq 1$

PVAL-inactive

You mean if you foliate the torus by quotients of lines of irrational slope?

There is certainly a compact transversal there.

DHMO

@Vrouvrou good. then is it closed?

Vrouvrou

no it must be finite , i think !

Balarka Sen

No, that's not what I meant. The leaves are not compact

also there's a compact transversal

DHMO

19:08

@Vrouvrou closed does not mean infinite

PVAL-inactive

The example I gave the leaves are not compact either.

Balarka Sen

I literally mean foliate by the circles fibers in the mapping torus of a diffeomorphism of S^1 with at least one non-periodic point, like the example Mike gave

PVAL-inactive

In fact each leaf is dense

Balarka Sen

But that was not my question. I wanted compact leaves but noncompact transversal

PVAL-inactive

Well the standard Reeb foliation should have a non compact transversal

but no compact one.

Balarka Sen

19:10

Lots of noncompact leaves though

PVAL-inactive

but somehow anything has a noncompact transversal because you have locally trivial charts

GSF

Could anyone help me with a question regarding algebraic groups and schemes?

PVAL-inactive

I think in lower dimensions all compact leaves probably implies a fiber bundle structure thought that's kind of unclear in higher dimensions.

Balarka Sen

Actually you can see the irrational line example from a different perspective too. Foliate te torus by the tranvsersals of the irrational leaves instead (which are just the meridians) :)

It's probably just the mapping torus example with diffeomorphism being rotation by irrational angle

PVAL-inactive

(which in particular has no periodic point).

Balarka Sen

19:13

Right.

OK, I'm happy. Thanks.

Paul Plummer

@PVAL-inactive Do these make it a lot easier to see what is happening to a curve?

PVAL-inactive

Measured laminations?

Paul Plummer

Yah

PVAL-inactive

I don't know how it works. I just know thats how the original proof of Niel-Thur goes. I am learning it passively now.

Balarka Sen

@PaulPlummer What's a pseudo Anosov map?

PVAL-inactive

19:17

I think I got something out of reading the last chapter of B&F though.

APTTMCG

I think they talk about Thurston's original proof there.

Paul Plummer

Yah I just started looking at it since you mentioned it, and it does look like that is basically what train-tracks are for (and they are connected to laminations)

A map is pseudo anosov if it acts on transverse measured foliations by scaling one by $\lambda$ and the other by $\lambda^{-1}$

Wow I forgot how to spell lambda

They are the "interesting" elements of the mapping class group

Balarka Sen

scaling which?

Paul Plummer

You have two measured foliations which are transverse, on the surface, and the measures are being scaled

Balarka Sen

Ah

PVAL-inactive

The mapping tori of pseudo-anosov maps end up being hyperbolic

Balarka Sen

19:25

Oh cute

Paul Plummer

That is an (appearantly, I don't know the proof) difficult theorem of Thurston (of course)

Vrouvrou

@DHMO i don't know if $\{\frac1n, n\in\mathbb{N}^*\}\cup \{0\}$ is closed , i just know that $\{0\}$ is closed

PVAL-inactive

I'll probably talk about that theorem

later in this semester

so maybe I'll figure out why its true

DHMO

@Vrouvrou what are the limit points of the set?

Vrouvrou

$\{0\}$

DHMO

19:28

does it contain its limit points?

Balarka Sen

These days whenever I ask a question, or hear a theorem, the answer to the question "Who proved it?" turns out to be Thurston

Vrouvrou

@DHMO i don't understand ?

DHMO

@Vrouvrou does the set contain its limit points?

Vrouvrou

no

DHMO

I thought $0$ is in the set

Vrouvrou

19:29

ohh yes yes

Paul Plummer

That is what happens when you start learning about foliations @BalarkaSen and basically anything in lower than 4 dim topology and geometry

DHMO

then it is closed

Vrouvrou

is there an other method , to prove that it is closed ?

DHMO

what method do you want?

you can prove that its complement is open... it's a tedious task trust me

Vrouvrou

i can calculat the adherence

it is the same set then it is closed

right ?

DHMO

19:32

yes

PVAL-inactive

@BalarkaSen A PA map is a map which is not isotopic to a periodic map, and does not fix any disjoint union of curves. (this is somehow the point)

Semiclassical

hi chat

Astyx

Hi @Semi

Paul Plummer

@PVAL-inactive I think the interesting thing is that all maps with those properties are pA in the way I mention, making the classification theorem more impressive, because theoretically there could have been other types of mapping classes

DHMO

@Vrouvrou bye

Semiclassical

19:37

hmm, I wonder if you'd know a name for the following scenario @PVAL-inactive

Balarka Sen

@PVAL Got it.

That's a good picture

Paul Plummer

classification: reducible (fixing some finite set of curves), periodic(finite order), things which aren't those are pA

PVAL-inactive

@PaulPlummer Ya certainly the fact that those maps can be described in a nice way is what makes the theorem meaningful.

Paul Plummer

@BalarkaSen pA are the things which act hyperbolically on the curve graph

Balarka Sen

That does make sense.

Semiclassical

19:41

Consider a family of complex projective curves such that the generic fiber is smooth, but there is a singular fiber where it is reducible to a set of projective lines in general position.

skullpetrol

How did the capacitor timing lab go?

Semiclassical

Hasn't happened yet. First one is tomorrow afternoon, and the other the following morning. @skullpetrol

skullpetrol

Ok.

PVAL-inactive

@Semiclassical Lefschetz fibration?

Semiclassical

Huh. Is that what a Lefschetz fibration is?

Balarka Sen

19:43

Is the family parametrized over CP^1?

PVAL-inactive

I don't know the higher dimensional stuff but the 4-dim real stuff, that is just a Lefschetz fibration. Locally you are saying near the singular point of the map the fiber looks like an intersection of complex lines

Semiclassical

Hmm. In principle it's parametrized over a larger projective space than that, but I can probably restrict to a CP^1 subspace.

@PVAL-inactive To be clear: When you say 'intersection' do you mean that all the lines intersect at a common point, or just a bunch of lines in general position?

I'm guessing the latter.

PVAL-inactive

ah I guess in the Lefschetz fibration case (the generic case) there's only two.

Semiclassical

Ahh.

PVAL-inactive

but that is what you mean, you have two intersecting locally over a given fiber.

Semiclassical

19:46

Yeah, for what I'm doing I've got at least four lines at the singular fiber.

PVAL-inactive

I think.

Ya somehow you can do similar arguments to the real case to get (generically) maps which locally look like z_1z_2=\epsilon with epsilon deforming to 0

Semiclassical

I should probably look at this: ams.org/notices/200508/what-is.pdf

PVAL-inactive

yah he only cares about the smooth case though

Semiclassical

Hrm.

PVAL-inactive

So somehow everything hes talking about is already pretty generic

Semiclassical

19:50

Yeah.

PVAL-inactive

You are aware thats my adviser right?\

Semiclassical

huh.

nooope, I did not know that.

PVAL-inactive

Ya if you are getting multiple (more than 2) lines the critical points of the map to the parameterization space are going to be higher degree than 2

Semiclassical

Hmm, makes sense.

That's true even if the multiple lines don't have a common intersection, I presume?

More than just the obvious double points, I mean.

PVAL-inactive

It shouldn't matter what the arrangement is

Jasper Loy

19:54

@AliCaglayan Did you get into Cambridge?

Semiclassical

Really? That's a bit surprising to me, since it would translate into the singular fiber having singularities that are more than just nodes.

PVAL-inactive

If you have 3 lines in a special position or in a general position the map to the space your parameterizing over is of higher degree.

I don't know and I have to leave.

Semiclassical

Okay.

PVAL-inactive

bye'

SAWblade

Do all Steiner Triple Systems of order $\geq 9$ satisfy the parallel line postulate?

Vrouvrou

20:19

@DHMO thank you very much @DHMO

Mahmoud

21:21

Hello everyone ! $:)$

I never saw this chat room is a state of inactivity, something strange is going on ..

SAWblade

0

Q: Do all Steiner Triple Systems of order greater than 7 satisfy the Parallel Line Postulate?

A Steiner Triple System is a set $\mathcal{S}$ of $v \geq 3$ elements together with a set $\mathcal{B}$ of $3$-subsets (triples) of $\mathcal{S}$ such that every $2$-subset of $\mathcal{S}$ occurs in exactly one triple of $\mathcal{B}$. As an example, the following forms a Steiner Triple System o...

Fargle

21:47

@Mahmoud Everyone seems to be engaging in their recreation. Which, based on this chat, I guess is also math.

Semiclassical

Eh, it occasionally gets into actual research stuff. (I'd count what PVAL and I were talking about in that category.)

Balarka Sen

i should be doing math but i am not

Mahmoud

@Fargle I just came from a gaming server, to be honest :P

Fargle

@Mahmoud I know that feeling. I can't stop playing Overwatch.

Mahmoud

@BalarkaSen Right, what's wrong with doing math there days :?

Semiclassical

21:49

My sin lately is old GB roms

Fargle

Luckily I've done all my math homework (some simple induction proofs).

Mahmoud

@Fargle My case is more complicated, it's Minecraft ...

Semiclassical

I don't have a big grading pile until tomorrow, though.

Fargle

Oh man, @Mahmoud. I didn't realize people still played that.

Any games in particular got your fancy, @Semi?

Semiclassical

Mostly ones I had when I was a lot younger.

Right now it's Final Fantasy Adventure.

PVAL-inactive

21:51

I just started playing online poker again (for play chips) and went from 2.5 buyins to 45 in the span of a weekend.

Semiclassical

Which is old even for GB, since it dates back to 1991.

Mahmoud

@Fargle It's very unique and precise, if you like details and creative gaming in a free platform, well you're going to love this one.

Balarka Sen

@Mahmoud Nothing? I'm just lazy

Semiclassical

Might download Link's Awakening at some point as well, or another one of the GB Zelda games.

Fargle

@Mahmoud I played for several years.

Semiclassical

21:53

(I'd do SNES ones, but I'm trying to resist going off the deep end.)

Mahmoud

@BalarkaSen Yeah .. that also counts.

@Fargle Minecraft ?

Akiva Weinberger

Happy Valentine's Day

Fargle

Indeed, @Mahmoud. I (read: my parents) bought it during alpha.

Semiclassical

Happy corporate-sanctioned relationship day :P

Balarka Sen

Also reading Dante but that's not relevant

Semiclassical

21:54

@BalarkaSen Had you been reading Eliot a bit?

Mahmoud

@Fargle It's way richer now, (especially during the recent updates).

Akiva Weinberger

What's the LaTeX…

Fargle

So I've heard. I don't know if I need to go down that rabbit hole again, lol.

Akiva Weinberger

$\huge\color{Red}{\heartsuit}$

PVAL-inactive

I can't imagine any math being more tedious than reading Dante.

Balarka Sen

21:55

@Semiclassical I did. Not much anymore.

Semiclassical

Mmkay.

Eliot liked Dante, which is evidenced by it showing up in multiple places in his work.

Mahmoud

@Fargle What do you mean ?

Fargle

@PVAL-inactive I found the Comedy to be a pretty engaging read.

Balarka Sen

In terms of poetry I carry around Tarkovsky's book with me all the time

Fargle

@Mahmoud I fear my studies will suffer quite a lot if I start playing again.

Balarka Sen

21:56

@Semiclassical Ahh

PVAL-inactive

Dante did interesting things. He just did it in the most boring way imaginable.

Fargle

[shrug] Cuique suum

Semiclassical

I know exactly one line of untranslated Dante because of Eliot.

Mahmoud

@Fargle Believe me, they'll surely do .. Any suggestions for getting outside the hole ?

Semiclassical

which I'll now butcher because I know how it's said rather than written:

PVAL-inactive

21:57

I have only read translated so thats all I know

Fargle

@Mahmoud Spam right-click while looking down.

PVAL-inactive

but I can't imagine calling the comedy engaging.

Balarka Sen

I'm reading the translated work of course.

Semiclassical

Poise escus nel foco chegli afina

Now to see how it's actually written...

"Poi s'ascose nel foco che gli affina"

Not terrible for a guess based on pronunciation

Transcript for

$Mathematics$ Mathematics