Mathematics - 2017-11-12 (page 1 of 4)

MatheinBoulomenos

00:06

I just realized that my username was wrong the whole time :(

So please call me Mathein now :D

Antonios-Alexandros Robotis

@ManolisLyviakis ti kaneis ;)

Adeek

00:51

@BalarkaSen I was wondering is there a way to geometrically explain difference between cotangent bundle and tangent bundle other than categorical notions ?

Antonios-Alexandros Robotis

@Adeek what do you mean by that

(sorry if I'm butting in)

Adeek

don't worry about it I mean that a covector is just the dual notion of a vector

Antonios-Alexandros Robotis

Indeed, the cotangent bundle is just the dual notion of the tangent bundle.

Adeek

so there is a lot of reasons to consider the dual notion such as integration on a manifold, but I just want to know if there is a geometrical way to explain what is exactly the cotangent bundle

in the same way as tangent bundle

Antonios-Alexandros Robotis

There's no real need for categories to explain it. You just have a vector space of linear functionals sitting at each point

well what would you say the tangent bundle is?

Adeek

00:57

For me I imagine the tangent bundle as different euclidean spaces where each eucilidean space is just all arrows eminanting at a point p.

that is how I imagine it

I am trying to build a mental picture for the cotangent bundle

Antonios-Alexandros Robotis

Hmm. That's a tough thing to address. I'm no expert, but I don't try to visualize it in that way. I stick to visualizing the tangent bundle, and just thinking of the cotangent bundle as being the functionals associated with it.

Adeek

I see

Antonios-Alexandros Robotis

It's an interesting question to ask, though. I think that John Lee's book on manifolds mentions this a bit and has some strategy to visualize covectors, at least. I'm not sure I found that so insightful.

Forever Mozart

I have 2 published papers now, so... yeah. I'm kindof a big deal.

in General Topology

Antonios-Alexandros Robotis

what on

Forever Mozart

01:08

strange connected sets... I discovered some counterexamples. I'm much worse at proving theorems.

user104729

@Adeek Why can't you think of the cotangent bundle in precisely this way?

Forever Mozart

although, maybe all of the big theorems I've tried to prove are actually false

Adeek

@user104729 I like having mental pictures for things

Forever Mozart

in trying to prove them, I learn what a counterexample must be like, and usually I can then construct one

Adeek

@ForeverMozart congrats

user104729

01:10

@Adeek Sure, but the mental picture is the same, right?

Adeek

yeah

yeah I guess it is the same but instead of points

it is functionals

Forever Mozart

I've still got about 3 big questions. If I could answer those, I'd quit math.

user104729

But in the fibre of the bundle, those functionals still give vector space structure, right?

Forever Mozart

problem seems to be, few people are interested in my work. General Topology has fallen out of fashion

Antonios-Alexandros Robotis

In some sense, vectors and functionals are the same thing. They are just dual notions.

Adeek

01:12

yeah exactly @Antonios-AlexandrosRobotis

user104729

Precisely what I am getting at

Adeek

yeah @user104729

yeah nvm I am just confusing myself you guys are right

Antonios-Alexandros Robotis

I guess somehow it doesn't seem very fruitful to think about it this hard.

user104729

You can view it, if you wish, as overlaying two vector spaces over eachother

One the vector space, the other the dual

Antonios-Alexandros Robotis

I would think of a bunch of arrows sitting on the tangent space, and their collection of functionals, the same way you might in linear algebra.

Adeek

01:13

yeah

Antonios-Alexandros Robotis

for what it's worth, the tangent space eventually just becomes thought of as point derivations and this all sort of falls out the window, but the intuition is still the same.

user104729

Yep, point derivations $\mathcal{O}_x\to K$

Forever Mozart

Does anyone know

Adeek

yeah it is true for $C^{\infty}$ manifolds @Antonios-AlexandrosRobotis

I got a really nice book which explains analytic geometry as ringed spaces and introduces stuff in terms of sheave theory

Antonios-Alexandros Robotis

Nice. Which book?

Adeek

01:18

I will read it but after I finish understanding analytic and complex geometry very well in its classic version

@Antonios-AlexandrosRobotis It is called manifolds sheaves and cohomology

it was published last year it seems pretty good

Antonios-Alexandros Robotis

hmm looks interesting

Adeek

yeah very interesting book

Akiva Weinberger

Cool problem I found (and looked up the answer to, and felt bad that I didn't try to finish it myself):

$x_0=0$ and $x_1=1$, and$$x_{n+1}=x_n\sqrt{x_{n-1}^2+1} +x_{n-1}\sqrt{x_n^2+1}$$

Find $x_n$.

Adeek

@Antonios-AlexandrosRobotis are you working in geometry ?

Antonios-Alexandros Robotis

I'm learning geometry, more or less.

Adeek

01:24

nice

Antonios-Alexandros Robotis

My professor keeps giving me problems to solve which appear to be from his past research papers. So, this is really something.

Is anyone particularly familiar with harmonic coordinates on a Riemannian manifold?

Probably a long shot.

Ted Shifrin

01:37

Ooooh, DogAteMy, I have a puzzle for you I can't do.

@Antonios-AlexandrosRobotis: You should talk with @EricSilva. I'm somewhat acquainted, but I'm not really a Riemannian geometer.

Antonios-Alexandros Robotis

Thanks @TedShifrin

Ted Shifrin

Interesting, Antonios. I did my Ph.D. in geometry at Berkeley. Who're you working with at NYU?

Eric Silva

i was called from the void

Antonios-Alexandros Robotis

Jeff Cheeger, though perhaps working with might be an overstatement.

For now, learning from.

Hi, @eric

Ted Shifrin

Awesome. I'm amazed that all these people who were big names when I was a student are still active and I've already retired :P

I know Jeff, but he won't remember me.

Antonios-Alexandros Robotis

01:40

He's a nice guy.

Ted Shifrin

Oh definitely.

Eric Silva

oh his book is nice

Ted Shifrin

He was actually a Chern student, I believe, a few years before me.

Antonios-Alexandros Robotis

Whoops. I've been trying to edit stuff.

@EricSilva I was wondering if you might say a thing or two about harmonic coordinates.

(sorry)

Ted Shifrin

DogAteMy and others: This appeared in the FB feed of an old math friend of mine.

Eric Silva

01:42

you have to be a bit more specific @Antonios-AlexandrosRobotis

Ted Shifrin

Fold this into a cube, folding only along creases.

Antonios-Alexandros Robotis

@EricSilva well, I'm trying to figure out the point. From what I've read, it seems they clean up the laplacian considerably, but I'm having trouble computing this for myself.

I'm also aware that they can be used to simplify the Ricci Curvature tensor

Somehow, I'm having trouble utilizing the definition I'm given: i.e. the coordinate functions satisfy Laplace's eqn.

Ted Shifrin

I totally misremembered=lied. Jeff Cheeger was a student of Bochner and Simons at Princeton. Agh.

Eric Silva

do you know the expression for Laplace-Beltrami in coordinates?

Akiva Weinberger

@TedShifrin Hm

Right so that's 9 there

Antonios-Alexandros Robotis

01:46

@EricSilva I don't think I'm familiar. I could look it up, though.

Ted Shifrin

Ayup, DogAteMy.

Akiva Weinberger

Oh ew do I need to fold one of them in half diagonally?

Antonios-Alexandros Robotis

Okay, I see the formula.

Ted Shifrin

No, you're only allowed the creases that are there.

Akiva Weinberger

Ah wait I see (why that would not be necessary)

Ted Shifrin

01:47

I haven't yet succeeded, DogAteMy. I figured I'd give it to my ninth graders tomorrow and they'd get it fast. Well, one or two.

Eric Silva

so you get a really nice expression for the Laplace-Beltrami in harmonic coordinates because the fact that your coordinates satisfy $\Delta_{g} x^{i} = 0$ lets you relate the Christoffel symbols w $\Delta_{g}$ @Antonios-AlexandrosRobotis

Antonios-Alexandros Robotis

I see. What's the best to derive this relation? Do I apply the definition of \Delta_g directly?

Ted Shifrin

@EricSilva: Is it clear a priori that you can always choose harmonic coordinates?

Eric Silva

I think what you should do is work through the computation for the coordinate expression of $\Delta_{g}$ and then try to see what you get for harmonic coordinates when you try to relate $\Gamma_{ij}^{k}$ to the fact that $\Delta x^{i} = 0$ @Antonios-AlexandrosRobotis, i dont want to give too much away, i think it's a good exercise

@Ted no it's an elliptic pde result

Ted Shifrin

Right, I figured it was nontrivial. I mean, I know that conformal coordinates on a surface is nontrivial.

Eric Silva

01:50

but you have them whenever the metric is $C^{k, \alpha}$ or something

Antonios-Alexandros Robotis

@EricSilva Okay, time to continue banging my head against where I was stuck ;) Thanks for the advice :)

Ted Shifrin

$k\ge 2$?

Eric Silva

@Antonios-AlexandrosRobotis yeah from the formula for $\Delta_{g}$ it aint too bad

Ted Shifrin

(Of course you know I am happy to be $C^\infty$ :) )

Eric Silva

@Ted maaaybe i havent looked at the result in a while lol

i would think you dont even need k \geq 2

i think $k \geq 1$ is enough

Ted Shifrin

01:51

You're taking second derivatives, say what?

Eric Silva

i mean when you start trying to interpret things weakly

Ted Shifrin

I should look back at Chern's proof for surfaces. It was cool.

Antonios-Alexandros Robotis

@EricSilva are you a researcher?

Ted Shifrin

I'm the old man here, @Antonios. Eric is a young'un. :)

Eric Silva

lol no

Antonios-Alexandros Robotis

01:53

ahh gotcha haha :) well thanks for the advice. I'm trying to learn more about Riemannian geometry.

Ted Shifrin

Eric is a geometry/analysis lover.

Eric Silva

i do love me some of dat

Antonios-Alexandros Robotis

I've definitely been changing perspective a lot recently. Somehow everything is of interest to me these days.

Ted Shifrin

I've taught the rudiments of Riemannian geometry, but I'm definitely a complex geometer.

That's fantastic, Antonios. Much better than the alternative.

Antonios-Alexandros Robotis

I'm sort of being encouraged to start exploring complex geometry. We'll see if that's what ends up happening.

Ted Shifrin

01:55

Well, if you start going a bit that way, I can help a bit. So can Danu, who's now doing his Ph.D. in Germany on that stuff.

Definitely should learn some Riemann surfaces and complex algebraic geometry :)

Eric Silva

i wanna learn some complex geo, idk if there are any faculty here who do a lot of it

Adeek

I am also starting to enter complex geometry as well @Antonios-AlexandrosRobotis

Ted Shifrin

Sid Webster, Eric. :P

Eric Silva

ya but he's unapproachable

he stalks around and mumbles a lot

he seems kinda scary

Ted Shifrin

That's so sad ;(

Antonios-Alexandros Robotis

01:56

@TedShifrin yeah, I'm rather interested in alg. geom, but it seems like complex is in the intersection of everything i'm interested in

Ted Shifrin

He was a truly nice guy, but sorta shy, in grad school.

Yup, @Antonios. Go for it. :)

Eric Silva

maybe he's just quiet

maybe ill show up to one of his SCV classes to see what's up

Ted Shifrin

@EricSilva: I just heard that one of my former colleagues, who used to be an award-winning teacher, is degenerating with dementia issues and having all sorts of serious problems. He refused to retire. Makes me so f***ing sad all around.

Eric Silva

:( that is really sad

Ted Shifrin

He is definitely quiet and shy, Eric. Or at least he was ...

Adeek

01:57

@TedShifrin very sad

Ted Shifrin

Yeah, Karim, totally. Now you know why I retired "early."

Eric Silva

ill show up to his complex variables class next quarter to have a look if it doesnt conflict with minimal surfaces/low dim top

Ted Shifrin

You could say hi for me, Eric, but I'm sure he won't remember me from the 70s.

heya @MikeM

Eric Silva

ya if i go ill say hi

Adeek

@TedShifrin yeah

Ted Shifrin

02:00

I'm somewhat pleased that DogAteMy didn't get the answer within a minute. I expected he would.

Antonios-Alexandros Robotis

@EricSilva One thing I'm noticing is that if I assume my coordinate frame is orthonormal then I recover the Euclidean notions of grad, divergence, and Laplace operator. Is this too strong of an assumption to be making on my coordinates?

Ted Shifrin

Yup. That can only happen if your manifold is flat.

But it's a good first step.

Eric Silva

way too stronk my dude

Akiva Weinberger

@TedShifrin To the cube thing?

Ted Shifrin

LOL ... Eric with his hip talk.

Yup, DogAteMy :P

Akiva Weinberger

02:05

In my defense I have things going on in real life at the moment :P

Antonios-Alexandros Robotis

Hmm. I guess it's time to recompute grad in general coordinates. Because with my orthonormal frame I don't really see much of a point for the harmonic condition, since everything is optimal haha.

Daminark

The thing I heard about Webster teaching was that he's really hard to hear if you're not at the very front

Ted Shifrin

That's ok, DogAteMy. I hope everybody/thing is OK.

Yeah, unless he changed, Demonark, he's super shy and that isn't great for teaching.

Mike Miller

hi

Daminark

According to some evals he can get super pissed when students ask questions

Eric Silva

02:07

@Antonios-AlexandrosRobotis it's good to compute all these things in various special coordinates

Ted Shifrin

what level, Demonark?

Daminark

Though others have said it was because the questions legit got out of hand

Complex analysis

Ted Shifrin

Hmm ...

I could see he wouldn't be a great calc teacher, for sure.

Eric Silva

the complaints ive heard were only from people who took riemannian geo with him and ive only ever heard that he's quiet and kinda rambly

Ted Shifrin

But it's hard to judge people based on 40 years ago.

Daminark

02:08

But yeah I'm not gonna say much along those lines, but really the problem that I've heard the most was that he made the class boring. Low energy, psets too easy/uninspiring, hard to hear

Antonios-Alexandros Robotis

How large are/were the classes?

Ted Shifrin

I believe most of that, Demonark.

He still did some beautiful mathematics, but ... yeah.

Eric Silva

i wonder if there's some systematic issue though, cause when i took complex analysis with charlie smart it was also boring even though charlie is great @Daminark

Daminark

But people felt it hurt the experience, he's teaching difftop next year and many people I know aren't happy about that since they wanted to take it

Eric Silva

but i always hear schlag and danny are great when they teach complex

so idk

Daminark

02:10

@EricSilva I'm doing it with Nori and it's pretty good

Antonios-Alexandros Robotis

I've had some shy professors before. I found that all of them were quite insightful in office hours.

Ted Shifrin

I loved teaching complex ... both undergrad and grad (which I did a bunch of times).

Yes, @Antonios ... for sure.

Eric Silva

i think complex would be dope to teach

Kevin Driscoll

What do you guys think is nice about the complex analysis course?

Ted Shifrin

It's such beautiful material, and those of us who are more geometric/topological can make all sorts of connections. And Banach algebraists can make different connections.

Daminark

02:11

Though it did slow us down a bit too much when he spent excessive amounts of time reviewing really basic stuff around the time of Cauchy integral formula

Kevin Driscoll

I think the results are interesting. But the proofs bore me to tears.

Eric Silva

all the wacky great results and interconnectedness

Ted Shifrin

Oh, the proofs are great.

Eric Silva

damn everyone is sniping me

Antonios-Alexandros Robotis

My undergrad complex analysis course wasn't great.

Ted Shifrin

02:11

And I love normal families.

Daminark

But yeah last class he finished up some stuff on max modulus principle and then started talking about the extended plane

Eric Silva

i think i really like that it's so easy to approach from different angles

Ted Shifrin

Lots of undergrad diff geo courses aren't great, either. That stuff is hard to teach, and some of the standard books overaim for most students.

Demonark, that's not nearly far enough.

Oh, you're still in the middle of the quarter, sorta.

Never mind.

Eric Silva

we're in 8th week

so close to the endish

Daminark

Well, essentially what happened was this

Ted Shifrin

02:13

I taught undergrad complex 5 days a week, but grad courses were only 3 hours a week, but they ran a whole year.

Anyhow, Antonios, welcome to chat and geometry-land :)

Eric Silva

yas bring on the GEOMETRY

Antonios-Alexandros Robotis

Thanks. I might be hanging out here a bit. Nice to have some like-minded people. Oddly enough, NYC can be a lonely place.

Kevin Driscoll

Math Chat: A Geometric Approach

Daminark

First week he reviewed the very basic properties of complex numbers, did differentiation, then C-R equations. Second week he finished that and did Cauchy's theorem, third he did branches of the log, winding number, FTA. 4th finished that, started Cauchy integral formula, and then midterm

Ted Shifrin

smacks Kevin

@Antonios: Big cities are more anonymous.

But I hope the NYU department is welcoming.

Maneesh Narayanan

02:15

2

Q: Pick out the correct choices -TIFR 2015

Let $f:\mathbb R\rightarrow \mathbb R$ be a continuous function and $A \subset \mathbb R$ be defined by $A=\{y \in \mathbb R:y=\lim _{n\rightarrow \infty}f(x_n),$for some sequence $x_n\rightarrow \infty\}$ Then the set $A$ is necessarily A.a connected set B.compact set C. a singleton set ...

Daminark

5th he sorta reviewed chapter 3 of Rudin and continued integral formula. 6th and 7th were stuff like argument principle, Liouville, open mapping, max modulus, similarities, and the Riemann sphere

Eric Silva

@Antonios-AlexandrosRobotis you said you were at NYU right?

Antonios-Alexandros Robotis

that's right Eric

Ted Shifrin

Demonark: This is the undergrad course?

Daminark

Yeah

Antonios-Alexandros Robotis

02:16

I'm not wild about NYU thus far, but there are some nice people in the dept. At least.

Ted Shifrin

Yeah, with your analysis background you could have taken the grad, but we talked about how that was degenerating into probability.

Antonios-Alexandros Robotis

Makes me want to go back to Berkeley.

Eric Silva

oh your school has cool colors

they're on my list for grad school apps

Maneesh Narayanan

How doe the the set is connected

Daminark

Well, Lawler isn't teaching it this year

Antonios-Alexandros Robotis

02:16

If you're into analysis and geom. this is your place.

Maneesh Narayanan

I tried to prove it path connected

Ted Shifrin

@Antonios: Well, Berkeley can be very intimidating and impersonal. When I was a grad student, a lot of the math majors talked to us grad students a lot more than to faculty.

Daminark

Marianna is, she's a geometric measure theorist but I've heard she does a lot of analytic number theory in complex

Ted Shifrin

Yes, Antonios, I have some good old friends there. Say hi to Sylvain Cappell :)

You should have taken her course, Demonark.

Daminark

She mentioned to me that she wants to do prime number theorem, among other things.

Eric Silva

02:17

@Antonios-AlexandrosRobotis those would be my jams

Daminark

I will, in the spring

Ted Shifrin

That's reasonable.

Oh.

Eric's getting pumped for grad school :P

Antonios-Alexandros Robotis

@TedShifrin How does Cappell know everyone !?!?! I have him for Alg. Topology, and he apparently has known everyone...

Daminark

Analysis here is fall real analysis, winter functional, spring complex

Ted Shifrin

We're old friends, Antonios, but he's super friendly.

Eric Silva

02:18

im p excited for mariannaplex

Ted Shifrin

Wait, Demonark, but undergrad complex and grad complex overlap a ton.

I co-ran the geometry/topology yearly conference many years at UGA, Antonios, and Sylvain came and spoke a bunch of times. I think I knew him from before that, too.

Daminark

I just felt like I was focused in the bootcamp on dynamics and the ND conference to the expense of complex, so I didn't feel comfortable going into Marianna's class blind since she's assuming people know undergrad complex

Ted Shifrin

@Kevin: The way holomorphic functions distribute their values is a fascinating thing. There are whole books and bunches of papers on this.

Kevin Driscoll

Complex analysis question: In what context does it ever matter that functions take on every value (minus 1) infinitely many times near an essential singularity

Antonios-Alexandros Robotis

@TedShifrin He's a great teacher. I thought I knew a decent bit of alg. top, but his explanations of Seifert Van Kampen, etc. have been so insightful.

Ted Shifrin

02:20

It's called value distribution theory, and it actually blends into differential geometry.

He's great, Antonios. Do say hi for me :)

Oh, essential singularity.

Big Picard.

Eric Silva

@Ted i imagine it's possible marianna would go crazy fast and get to cool things in her complex class, although ive seen things like pnt before id like to take complex basically for the third time if it's with her

unless there are cooler math electives

Ted Shifrin

I totally endorse the idea of taking courses from great teachers.

But you get stretched pretty thin :P

Eric Silva

true but im not gonna take 3 math classes again this year

im dropping to two a quarter

Daminark

@EricSilva she said she's gonna start from the middle, assuming people know what's going on, so yeah she'll definitely get to the fun stuff

Ted Shifrin

You can wait 'til grad school to be a glutton. :)

Eric Silva

02:22

@Ted I really can't i just see all these classes and get waaaaay too excited

Ted Shifrin

LOL

like a child in a candy store

Daminark

Yeah same. Next quarter I'm taking at least 3, possibly 4

Eric Silva

Schlag's teaching a harmonic analysis class in spring and i literally cant wait

Maneesh Narayanan

2

Q: Pick out the correct choices -TIFR 2015

Let $f:\mathbb R\rightarrow \mathbb R$ be a continuous function and $A \subset \mathbb R$ be defined by $A=\{y \in \mathbb R:y=\lim _{n\rightarrow \infty}f(x_n),$for some sequence $x_n\rightarrow \infty\}$ Then the set $A$ is necessarily A.a connected set B.compact set C. a singleton set ...

real-analysis metric-spaces

My Argument: image of connected set is connected. so A is true

Am I correct?

Eric Silva

@Ted i think child in a candy store is a very apt comparison

Daminark

02:24

Are you gonna do low dimensional topology?

Ted Shifrin

LOL, Eric, I'm not as dumb as I look :P

Demonark, so you're obviously not giving up math for computer science :P

Eric Silva

@Daminark im currently deciding between Minimal surfaces and low dim top, and ill drop by scv to see what's up but im not gonna take that

Daminark

I'm counting algorithms as math in that count @Ted

And this spring I'll do combinatorics+formal languages

Eric Silva

low dim top is cool and different, minimal surfaces is something im definitely learning regardless but Neves already talked to me abt it so idk

Kevin Driscoll

@TedShifrin Yea is Big Picard important for something? I now its sort of weird, I just dont know anything about what it buys you in terms of understanding

Daminark

02:26

It's more like, I'm worried that in stuff like calculus and differential geometry I'll just get eaten alive since I can't navigate well

Ted Shifrin

@Maneesh: I don't see how you get it out of "[continuous] image of connected set" ... but I believe it's correct. You need more explanation.

@Kevin: In general, one wants to understand how holomorphic functions distribute their values. Essential singularities give wild behavior ... Casorati-Weierstrass is the baby theorem, and this is the ultimate result.

Daminark

So I'll see if I can get by more on the lines of functional analysis, algebra, topology, and CS theory.

Ted Shifrin

Demonark, if you go on in math, you will need to acquire more mastery of standard stuff, yes. (In particular, for the GREs.)

Daminark

Is there a field that does all of those subjects in one shot?

Ted Shifrin

No.

Daminark

02:28

Darn

Ted Shifrin

There's $C^*$ algebra stuff in which you can combine functional analysis, topology, and algebra, but ...

oh, and linear algebra (representation theory)

But you can't be a dope about standard calculus and multivariable calculus.

Eric Silva

rep theory so stronk

Maneesh Narayanan

@TedShifrin. Given that $f$ is continuous, so any $x \in \mathbb R$ $\exists$ a sequence converging to x. so $f(\mathbb R)$ is connected. I am correct no?

Ted Shifrin

Whoa, @Maneesh. That's not relevant to the question

We're looking at limits as $x_n\to\infty$

Maneesh Narayanan

@TedShifrin Oh!!sorry. How to prove it is connected?

I tried to prove it is path connected. but I am not able to do that.

Ted Shifrin

02:31

Suppose $y = \lim f(x_n)$ and $z=\lim f(x_n')$. Choose $y<w<z$. Can you find a sequence $x_n''$ so that $f(x_n'')\to w$?

Daminark

@EricSilva "stronk"

Eric Silva

it's so good

triple A stuff

Maneesh Narayanan

@TedShifrin. Give me time. May I think. If I am not able to find. Please help me.

Ted Shifrin

I am leaving in a few minutes, actually.

Think about intermediate value theorem, @Maneesh.

Antonios-Alexandros Robotis

@TedShifrin you mentioned you retired, what are you doing in your newfound free time?

Kasmir Khaan

02:36

in example 2

Cent_Q8 (i) is just <i>

but isint the identity also there?

Maneesh Narayanan

Ok. Now it is clear. Thank you for helping to catch a daemon(IVT) inside the problem@TedShifrin

Kasmir Khaan

or what is it am missing?

Ted Shifrin

You're welcome, @Maneesh.

Antonios-Alexandros Robotis

@KasmirKhaan note that <i> denotes the subgroup generated by i, which implicitly includes e

Ted Shifrin

@Antonios: Still "teaching" a bit here. Also teaching a class for Art of Problem Solving. Also doing non-math things for fun. :)

Antonios-Alexandros Robotis

02:38

Ah that's nice. AoPS is cool. Didn't know they offered classes.

Kasmir Khaan

@Antonios-AlexandrosRobotis oh right >< thanks

Ted Shifrin

@Kasmir: The identity is in any subgroup.

Kasmir Khaan

so it is of order 4 not 2

Ted Shifrin

Lots of stuff on-line, but they just opened a brick-and-mortar school 15 minutes from me in San Diego, Antonios.

Antonios-Alexandros Robotis

Cool!

Daminark

02:40

AoPS: a geometric approach

Antonios-Alexandros Robotis

finally worked out this gradient computation :)

Daminark

Which?

Antonios-Alexandros Robotis

Finding the coordinate formula for the gradient v.f. on a Riemannian manifold

Ted Shifrin

Yeah, @Antonios.

smacks Demonark ... Actually, I am trying to influence them to put more geometric stuff in their trig, which is too sneaky/contest-oriented. So, yeah.

OK, I'm off to a potluck. Y'all have fun.

Antonios-Alexandros Robotis

Enjoy.

Eric Silva

02:43

@Antonios good do div and $\Delta_{g}$ too

unless you've already done em

Antonios-Alexandros Robotis

those are next!

Ted Shifrin

Bye, Demonark, ERic, Antonios, Kasmir, et al.

Antonios-Alexandros Robotis

See you @TedShifrin.

Daminark

See you!

Eric Silva

bye @Ted

Antonios-Alexandros Robotis

02:44

@EricSilva, there's quite a learning curve here with the amount of objects one manipulates.

The indices can be quite interesting.

Eric Silva

yeah it can be tedious but it's also good honest work

Antonios-Alexandros Robotis

They're somewhat satisfying, to be honest haha.

Eric Silva

I like em too

computations like this are all over riemannian stuff

Antonios-Alexandros Robotis

That, I have seen.

I'm relatively experienced with manifolds and so on, but I hadn't done much Riemannian business until this sem.

Eric Silva

I got into manifolds through classical diff geo so this type of stuff was really my starting point

Antonios-Alexandros Robotis

02:47

Definitely the opposite route to me haha

I went Euclidean Diff top ---> Abstract Manifolds ---> de Rham ---> Geometry

@EricSilva Do you know a good book of classical diff geo? I've been thinking about picking up do Carmo Curves and Surfaces for my spare time

Eric Silva

I mean I really like do Carmo, but Ted also has a book that's on the internet

Antonios-Alexandros Robotis

Oh, neat.

Eric Silva

I also really like Montiel and Ros

Antonios-Alexandros Robotis

I've been mostly reading do Carmo's Riemannian geometry, so it sounds like it might synergize nicely.

I guess div. is just nasty without normal coordinates, huh?

Eric Silva

i dont remember the expression not in normal coordinates

montiel and ros is a bit more sophisticated than Ted's book or do Carmo, it assumes you know about Lebesgue stuff for instance

Antonios-Alexandros Robotis

02:54

Measure theory is a nonissue

(though I find it so dry now...)

Eric Silva

sometimes you need to know a loooot of measure theory for some diff geo applications depending where you're working

Antonios-Alexandros Robotis

Yeah, I saw some in a paper a few days ago. (just skimming)

I presume the Riemannian volume form gives you a natural notion of measure?

Eric Silva

i mean that's not really where the heavy duty measure theoretic stuff comes up

Antonios-Alexandros Robotis

Fair, but my point is that there is definitely intersection. Where does it come in in full force, so to say?

Eric Silva

say if you're studying submanifolds that extremize some sort of geometric functional

Antonios-Alexandros Robotis

02:58

Sure

Eric Silva

like minimal surfaces have a lot of GMT and PDE bouncing around

Antonios-Alexandros Robotis

Hmm, alright. (I'm just sort of flipping through my lecture notes haha)

Eric Silva

plateau's problem, isoperimetric probs etc etc

Antonios-Alexandros Robotis

gotcha.

cool, thanks.

Eric Silva

the folk around here are big on that sort of stuff so ive gone a little bit down that rabbithole

Antonios-Alexandros Robotis

03:10

I'm not sure quite how far I'll go with Riemannian geometry, but it's quite enjoyable for the moment, so no need to worry about it.

I think I want to work more on the algebraic side at the moment.

Eric Silva

it's good to learn widely at least

Akiva Weinberger

So there's apparently a combinatorical way to show that the trefoil is not its mirror image

You color its edges in a certain way, and then you give each crossing a number based on the colors going into it

and show that that number doesn't change with the Reidemeister moves

but the two trefoils end up with different numbers

It's weird though 'cause the colors can change at overcrossings as well as undercrossings

Kasmir Khaan

05:55

@TedShifrin Hello Ted

I hope you are still awake :D

user104729

Ted has vanished identically

Kasmir Khaan

haha

what does identically mean here

user104729

(It's not true what I said, hopefully)

Kasmir Khaan

Anyway, how does one find the conjugacy classes of some Groups?

user104729

A function vanishes identically on a domain, if it is equal to zero everywhere on this domain

Well, look at an element, and then conjugate it with everything

Kasmir Khaan

05:57

oh ._.

I meant more like, is there a systematik way ?

user104729

Is that not systematic?

Kasmir Khaan

not very practical ><

user104729

Take an element, conjugate it with everything. Take another element, not in this class, and conjugate it with everything.

Well what's an example group?

Kasmir Khaan

hmm

user104729

With most of your favourite groups, you'll find a 'pattern' and then you can be smart.

Kasmir Khaan

05:59

I was thinking about looking for stablizers

user104729

Consider $\text{GL}_n$ and the Jordan decomposition

Kasmir Khaan

then use orbit stab theorem

user104729

(The invertible matrices of size $n\times n$ from yesterday)

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$Mathematics$ Mathematics