Mathematics - 2017-05-02 (page 5 of 7)

Ilya_Gazman

20:01

Hey, guys, can you help me please. I need a direction: math.stackexchange.com/questions/2262833/…

@Smple_V Mr lion can you help me?

s.harp

@MikeMiller what are you referring to?

(oh I see you put a comment on my question :=) )

SoumyoB

@AlessandroCodenotti about time ;)

what opinion would you guys have of someone who is doing a PhD just for fun and interest? As in he may not be going into academics, and may go into industry and research?

s.harp

@SoumyoB that is an incredibly common situation just so you know

SoumyoB

wow really?

MickLH

I've been thinking of pursuing a PhD purely for ego purposes

SoumyoB

20:09

I'm so confused about my career at this point

s.harp

Atleast in physics (fundamental, experimental, and theoretical)

Daminark

I'd say that's a rather admirable way of going about things. And I imagine it can be pretty rewarding when you're not doing your PhD with all the fright about whether or not academia will crush you immediately upon exit

SoumyoB

@Daminark I suppose that can happen, and I hope it does

my career choices were driven by some fundamental beliefs I had about happiness in life, but those beliefs were contradicted by research in psychology very recently

MickLH

lol

SoumyoB

life is so frickin convoluted

Eric Stucky

20:12

hey dami

Balarka Sen

everything's an illusion (disclaimer: not helpful)

MickLH

...unless it is helpful

Eric Stucky

Mick, idk how much you're joking, but unsavory motivations like those are not so uncommon.

MickLH

hahahaha are you calling me unsavory?

Eric Stucky

For instance, I'm finishing my 2.5-year blog project mostly out of spite

SoumyoB

20:13

@BalarkaSen do you believe we're all a simulation?

Daminark

Not you, your motivations

:P

And hey @EricS!

SoumyoB

and that some computer geek sitting out there is our god?

MickLH

don't worry I take it as a compliment, a convenience that goes with the territory I think

Eric Stucky

jeje

Balarka Sen

@SoumyoB i don't even know what that means, except that's the story in a pretty good sci-fi film

SoumyoB

20:15

I've heard this several times on the internet-scientists concluding this

Daminark

But yeah my idea is to do whatever you think will work out best for you. The main reason I want to do academia is that I like teaching (not babysitting or teaching to a test as high school teachers must do, real teaching) and thinking about clever proofs

Balarka Sen

lol

Semiclassical

simulation hypothesis : decent short story premise, pointless IRL

Balarka Sen

r/im14andthisisdeep

MickLH

I think the polytheistic formulations are more reasonable

I mean come on, one engineer to look after the entire simulation?

SoumyoB

20:16

@Semiclassical it may not be, after all

Daminark

@SoumyoB I'm interested to know, what were your thoughts on happiness that have been contradicted?

Eric Stucky

^

Balarka Sen

@Semiclassical there has been so many stories based upon simulation that it's a bit boring now

Semiclassical

can you prove that the universe is a simulation? No? Then the endeavor is entirely pointless.

MickLH

Well I don't think you can prove that there is a meaning to life either but it's fun to look

SoumyoB

20:16

for example if the universe becomes too complicated for the computer to handle, and we're close to that point at some time or other, the computer might crash and it's horrifying to think what would happen to us if that happened

MickLH

@SoumyoB I believe this happens regularly

Semiclassical

...

MickLH

You would have no way to notice it unless they broke something or lost data

Daminark

Well, thinking about it now will not decrease the horror that would come upon us if it does happen

Semiclassical

I'm trying to find words that capture how much that makes my eyes roll.

MickLH

20:17

But then clearly they would just rewind the sim and fix it

Balarka Sen

This is a much better science fiction short story, probably my all time favorite: trillian.mit.edu/~jc/humor/Ms_fnd_in_a_Lbry.html

Daminark

@Semi Pull a Ted and sprout new eyes to roll

SoumyoB

@Daminark not suited at all for this chat room. It will upset many in here

MickLH

@SoumyoB Well now you have to share.

Semiclassical

I find simulation hypothesis stuff to be more annoying than religion, tbh. At least religion has stories and mythologies and worldviews built into it. Simulation hypothesis manages to be both speculative and purposeless.

Balarka Sen

20:20

it's amusing crackpottery @SemiC

pseudophilosophy is good laugh

Semiclassical

Depends on ones tolerance for such things, I think.

And how much one has been exposed to it.

Anonymous

It's just another type of conspiracy theory =P

Balarka Sen

yeah I'd expect a serious physicist to be really annoyed by that stuff :P

Semiclassical

Yup.

Oh, I did run into a fun conspiracy theory the other day.

Daminark

Yeah I find it funny if nothing else. And I mean, I haven't seen much of either type of mindset to be at all... intrusive?

Astyx

20:21

What are we discussing ?

Semiclassical

A guy holding up a selfie stick and the sign "Satellites do not exist"

3

Anonymous

I ran into a cube earther few weeks back on youtube :D

Balarka Sen

cube? holy hell

Anonymous

And plenty of flat earthers (as usual)

Anonymous

lol

Semiclassical

20:22

Who, when the guy walking next to me was snickering after passing bye, complained about us mocking him but not engaging.

Anonymous

It is hard to differentiate between trolls and actual conspiracy theorists nowadays :D

Semiclassical

To wit: aplanetruth.info/2015/11/24/satellites-dont-exist

MickLH

@blue ...what balarka said lol

Semiclassical

I mostly laugh at it, but

Ted Shifrin

I have to add this (not brand new). I posted it on Facebook with the roll of 7 eyes. (They're not as used to my eyes as you are.) golfdigest.com/story/…

Daminark

20:24

If I were to get bored/tired of thinking about simulation stuff, I could easily just stop engaging with it, so I never find it annoying, and when I do want, it's amusing. Just my 2 cents.

Semiclassical

the extent to which such conspiracy theories deny the genuine achievements of physics, engineering, and technology actually does rankle me.

Anonymous

@MickLH He said that cube earth theory is only viable theory that unifies both the flat earth theory and spherical earth theory...and I was like...wtf :P

MickLH

no lol

Anonymous

I didn't even dare to debate with him

Anonymous

lol

Balarka Sen

20:24

pretty good idea tho

Eric Stucky

"How would they know? Were they there?"

Astyx

Well a cube does have the same topology as a sphere .. kinda

Balarka Sen

and it's flat almost everywhere

MickLH

I sat down and specifically tried to reconcile it at all... the best I could come up with is a toriodal earth that the world governments conspire to masquerade as a sphere by not allowing anyone to realize the poles are connected

Ted Shifrin

I guess fake history and fake geometry and physics are not quite the same ...

Semiclassical

20:27

the other reason why conspiracy theories irritate me is best summed up by this bit of a "the moon landing could NOT have been faked" video: youtu.be/sGXTF6bs1IU?t=662

Ted Shifrin

@Astyx: Same topology, different geometry.

Semiclassical

(the rest of the video is good too, but that part in particular)

Anonymous

In case someone is running out of good jokes, use this (theflatearthsociety.org/home) :D

Daminark

I guess I'm fast reaching the stage where it is beginning to feel like any sense of appreciation for even good people's amazing accomplishments feel like an expensive luxury, so the lack of which is becoming normal somehow? I dunno

Balarka Sen

no

Daminark

20:29

@Ted Lol at the article and eyes

Ted Shifrin

I dunno what's going on around here. We have numerous people posting essentially the same questions repeatedly (latest) ... and then some of these people try to remove them to remove traces ... Am I missing something on my snarky comment on this one?

Danu

Probably homework stuff

Ted Shifrin

Beyond scary, Demonark.

Hi @Danu.

What did you learn today? :)

Astyx

I've read they were people "trolling" and posting spam questions on MSE. However this does seem a bit too structured to be pure spam

Danu

although: math.stackexchange.com/questions/2260172/…

Semiclassical

20:30

@ted Eh, it sounds from the comments to the previous question that they wanted to split the two questions up?

Danu

Today I learned how to solve Calabi's conjectures

It sounds better than it was.

Semiclassical

I mean, the comment from Daniel Fischer makes it sound like that was positively encouraged.

Danu

I am now pretty sure I don't want to do geometric analysis.

Ted Shifrin

Actually, the OP just edited to make it slightly different, I guess, so I'll remove my comment.

Danu

0 insight 100% calculation :P

MickLH

20:32

Isn't that effectively the edison quote?

Ted Shifrin

Oh, I listened to lots of Yau lectures on that. It's very hard third-order estimates in PDE, as I recall.

Daminark

Today was manifolds, so we finished the proof from last time about how transversality was a stable condition, and then did the proof where basically, if $X$ is a manifold with boundary, $f:X\to\mathbb{R}^m$, and $f(\partial X)$ intersects $Z$ transversally, then you can homotope $f$ to $g$ such that they agree near the boundary and so that $g(X)$ intersects $Z$ transversally

Ted Shifrin

Well, @Danu, there's lots of room for other things. Look at some of Rafe Mazzeo's geometric analysis work. Lots of really cool geometry.

Yup, Demonark, important theorem.

Daminark

Once we did that, we did a naive definition of orientable manifolds

Ted Shifrin

Although it's more general than boundary. If you are transverse along a submanifold (in general), you can homotope away from that submanifold to make it everywhere transverse.

Balarka Sen

20:34

That's known as genericity by the way.

Daminark

What exactly is being referred to by "that"?

Danu

@TedShifrin The one that we discussed explicitly was pretty bad but nothing really advanced, it seemed. Mostly just shitty calculations.

I think that was supposed to be "Calabi's second conjecture" in the case $\lambda\leq 0$ (hope that this terminology is standard). We did something like 1/2 or 2/3 of the proof explicitly.

Balarka Sen

@Daminark That you can homotope maps to be transverse to manifolds. The theorem you quoted.

Ted Shifrin

You asking me, Demonark?

I actually don't know that, @Danu.

Daminark

I was just not sure which of the two

Akiva Weinberger

20:37

What's the most flattering city in Hungary?

Balarka Sen

In general you'd prove if $f : X \to Y$ is so that $f|_{\partial X}$ is transverse to $Z \subset Y$ then $f$ can be homotoped to $g$ fixing boundary such that $g \pitchfork Z$

Daminark

I was asking Balarka :P

Akiva Weinberger

Yudabest

Ted Shifrin

Demonark: $f\colon X\to Y$, $Z\subset Y$, $W\subset X$ (both closed). If $f$ is already transverse to $Z$ at all points of $W$, we can perturb $f$, keeping it the same in a neighborhood of $W$, so that it is globally transverse to $Z$.

howdy, DogAteMy ...

Balarka Sen

The way to do it is to compose with an embedding $Y \to \Bbb R^N$, perturbing that inside $\Bbb R^N$ by your theorem to be transverse to $Z$ carefully so that everything's happening inside a tubular neighborhood of $Y$, and then projecting back.

(Embedding comes from Whitney)

Danu

20:38

@TedShifrin Hmm? The proof, or the statement?

Ted Shifrin

@Balarka: You are not addressing this particular theorem.

@Danu: Either.

Balarka Sen

@TedShifrin Which particular theorem??

I have no idea what you are referring to.

Ted Shifrin

The one Demonark and I are discussing. You're doing the general case without any submanifold in the domain.

Danu

@TedShifrin From Wikipedia: "[...] states that if a compact Kähler variety has a negative, zero, or positive first Chern class then it has a Kähler–Einstein metric in the same class as its Kähler metric, unique up to rescaling. This was proved for negative first Chern classes independently by Thierry Aubin and Shing-Tung Yau in 1976"

Balarka Sen

@Ted No, I have boundary on the domain.

Ted Shifrin

20:40

I'm talking about what G&P call the Extension Theorem.

Danu

This is the one we did most of, explicitly. Then we also discussed the one that the Wikipedia page takes as the main topic: en.wikipedia.org/wiki/Calabi_conjecture

Balarka Sen

I don't remember G&P's names. I just quoted my theorem here: chat.stackexchange.com/transcript/message/37120297#37120297

It's a generalization of what Daminark said with codomain $\Bbb R^n$.

Ted Shifrin

@Danu: That theorem I know (and I saw Yau's proof). But you were mentioning a $\lambda$?

Danu

On now we aim to discuss the Fano case in the coming days. There have been big breakthroughs in the early 2010's by Chen, Donaldson, Sun (and Tian; there is a controversy regarding priority claims)

@TedShifrin $\lambda$ is just the constant for the Einstein metric :P

Ted Shifrin

You have to do partition of unity games in the case of boundary or what I'm talking about, @Balarka.

Oh, never mind, @Danu

Balarka Sen

20:42

I know, Ted.

Ted Shifrin

Yes, this is going to be the famous differential geometry Chinese mafia fight, @Danu.

Danu

...right :P

I wonder if there has been any resolve

well y naught

shiing-shen wins

Danu

I just know that thing Chen put on his webiste and Tian's response.

@wellynaught ?

SteamyRoot

"A guy holding up a selfie stick and the sign "Sattelines do not exist"

Ted Shifrin

20:43

You're decades out of place, @wellynaught.

SteamyRoot

You're not talking about Mark Peeters are you? :O

Danu

Chern was dead by 2012

well y naught

his lineage prevails

Ted Shifrin

However, I know stuff about how S.T. Yau treated Chern, which I'm not going to talk about in here.

Daminark

Mafia fight?

Ted Shifrin

20:44

Chinese math mafia, yeah, Demonark. I'm not going to elaborate.

Danu

@TedShifrin I've also heard some stuff regarding some students of Yau and the Poincare conjecture...

Alessandro Codenotti

@Ted @Balarka remember that I asked you a few days ago whether I can just plug in a complex variable in a real power series and hope it converges? I'm pretty sure it even has the same radius of convergence as the real series now

Daminark

That's... an interesting way of putting it

Yeah fair

Ted Shifrin

Yes, @Alessandro.

In fact, it's singularities in the complex setting that usually explain odd radii of convergence in the real case.

SteamyRoot

@Danu Yup... One of the reasons Perelman "quit math"

Alessandro Codenotti

20:44

Because that should always be $1/\text{limsup}(a_n)^{1/n}$

Ted Shifrin

@Danu: Despite our being essentially mathematical brothers, I'm not a fan of Yau's (his brilliance notwithstanding).

Balarka Sen

@Ted I have a silly question. $G$ be a Lie group acting smoothly on $M$; how do I prove $f : G \to M$, $f(g) = gp$, is an immersion?

Danu

Manifold Destiny

"Manifold Destiny" is an article in The New Yorker written by Sylvia Nasar and David Gruber and published in the August 28, 2006 issue of the magazine. It claims to give a detailed account (including interviews with many mathematicians) of some of the circumstances surrounding the proof of the Poincaré conjecture, one of the most important accomplishments of 20th and 21st century mathematics, and traces the attempts by three teams of mathematicians to verify the proof given by Grigori Perelman. Subtitled "A legendary problem and the battle over who solved it", the article concentrates on the human...

Ted Shifrin

Ah, yes, Nasar writes math history so well.

It's probably false without further hypotheses, @Balarka?

Balarka Sen

Free action, I am sorry.

Ted Shifrin

20:46

Oh.

Danu

Oh btw

something more topological/geometric for you too, @Balarka

So consider an oriented surface of say genus $\geq 2$ and take $\Sigma\times \Sigma$. Remove a tubular neighborhood of the diagonal.

Now a friend of mine is interested in the homology of the boundary of "what's left".

It's an $S^1$-bundle over the surface.

Balarka Sen

The boundary is the unit tangent bundle of $\Sigma$.

Danu

Sure

Or the unit normal

In any case

Balarka Sen

Normal bundle of diagonal $\Sigma$ in $\Sigma^2$ is tangent bundle.

Danu

So $H_1$ is pretty easy to understand. Just the usual circles plus one from the $S^1$ bundle

Ted Shifrin

20:48

@Balarka: I guess the point is that if you have $X\in\mathfrak g$ in the kernel of the derivative, then you'll get a trivial action of the one-parameter subgroup you get by exponentiating $X$.

Danu

@BalarkaSen ...which is why I said that.

But what to picture for $H_2$? All the products of the usual circles with the $S^1$-circle (ha-ha) are not so bad to imagine. But what about the class represented by the fundamental class of the surface.

How does that look? It seems like, if we were able to make a surface like that, i.e. something like $\Sigma\times \{p\}$, $p\in S^1$, we'd get a globally nonvanishing section of the tangent bundle...

Balarka Sen

The fundamental class of the surface self intersects at $\chi(\Sigma)$ many points I think.

Ted Shifrin

Correct. There is no global section.

Danu

So how to picture it

Balarka Sen

(By Poincare-Hopf)

Ted Shifrin

20:51

But Danu's asking how even to get that class "lifted" to the bundle.

Danu

I know there is no global section, obviously. I just want to understand how to look at this

Ted Shifrin

We have to take the Poincaré dual of a fiber. What is that?

What does the Leray-Hirsch spectral sequence give us for $H_2$?

Danu

A fiber, for you, is an $S^1$? Or a copy ofthe surface?

Ted Shifrin

$S^1$.

Danu

@TedShifrin I don't know spectral sequences. But I'd hope you just have it being the same as $H_1$.

Ted Shifrin

20:52

Well, let's try an example. We all know the unit tangent bundle of $S^2$.

Right?

Danu

ok

Ted Shifrin

What is it?

Danu

am I supposed to say somehting? :P

Ted Shifrin

Yes.

Astyx

(I think you're supposed to say what it is)

Ted Shifrin

20:53

LOL

smacks Astyx for fun

Balarka Sen

I know that one.

but i won't say it

Daminark

"It's the set of $(x,v)$ where $x\in S^2$ and $v\in T_xS^2$ such that $\|v\| = 1$"

Danu

lmao

Ted Shifrin

@Balarka: You saw my off-the-top-of-my-head answer to you?

OK, Demonark. What well-known manifold is that?

Didn't you have homework on this, actually?

Balarka Sen

Not yet, but I am bookmarking to read that after this conversation :D

Daminark

20:54

Nah we did $S^1$ and $S^3$, I think

Balarka Sen

don't want to miss out on this

Daminark

Along with $S^1\times S^1$

Ted Shifrin

I thought you had an exercise on $SO(3)$.

Daminark

Proving that it was homeomorphic to $\mathbb{RP}^3$

Danu

I haven't seen this before but I guess you can just go through the clutching construction and see there aren't so many options. It should n't be trivial.

Ted Shifrin

20:55

Uh huh ...

@Danu: $SO(3)$ is the frame bundle for $S^2$, hence the unit tangent bundle.

Balarka Sen

@Danu It's not trivial, by hairy ball.

It's RP^3, topologically

Danu

@BalarkaSen duh balarka

Ted Shifrin

So, now, let's do homology of $SO(3)=\Bbb RP^3$.

Daminark

And here's where I back out

I should probably get back to my pset anyway, I have a lot to do for tomorrow

So see you guys around!

Ted Shifrin

Yes, integrals to do. Bye, Demonark.

Daminark

20:57

(Only one integral, the others are again, theoretical problems)

:P

Danu

@TedShifrin OK that's just a standard result

Balarka Sen

@Ted Can you describe the unit tangent bundle of $\Sigma$ as a mapping torus of some diffeomorphism? What would that diffeomorphism be?

Danu

polynomial ring over one generator with $\Bbb Z_2$ coefficients

Ted Shifrin

I don't want $\Bbb Z_2$ coeffs, @Danu.

Every student taking a topology qual in the US has to know this :P

Cellular homology?

Danu

Still the same I guess

just 2-torsion generator

Ted Shifrin

20:59

Huh

Danu

It's not Z[a]/2a?

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$Mathematics$ Mathematics