Mathematics - 2017-07-09 (page 5 of 7)

Ted Shifrin

17:00

yes, @Semiclassic. Geodesics on any sphere are great circles.

Fargle

The research I'm doing right now concerns Lie groups and algebras, and I kinda wish I hadn't been given the research since I only know the basic definition and some basic examples of Lie groups and algebras.

Semiclassical

Mmkay.

Ted Shifrin

Eric got quiet — I guess that means he's pondering.

Secret

[Chemistry]
Today's progress in chemistry: One of the software technicians have reported back on the error and they said the current software is too old. I have just forwarded that reply to the computing cluster's help desk as they have the new version in the cluster, but it is not installed properly (the executable script is not present in the binary folder while it should)

so much for chemistry for 5 hours, I need to rest a bit with some maths

Semiclassical

Don't know what totally geodesic means, though.

Though the S^2 is a bad mental example since it's presumably not an example of that.

Ted Shifrin

17:02

Totally geodesic submanifold means that any geodesic starting tangent to that submanifold stays in the submanifold.

Spheres won't be good for a counterexample.

Semiclassical

Am I right in thinking that exp in the case I sketched earlier would just give the sphere itself?

Eric Silva

i remember working out an example to this before but im blanking now

Ted Shifrin

you need to be thinking in higher dimensions than 2, Semiclassic.

Semiclassical

Yes, well

Then I've got no hope of saying anything useful at all :P

Ted Shifrin

Not an interesting question for a surface.

Balarka Sen

17:04

Ok, I got too many questions to ponder on

Ted Shifrin

It's all your own fault, Balarka'.

Balarka Sen

lol

I don't mind it at all

SteamyRoot

Fixed point theory is a good way to find totally geodesic submanifolds :D

Balarka Sen

Right, @SteamyRoot

Ted Shifrin

We want something that's not totally geodesic.

Fargle

17:05

Would it be necessary to understand a decent amount of analysis before trying to understand Lie groups/algebras? (Namely, I'm asking, "Should I work through a lot of Rudin before worrying about them?")

Balarka Sen

Fixed point locus of isometry f such that f^2 = id => totally geodesic

I don't think this helps here though, because what Ted said

Ted Shifrin

Not that much, @Fargle. There are algebraic, geometry, and analytic parts of this subject.

SteamyRoot

Oh

Fargle

Hmm. I guess I'm just trying to understand the prerequisites overall.

SteamyRoot

Well, in that case, I know a lot of exampes, but they're way too specific, most likely.

Semiclassical

17:05

I'm having a hard time even understanding the claim, though. But that's presumably b/c I am teh ignorant when it comes to Riemannian geometry.

Ted Shifrin

Depends what aspect you're supposed to think about, @Fargle. I can't advise without lots more details.

Fargle

@TedShifrin I can email you with relevant details.

SteamyRoot

@BalarkaSen Do you even need $f^2 = \operatorname{id}$ ?

Ted Shifrin

OK.

Semiclassical

$\exp(\Lambda)$ is generated by geodesics, but you want an example where one of those geodesics doesn't remain in $\exp(\Lambda)$? That doesn't seem like the right way to understand it.

Ted Shifrin

17:07

@Semiclassic: The issue is geodesics starting somewhere other than $p$.

Balarka Sen

@SteamyRoot Ah, probably not.

No, actually you don't

Semiclassical

Hmm.

Balarka Sen

easy to see that

Semiclassical

One thing which is confusing me, I suspect, is notation.

Where is $p$ in $\exp(\Lambda)$? I presume it's implicit in there and I'm just not aware of it.

Ted Shifrin

We were talking about $\exp_p(\Lambda)$, to be precise.

I said $\Lambda\subset T_pM$.

Semiclassical

17:08

Fair.

Ted Shifrin

Fair for you to query.

Semiclassical

Okay. So one is looking for a case where $\exp_p(\Lambda)$ depends on the choice of $p$? (That doesn't seem quite right either but it seems closer.)

Ted Shifrin

It almost always does. We want to understand the geometry of that surface away from $p$.

Semiclassical

Hmm.

Balarka Sen

We want an example where not every geodesic of exp_p(Lambda) is also a geodesic of M

M being the ambient manifold

Semiclassical

17:11

I think I can see why the surface case would be uninteresting, at least. Each exp_p would just give the entire surface?

Ted Shifrin

That's differently stated, but I guess correct, Balarka'.

Right, Semiclassic.

Eric Silva

you can also find an example where the shape tensor is nonvanishing

Ted Shifrin

second fundamental form, you mean? yeah

so do you have an example?

Eric Silva

yeah

not yet im still pondering

Semiclassical

If I go to M=R^3, would the point be something like: If I pick a point on the z-axis, then the tangent vectors of the form $a\partial_x+b\partial_y$ would give a horizontal plane through the z-axis

Eric Silva

17:14

it didnt take me this long last time hmm

Secret

Oh, great, the cluster have also mistyped one of the paths in the script, thus even if an alternate way to run that calculation is not possible. Guess I cannot do much chemistry calculations today ::goes back to read journal articles::

Balarka Sen

great so the whole chat is now thinking about this problem lol

(I am not complaining)

Ted Shifrin

I still remember that when I taught the beginning manifolds/geometry grad class my first semester teaching at MIT, I put the question on the take-home final to show that sectional curvature was curvature of $\exp(\Lambda)$. My solution (which I handed out) was wrong, because it assumed that that would be a totally-geodesic surface. So this has become one of my favorite challenges.

Semiclassical

lol

Secret

I would have been able to participate in this discussion but I knew nothing about algebraic geometry

Ted Shifrin

17:15

@Semiclassic: In a flat space, you just get linear spaces, essentially, yeah.

This isn't algebraic geometry at all, Secret.

Semiclassical

Riemannian geometry.

Secret

ah ok then

Semiclassical

In that case I can see why different p's and Lambda's would give different exp(Lambda).

Two parallel planes won't intersect etc.

Ted Shifrin

Well, in general, you can't even compare $\Lambda$s at different points!

Semiclassical

But I feel like this is missing the point somewhat.

Yeah.

I think I don't have the toolkit (either analytically or in terms of known examples) to see why a geodesic of $\exp_p(\Lambda)$ would fail to be a geodesic of the ambient space.

Ted Shifrin

17:21

Well, as I said, of course the ones going through $p$ are just fine.

Semiclassical

Hmm.

Eric Silva

maybe you could use a product

Semiclassical

In my R^3 example, is that point that any geodesic in R^3 (i.e. a straight line) intersects a horizontal plane either as a point, as a line in the plane, or nothing.

Ted Shifrin

No, the $\Bbb R^3$ example isn't an example. Nor are things in spheres or any other space of constant curvature.

Semiclassical

No, I'm not trying to say R^3 is an example.

I'm trying to figure out what it means that R^3 is not an example.

Though even on those terms I think what I was saying didn't make sense.

Ted Shifrin

17:25

Any straight line starting tangent to your surface (anywhere) stays in the surface.

Semiclassical

I think in R^3 the reasoning would be that, if I pick two points on a given horizontal plane, then the straight line between them is a geodesic for both the plane and for R^3 overall.

Ted Shifrin

Shrug. Same thing.

Fargle

Thanks, @Ted. Looks like I can no longer tacitly avoid using your book. >_>

Semiclassical

Mmkay. Just trying to put it in my terms.

Ted Shifrin

Sorry, @Fargle :)

Fargle

17:27

Haha, no big deal. I do like your book. I'm just very scattered.

Secret

$\exp_p(\Lambda)$ is the set of all geodesic at $p$ of the $2-$plane $\Lambda \subset T_pM$ and we want to find an $M$ such that $\exp_p(\Lambda)$ don't form geodesics of $M$ but do in $\Lambda$ at $p$?

Semiclassical

I am not trying to solve the problem so much as understand it in the first place :P

Ted Shifrin

Secret. That's sorta muddled. The question is: if you pick $q\in\exp_p(\Lambda)$, $q\ne p$, must geodesics starting tangent to the surface $\exp_p(\Lambda)$ at $q$ stay in that surface? Or, as Balarka said, must the geodesics through $q$ in the surface coincide with geodesics in the ambient manifold?

Semiclassical

As I said earlier, alas, my toolkit of relevant examples is narrow as heck.

Ted Shifrin

You're not alone. This is a non-obvious question, I think. Presumably @EricSilva has an example now.

Secret

17:40

I am thinking about how to make geodesics in $\Lambda$ away from $p$ to "diverge like crazy"...

(assume I understood correclty) here, only those geodesics through p will coincide with the manifold M. Now to figure out what M is

Eric Silva

@Ted shouldn't there be such a plane in any manifold of nonconstant curvature

Secret

Are the only geodesics on a torus the horizontal and vertical circles?

SteamyRoot

Definitely not

Eric Silva

i think there's a theorem of Cartan to the effect of: if there's a totally geodesic submanifold tangent to every $\sigma \subset T_{p}M$ then $M$ is constant curvature

lol suuuuuper no

Mike Miller

@TedShifrin I never caught that subtlety before

Eric Silva

17:43

in which case there are loads of examples

SteamyRoot

@Secret You have geodesics spiralling across the torus, or wavy lines contained in an annulus on the surface

Secret

ah I was suspecting about the spirals, but I never though of the wavy annulus ones

SteamyRoot

and also geodesics that converge to parallel circles...

Mike Miller

@Ted Can you change the metric to third order to force the existence of totally geodesic submanifolds?

Secret

Trying to find a manifold with very wavy geodesics...:

Ted Shifrin

17:53

EricS: yes, and I think you'll find that result on one of my homework assignments :P

I doubt it, @MikeM.

Eric Silva

yeah last time i did this i thought of products immediately, they're usually good for this kind of counterexample

Secret

Is it possible to find a manifold whoose geodesic behaves like $\sin(\frac{1}{x})$ (made continuous by specifying f(0)=0$)?

Eric Silva

ah i think it's not hard to show that theorem just playing with jacobi fields

Ted Shifrin

@EricSilva: See problem 4 on assignment 3 from my Riemannian geometry homeworks.

Eric Silva

ah i see

Balarka Sen

18:02

Sorry I had to afk for a while

so, did we figure it out?

Ted Shifrin

I don't know. Did "we"?

@Secret: You don't mean what you wrote there with $\sin(1/x)$.

Balarka Sen

@EricSilva Oh cool

Ted Shifrin

But remember you have to have geodesics in every direction at every point, and things vary smoothly.

@Balarka': That's the theorem whose proof is the exercise I mentioned above.

Balarka Sen

I see

Secret

O smooth, right, then geodesic which "shape like" a continuous version of $\sin(\frac{1}{x})$ is not possible since it is not differentiable at $x=0$

Balarka Sen

18:05

Ah, I asked MM, @Ted. He says the complex hyperbolic plane works.

Ted Shifrin

You meant $x\sin(1/x)$ or some $x^k\sin(1/x)$, but ... yeah, smooth.

Anything not of constant curvature will work :)

You just need to figure out an explicit example.

Balarka Sen

Right, I wanted a visual example. He says if I exp project a 2-plane which is not totally real or a complex 1-dimensional subspace, then I get something that isn't totally geodesic.

Though of course it's very interesting that literally anything works

Ted Shifrin

The same should work in $\Bbb CP^2$.

We know what the geodesics are, etc.

Balarka Sen

Ah, good point

Secret

I suspect a torus is not going to work. Even though it has regions of positive cuvature, regions of zero curvature and regions of negative curvature, in each region where the curvature are of the same sign, it is constant

Ted Shifrin

18:09

Both Eric and I had the idea that you should think about a simple product (that is not constant curvature).

Eric Silva

the first place my mind went to prove the theorem was to use $J'' + R_{J, \dot{\gamma}}\dot{\gamma} = 0$ and then Schur's theorem

cool theorem tho

Ted Shifrin

Not so, @Secret. I really don't understand.

EricS: I have never been a Jacobi field person :P

More evidence that I'm NOT a Riemannian geometer.

Eric Silva

i think the jacobi equation is p cool

Ted Shifrin

Sure thing.

Secret

Ted Shifrin

18:10

Sturm comparison, etc.

@Secret: But totally NOT constant anything.

Balarka Sen

I am very content with CP^n as an example. Thanks all, for all the very interesting facts

Eric Silva

i like to keep $S^{2} \times S^{1}$ and $S^{2} \times S^{2}$ high on my list of possible counters to things

Balarka Sen

@Eric I am not sure if I understand the geodesics of either very well

Nah, for the first, I do. It's covered by S^2 x R

Secret

$S^1 \times S^1$ is the flat torus, thus should have zero gaussian curvature everywhere?

Semiclassical

What I always wonder when I see "Jacobi field" is why it's called that.

Ted Shifrin

18:13

I can cheat for $S^2\times S^2$ and write it as $\tilde G(2,4)$, so it's a symmetric space.

Balarka Sen

OHHH, right

Eric Silva

$S^{2} \times S^{2}$ has a totally geodesic torus @Balarka so that's cool

Ted Shifrin

@Secret: Depends on the metric!! If you are thinking of it in $\Bbb R^2\times\Bbb R^2$ in the obvious way, then, yes, flat.

Presumably because Jacobi thought of them in terms of varying geodesics, @Semiclassic.

Semiclassical

Hmm.

Eric Silva

I assumed it was cause they satisfy the Jacobi equation but that might not be true

Semiclassical

18:14

Mostly I wonder about the prehistory of Riemannian geometry.

Ted Shifrin

well, why is the Jacobi equation called that?

Eric Silva

good question

MrAP

How do you find the range of $f(x)=\log_e \sqrt(4-x^2)$. I am getting $\R$ but the correct answer is $(-\infty,\log_e 2]$.

Secret

I see, yes, I tend to think of it as being embedded in $\Bbb{R}^4$

Eric Silva

i am not usually confident that things are named after the right people

Ted Shifrin

18:15

Spivak addresses some of it, Semiclassic. After all, he includes translations of Riemann, etc.

True, Eric.

Semiclassical

Hmm, nice.

Balarka Sen

@EricSilva I'll bite you for not knowing the history

Ted Shifrin

Well, Secret, most of us think of it in $\Bbb R^3$. :)

Semiclassical

Yeah, Arnold's principle and all that.

Danu

^^

Ted Shifrin

18:15

heya @Danu !

Danu

Hi Ted

Las Vegas Raiders

Hi chat

Ted Shifrin

Wow, Danu and Manu ...

hi skull

Eric Silva

@Balarka tbf when i say we should know history i mean more history of ideas than i do history of who named things what

Semiclassical

I mostly wonder to what extent classical mechanics served as a historical motivation/inspiration for Riemannian geometry.

Balarka Sen

18:16

@Eric i know i was pulling your leg :P

Eric Silva

i am aware

Balarka Sen

why so serious

Ted Shifrin

Thinking of geodesics in terms of the energy functional, sure, Semiclassic.

Danu

@TedShifrin Who's Manu?

Ted Shifrin

I'm not sure if that was the first approach.

Manu Schiller just entered, @Danu.

Danu

18:17

@Semiclassical Hint: Ask on History of Science and Mathematics!

Eric Silva

whooooaaaaa

Semiclassical

lol

Secret

I can visualise the tesseract, the pentachoron, $S^1 \times S^1$, $S^2 \times [0,1]$, Klein bottle, $S^3$ and $T^3$ as close as possible to it in $\Bbb{R}^4$, but there's a lot of mathematics and geometry I still don't understand, and I expect to have those gaps all filled in once I get into differential geometry

Ted Shifrin

EricS: Such utterances are so informative.

Danu

The real question is: Why are there no tennis matches today :(

Eric Silva

18:18

i was informing you all of my delighted surprise

Semiclassical

What I have in mind is the dimensionality of phase spaces in classical mechanics.

Ted Shifrin

Because Wimbledon always gives the lawns a rest on the middle Sunday. (Unless there have been massive rain backups.)

Danu

@TedShifrin So it turns out the Riemann surfaces course is kinda... ehh... We're not even going to do Riemann-Roch :\

Ted Shifrin

Bah, @Danu. I fail him.

Eric Silva

@Balarka i default serious bc internet unseriousness is hard to read

Semiclassical

18:19

I mean, if you've got a particle confined to the surface of a sphere, the configuration space is 2D but the phase space is 4D. So just by thinking in that way it seems like one is forced to go to spaces of higher dimension than 3.

MrAP

How do you find the range of $f(x)=\log_e \sqrt(4-x^2)$. I am getting $\mathbb{R}$ but the correct answer is $(-\infty,\log_e 2]$.

Danu

No RR, and no surfaces associated to algebraic functions

Las Vegas Raiders

@Danu good question

Danu

That leaves us with the torus and the sphere :D

Ted Shifrin

This is what happens when someone too removed from algebraic geometry teaches it, @Danu.

Danu

18:19

I even tried to sneak in an exercise showing that any oriented surface will do but it was veto'd :P

Ted Shifrin

It's like people who teach undergraduate curves and surfaces and don't get to Gauss Bonnet. Makes me so angry.

Eric Silva

omg why would you ever do that

Ted Shifrin

(Or multivariable calc and don't get to Green's/Stokes's, etc.)

Danu

Everybody will be left with the feeling of: Why did I do this?!

Ted Shifrin

Because people don't plan or time things right, and they may or may not be experts on the material.

Secret

18:20

Most other 4D shapes, I cannot visualise or draw their projections without incuring significant error

Danu

By the way, Ted, I have finally come to appreciate the Weierstrass P :D

Ted Shifrin

Ah, cool, @Danu.

Danu

I have done so many exercises on that this semester

The only negative feedback I got on the student evaluations was a one-sentence critique saying "Less Weierstrass P".

Eric Silva

gauss-bonnet and theorema egregium are like my must do theorems i think

Ted Shifrin

I usually did it in the regular part of complex analysis and then reintroduced it during Riemann surfaces.

Secret

18:21

@MrAP think where log will be undefined

Las Vegas Raiders

@JyrkiLahtonen welcome

Ted Shifrin

I'm just saying, Eric. There are lots of badly constructed/taught courses out there.

Eric Silva

so sad :(

Danu

The problem in this case was really that the teacher didn't know the theory himself

It's not hard for him but he had no overview at all before starting... No plan

Ted Shifrin

@Danu: One of the best courses I ever taught was a year-long applied math course back in 1986-87. I learned over half of it as I taught the course. But I guess I worked a lot harder at it than the average university prof bothers to.

Danu

18:22

He was just reading the material from the book a few weeks before teaching it

@TedShifrin Hmm.. Maybe.

You are probably way more into teaching than most, yeah.

Ted Shifrin

Yes, but I was also excited to learn that stuff and teach it well. ... Surely reading the table of contents or preface should give someone perspective on the Riemann surfaces course. Or talking to colleagues ...

Eric Silva

we should have a riemann surfaces course here

but alas

Danu

I think I was probably among those who learned most this time :)

Ted Shifrin

Well, if you're going to have probabilists teaching complex analysis ... :)

Danu

I found this super amazing complex analysis book (2 volumes) by Freitag

Ted Shifrin

18:25

I think they recently used that at UGA.

Eric Silva

when i take the grad complex course it will be a geometric measure theorist teaching it

Ted Shifrin

I don't know the book.

Danu

I really like it. It sorta does Riemann surfaces but completely without "technology" so no sheaves and stuff

Very classical... I like it

Semiclassical

Sounds like the kind of book I'd appreciate.

Danu

It does applications to number theory, also.

Ted Shifrin

18:26

I remember when one of our analysts complained back in the 80s the first time I taught graduate complex analysis that I wasn't competent to teach it.

Um, complex geometer? Really? ... :D

Danu

I'm finding reasons to be interested in number theory/algebra recently. That's good becuase I probably have to spend time learning algebraic geometry...

Eric Silva

rip

Ted Shifrin

Well, so does Stein/Stakarchi, Danu. Meh.

Daminark

@TedShifrin Yikes

Also hey everyone!

Balarka Sen

hihi

Eric Silva

18:27

i think im not gonna have a mathy day today

Danu

If I don't actually find any PhD positions I'm gonig to work my ass off on algebraic geometry stuff in the coming months.

Ted Shifrin

Keep me posted on news, Danu.

Typhon

0

Q: Tangential proof regarding the Collatz Conjecture formula.

Let us start with the set: $\{1\}$. Every iteration replace every element $x$ in the set with two elements $2x$ and $\frac {x-1}{3}$. Conjecture: The sum of every element of the set formed by the Nth iteration is $2^N$. I was just messing around with reversal of the Collatz Conjecture and ...

Danu

@EricSilva Same here! Got up at 7:30 to play tennis, arrive at the courts at 9, started raining. Moved tennis to 12:30, went to play squash 11:15-12:15 then played tennis 12:30-2:30

Haven't managed to be productive since getting home :\

Ted Shifrin

Rain again, Danu? Geez ...

Danu

18:28

Yeah, it's been super unlucky: Every day but Sunday seems fine!

Ted Shifrin

LOL

I thought I was the only jinx.

Eric Silva

i just have a migraine

Danu

I lost that tournament match btw Ted

but the guy is now a training partner of mine

Ted Shifrin

Sorry if it's my fault for asking you questions, Eric :)

Danu

and I beat him for the first time today!

Eric Silva

18:29

nah i woke up with it

Ted Shifrin

aww, well that's a cool result, Danu.

Danu

6-3 7-5, came back from 1-4 in the second

Semiclassical

I gotta wonder how the tennis player with the knee injury is faring.

Eric Silva

usually it helps to cook good food but i have no groceries

Semiclassical

But that probably won't be clear for quite a while.

Ted Shifrin

18:29

She has more doctors to see in the US, last I heard, Semiclassic.

Well, go get some and cook, Eric.

Balarka Sen

I have a fractured finger. I am coming up with impressive stories to tell about it's origin.

Semiclassical

Makes sense.

Danu

Had an epic point serving at 4-5, 15-30 down. He managed to come to the net and I handed him a smash, but then hit a forehand winner down the line, on the run. That really swung the momentum in my favor!

Ted Shifrin

Not to mention ITS origin, Balarka'.

Typhon

@BalarkaSen does it hurt?

Eric Silva

18:30

that's the plan @Ted

Danu

...would've been two set points down had I lost that point.

Balarka Sen

A deadly fist fight with a neobourgeois on Hegelian dialectics is my current story

Typhon

...

Ted Shifrin

Danu, no one can accuse you of lack of self-confidence :)

Typhon

the truth?

Danu

18:30

@TedShifrin Well... That's just in tennis ;)

Eric Silva

@Balarka i broke my finger free climbing a cliff once

i didnt realize until i was writing an essay the next day

Typhon

@EricSilva bad or minor?

Eric Silva

relatively minor

Typhon

oh ok

Semiclassical

"The thesis was his smug face, the antithesis was my fist, and the synthesis was a broken finger."

8

Typhon

18:31

currently dealing with a radial neck fracture in the e;box

Balarka Sen

Gold

Thanks for supplying me with this

Typhon

it doesnt hurt except when i do something I shouldn't and then that's literally the equivalent of my body slapping me in the back of the head for being an idiot

Eric Silva

im out

bye chat

Ted Shifrin

Bye, Eric. Save me some food :)

Las Vegas Raiders

Cya

Ted Shifrin

18:33

OK, I'm going out to lunch with a friend. See y'all anon. Oh, haven't seen tern in a while!

Las Vegas Raiders

Bye bye

Ted Shifrin

see ya, skull

Akiva Weinberger

For 3B1B fans

Balarka Sen

Bye geometers

Akiva Weinberger

(Q&A)

Las Vegas Raiders

18:34

Cya

Typhon

@AkivaWeinberger scroll up to see a neat number theory question I posted

Secret

Given $f(x)=\frac{x}{3}-\frac{1}{3}$ $$f^n(x)=\frac{x}{3^n}-\sum_{k=1}^n\frac{1}{3^n}=\frac{x}{3^n}-\frac{\frac{1}{3}‌}{1-\frac{1}{3}}=\frac{x}{3^n}-\frac{1}{2}$$

Given $g(x)=2x$ $$g^m(x)=2^m x$$

Any cross term has the form: $(f^n \circ g^n) (x)$ or $(g^m \circ f^n) (x)$

Akiva Weinberger

@Typhon It's in a random order

It shuffles every time you load the page

Daminark

Back to algebra finally

SteamyRoot

Woo algebra

Daminark

18:40

By which I mean dynamics because holy crap this problem

SteamyRoot

@Daminark Way to ruin my excitement :(

Balarka Sen

lol

Daminark

I mean also algebra

Dynamics has a different vibe than I thought it would

Secret

$(f^n \circ g^m) (1)= \frac{2^m1}{3^n}-\frac{1}{2}$

$(g^n \circ f^m) (1) = \frac{2^m1}{3^n}-2^{m-1}$

Daminark

There are a few main examples we're looking at, the automorphisms of the torus, solenoid (which I don't really get), horseshoe which is apparently conjugate to symbolic dynamics and shifting, and rotations of the circle

Semiclassical

18:43

symbolic dynamics are neat

SteamyRoot

"automorphisms of the torus" ?

As in, of the fundamental group of the torus?

Balarka Sen

he means self-diffeomorphisms

automorphisms in the DIFF category

Daminark

Think of the torus as $\mathbb{R}^2/\mathbb{Z}^2$

Semiclassical

Though by that I mostly mean "it's neat that numbers in the Cantor set = numbers that don't need 1 when written in base 3."

Daminark

Literally a quotient group

SteamyRoot

18:45

Ew, diffeomorphisms

I'll stick with homeomorphisms :3

Daminark

Then if you have an integer matrix, you get an induced action on the torus

Akiva Weinberger

@Secret You evaluated the finite sum like an infinite one

Daminark

Now, if we ask that it's got determinant 1, it becomes an automorphism of the torus

Akiva Weinberger

which is why it fails at $n=0$

Secret

ooooooops....

SteamyRoot

18:45

@Daminark Well, yeah, that integer matrix will just be the automorphism of the fundamental group of the torus, no?

Also, shouldn't determinant -1 be allowed too?

Daminark

Oh right, yeah

Balarka Sen

@SteamyRoot homeomorphisms are yuckier than diffeomorphisms. but it seems Daminark means group automorphisms of the torus

Daminark

And I mean, I guess that is true as well, but we're looking at the torus itself as a group also

Induced by automorphisms of its fundamental group, much as we aren't quite framing it in those terms

Though we're considering those that are called hyperbolic toral automorphisms

Balarka Sen

@Steamy I don't get it. There are distinct automorphisms of T^2 which induce the same thing in pi_1

Daminark

Meaning, we don't want any eigenvalues on the unit circle. This has the effect of making them real if we ask $|\det(A)| = 1$

Balarka Sen

18:47

why do you want to identify it with anything happening on pi_1?

Alessandro Codenotti

hi chat

Sha Vuklia

hi chat from me too

Daminark

And also apparently the eigenvectors have irrational angles?

Secret

but still, computing those don't really help because $f$ and $g$ don't commute thus $fffffgggg$ differs from $ffggggfff$, and to compute the sum in typhoon's question, you need to summ up all possible strings of f and g. Otherwise, some clever number theory stuff is needed which I don't know

SteamyRoot

A lot of stuff will be the same up to homotopy

Balarka Sen

18:48

You don't want to do that

SteamyRoot

Restricting to the fundamental group means you merge a lot of those homotopy-equivalent stuff

Also, calculations on the fundamental group are easier :P

Daminark

Anyway, so what happens is that you can take a point $x$ and define $W^u(x) = \{x + tv_{\lambda} \mod 1 | t\in\mathbb{R}\}$

Balarka Sen

Unless you want to study the mapping class group, which is an entirely different topic

@Steamy Yes, but utterly useless in dynamical contexts

Daminark

Where $v_{\lambda}$ is the eigenvector associated with eigenvalue $\lambda > 1$, the other one being $v_{1/\lambda}$

SteamyRoot

If $f$ is a self-map of the torus, and $f_*$ is the induced automorphism on the fundamental group, then the number of $f_*$-twisted conjugacy classes is a lower bound for the number of fixed points of $f$ :D

Semiclassical

18:50

This starts to sound like KAM theorem stuff

Daminark

@Balarka I think the fact that we're not framing it in terms of fundamental groups is that the book doesn't want to assume you know any algebraic topology background? It may very well be possible that, as is often the case, weird shit connects fields

Balarka Sen

MCG(T^2) on the other hand (which is homeomorphisms of T^2 upto isotopy) turns out to be isomorphic to Aut(pi_1(T^2)) = SL2(Z), yeah

SteamyRoot

GL_2(Z)

Balarka Sen

@Daminark Well, it's not terribly useful to identify homotopic diffeomorphisms if you want to study the dynamics of a map like Steamy said

That's not relevant to the kind of business you want to do

In general maps with interesting dynamical properties tend to get dumb if you perturb it a little

Semiclassical

For reference, KAM is this: en.wikipedia.org/wiki/…

Daminark

18:55

I think I'll just table the torus problem though, even when not braindead (or at least not due to sleep) this isn't registering well with me at all

SteamyRoot

I'm guessing you'll learn about Anosov diffeomorphisms and Arnold's Cat Map, if you're already working on the torus?

Daminark

Potentially

Right now, we're doing topological dynamics, so stuff like transitivity, mixing, expansiveness, entropy, etc

Balarka Sen

@SteamyRoot Oh yeah right. I meant the orientable MCG I guess

Daminark

This Wednesday I'm gonna give a lecture on applications of topological recurrence to Ramsey theory

Especially this one theorem called van der Waerden

For each finite progression $\mathbb{Z} = \bigcup_{k=1}^m S_k$, one of the sets $S_k$ contains arbitrarily long finite arithmetic progressions

So that's something to look forward to for sure

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$Mathematics$ Mathematics