Interesting

Geometric topology is pretty encompassing.

Yeah. Do you have a good delineation of those currently doing differential topology?

I mean, the days of Hirsch-Smale theory are pretty much gone, right?

Well so in the 50's-60's Smale, Kervaire, Milnor and others reduced lots of problems in "differential topology" to purely algebraic topology. People are still working on those questions but it doesn't look so much like what those people did.

Like there was a paper by students of May I think talking about exotic ness of odd dimensional spheres.

Well, I guess we have to call that diff top.

Even though it's orthogonal to the mathematics I ever thought about.

3-d topology really started with Dehn in the 1900's.

and was formalized by people like Kyriakopolous around the same time of those other guys.

Alexander in the early 20th century had algebraic invariants of knots and 3-manifolds that continue to be studied today.

Well, sure, that's what I think of as true geometric topology (and not working so much with derivatives and tangent bundles, etc.).

Where do the people working on classifying 4-manifolds fit in?

That's a mix of differential geometry techniques with topology techniques.

I think in the 70's some important things happened for 3-manifolds and knots particularly on the study of surfaces inside of them.

But that's PVAL's bailiwick.

but I don't know that much.

Then in the 1980's a bunch of stuff happened.

LOL, s**t happened.

Donaldson constructed invariants of four-manifolds arising from PDEs in quantum mechanics.

Lots and lots of papers and conferences :)

Jones constructed a new invariant for knots (with unbelievable utility) coming from operator algebras.

Gromov created the essential tool for symplectic geometry and invented geometric group theory.

I hope everyone's taking historical notes.

and Gromov and Thurston revolutionized the study of hyperbolic manifolds and Thurston really I think first showed that in order to understand 3-manifolds understanding their hyperbolic geometry was crucial.

Haha, this reminds me of the professor I'm TAing for. Every class is at least 25% historical recollections

Floer constructed homology theories from both Donaldson and the symplectic geometry part of Gromov ideas using some hard analysis.

Lots of people (e.g. Mike his advisor and his academic syblings) look at those today in different settings.

25% is probably too much, but I think some history is fabulous and exciting to some students.

Lots of people work hard on these hyperbolic problems in 3-manifolds and generalizations in higher dimensions.

I tried to include some, but I wouldn't put it at more than 5% of total classtime.

So PVAL is emphasizing the interplay with geometry in topology (which really started in the 70s ...) ...

Lots of people do symplectic geometry.

And there's symplectic topology.

Lots of people do geometric group thoery.

So, do we delineate something that's specifically "modern" differential topology? :)

I think symplectic geometry is modern differential topology.

Interesting.

In four dimensions it's a theorem of Akbulut that every 4-manifold has the structure of a broken lefschetz fibration. This is a sort of a 2-dimensional analogue to a Morse function. The way that these are studied is modern diff topology.

I agree that Lefschetz fibrations is one of the overlaps of diff top and alg geo to which I alluded earlier.

collecting so much hw

In 3-dimensions one "Floer theory" you can define is essentially coming from considering a Morse function (Heegard diagram) on a 3-manifold, and the study of the algebra arising is really diff topology.

welcome back to school, Faust :)

i got like 1.5 of 4 done and anthor coming sometime this week

Another thing that happened in the 1980's

Just be thankful you never had me as a prof, @Faust :P I was renowned for tough homework.

is that Thurston and Gabai really showed the obstructions to certain foliations are mainly topologyical.

@PVAL: generalizing the fact that $\chi(M)$ is the obstruction to a $1$-dimensional foliation.

no not really

@TedShifrin well so far all 4 assingments amount to about a workload bi weekly of less than half what i had to do every week just for differential geometry so i think ill be ok

LOL, ok, Faust.

a detail, @PVAL?

i spent 20-30 hrs a week on diff geo hw

and despite my spelling im not actually stupid

:p

My students never did half of that for that course, Faust.

That was one of my easier courses :P

I only required 5 problems every week and a half.

Gabai proved 0-surgery on a knot was foliated by surfaces one of which was the seifert surface glued to the glued in disk.

And they could choose (most of the problems) to do computation, middle, or hard theory.

i did alittle excess i didnt even have to do the last assinment cause i already had all the marks i needed for hw

and the bonus questions were really hard

Old restrictions of Novikov and others (stuff that happened in the 60's) showed that this could distinguish the 3-manifolds.

But i liked the subject makes it alot easier to slog through it

I see, @PVAL, so really not a characteristic class thing at all.

@Faust: Liking a subject and/or liking a teacher are a big deal.

yeah my fav teacher is alittle heavy on hw but hes so awesome that i don't care i still take him over other profs

I'm proud of you for that.

And you learn more.

The open question I hear the most about is actually about the relationship between the Heegard splitting Floer theory, this foliation theory, and group theoretic ideas of the fundamental group.

So the algebra comes to the fore, PVAL.

The first proof of the known direction actually uses contact and symplectic geometry.

That's really cool. Some day I should try to read a survey article on this.

or is contact annd symplectic geometry.

Unfortunately a lot of the surveys seem to omit stuff with geometric background required.

so the connections aren't as evident.

yeah =) also thanks for the help yesterday i did eventually get the whole thing and proved all the little results it was arguably more work than just redoing the proof i already had but i think i learned alot from the exercise.

Floer homology is connected to some of the Witten math-physics type stuff, isn't it?

@PVAL, if you find something appropriate for me, please let me know.

@Semi It's a homology theory associated to equations coming from Witten type stuff as far as I understand.

That's pretty insane

hey @TedShifrin

Hi Nate :)

I need to run away in a minute, but great to see you!

I'll be back later tonight.

No worries, I'm doing some readings tonight!

You hanging in there?!

Hopefully.

Remains to be seen. :)

see you in a bit.

Good. Keep a positive attitude!

Bye all.

oh, $x=x$ is not a statement, but a formula

@AlessandroCodenotti

Does PA preclude the existence of 0.5?

I hope somebody can help me out with applying Bayes theorem. This is the problem I am trying to solve mathb.in/154684, but I can't wrap my head around the incremental updates to the "belief". Not looking for an answer, but I want to understand what is the approach.

I have watched every video on Bayes, and I have defined the events as I see them, but when I put it all together it doesn't make sense

@SteamyRoot On the one hand, I'm so glad that's no longer on the main site. On the other hand, I'm so glad I'm a 10k user and can see deleted questions :)

ams.org/publications/journals/notices/201708/rnoti-p876.pdf

topological structure of 3x3 matrix using singular and polar value decomposition

@Secret fancy

@Semiclassical I looked over the chapter sections and they're just review

well, that should be straightforward then

yeah he didn't assign any of the problem sets, but we have an assignment tomorrow.

he'll be assigning one then? @dodsy

yes, but it probably won't be from a textbook.

neat.

physics lab stuff is ridiculous.

welcome to the land of tedium

ew

Can't say I'm surprised, though. There's a reason I hated TAing for the first semester intro physics: you can only do so many motion problems before it becomes tedious

The way we do motion problems in lab is worth mentioning, though.

We have them use cameras to film the motion on the screen, then go into a separate program. In there they can assign a coordinate system to the video; we then have them go frame-by-frame and mark the location of their image in each.

well so I have to do an online test on measuremnts, if i get bellow 70% I fail outright, then before every lab I have to do a little assignment which for the first assignment is measurement errors and significant digits.

oh that's cool @Semiclassical

in principle, yes

that actually sounds interesting.

I had to pay 45 dollars for this book that you could pay 15 dollars at 10 cents a sheet for.

I think it might be a bit easier this year because now we're able to have them use iphones for that purpose rather than old mini cameras

ugh. reminds me of how we're technically supposed to have students use the official uni lab notebook

but that's stupid.

that's what it is

it's the official uni lab notebook haha

roll eyes

gawd

do you always have to do these labs?

I just stopped caring after a while. (I did ask them to have it be a quadrille bound notebook, though.)

or only first year?

"do" as TA, or "do" as part of the physics major?

as part of physics major

there's a couple physics courses that sound interesting in principle

yes and no. you only do the intro-style labs in the intro courses

but the lab component sticks around in various ways.

:(

for instance, there's a lab component of the sophomore-level quantum physics course. (It's separate from the lecture component, so I never had to interact with it.)

in the version at my undergrad, that involved stuff like the millikan oil drop experiment (which is pretty miserable) and electron diffraction (which is pretty cool)

gosh darnittttt

Eh, if you're thinking you like physics but don't like the intro labs

talk to your TA and see if they know what the higher level labs are like

they probably will be rather different.

(or they can put you in touch with a physics major who knows better, since a TA may be a grad student)

I think I have a hobbyist level interest in physics tbh.

That's fair.

See how you feel at the end of it, though. That'll give you a better perspective.

yeah.

@LeakyNun Sorry for the late post. I still don't get how $((\cos{t})^{2/n}, (\sin{t})^{2/n})$ is equivalent to $(\operatorname{sgn}(\cos{t})(|\cos{t}|)^{2/n}, \operatorname{sgn}(\sin{t})(|\sin{t}|)^{2/n})$ -> desmos.com/calculator/e5tk50h0af

woah haha

hey

againnn

oh hey dude

didn't recognize your picture

lol

how have you been?

shitty-ish

high school just strted

you?

wow hs.

Now you've gotta update your profile from "middle school student"

:P

University just started, pretty cool so far!

what classes are you taking?

Enriched calc, enriched physics and lin alg.

calc is pretty awesome, the teacher is really cool

but it's awesome to just sit around and listen to people talk about math.

yeah, that's fun

how is linear algebra?

Eh, the prof still isn't very organized, but today he did some vectors and some of their algebraic properties. He lied and said that a mistake he made on friday was on purpose to keep us on our toes.

lol

@Dodsy riiiight

i know i've used that link before, but i never get tired of it

Yeah do find a linear algebra book that treats it well

but somebody brought up finding a vector's magnitude by doing the old $\overrightarrow{\rm AB} = \sqrt{a^2+b^2}$ and he said "square root property of vectors? I don't think so"

The subject is like, so much fun, I'd hate if a badly organized prof left a bad aftertaste

@Semi same

eh, that'd work if the displacement vector had components $(a,b)$

yeah it did

@Dodsy i do a lot of programming now

:/

@MeowMix that's awesome dude.

@Semiclassical yeah there's a lot of miscommunication going on.

then $\|\vec{AB}\|=\|(a,b)\|=\sqrt{a^2+b^2}$ is perfectly valid

am not impressed so far

@Daminark the text is some text by poole.

"Linear Algebra, A Modern Introduction"

I want to say that's the book we used, but I'm not upstairs right now

Is anybody here well versed on bayes inference?

christopher poole?

David, I believe.

So Zach, are you taking any computer courses this year, then?

lol

Oops, that one was over my head, sorry zach :)

yea computer programming and i have to stop myself from sleeping and correcting her :(

first day she implies all programming languages are object oriented

Interesting notion.

did you bring up a counterexample?

no, didn't want to be disrespectful

one of the languages she brought up was a counterexample

haha

ew

ew?

Was it LISP?

but i recently picked up node.js and i intend to drop it soon since i finished developing my web application

re: her (counter)example

i never correct a prof during class

i just ask them about it later

I wish I could do compsci but I get so overwhelmed

why is that?

@Faust all of my profs have encouraged us correcting them during class to avoid confusion.

@MeowMix just so bad with technology, tbh.

It comes down to whether you're willing to be wrong in front of the entire class

When my microsoft word crashes I almost throw my laptop across the room.

had one prof for a dynamical systems class i had to show him where he went wrong after every class with his picture and hed correct it at the beginning of the next class =)

use libreoffice ;)

not a geometric individual but really good at what he did

wish he was still around passed away rather suddenly last year :(

@MeowMix my calc prof said we can't use any electronic resources other than the texts he's provided to do any assignments, we aren't allowed to discuss assignments with anybody.

Makes me wonder if he frequents this chat :o

thats not a very good way to learn

i learn alot from teaching/asking questions of my fellow students

yeah.

it's interesting, but I understand why he's doing it.

Yea I can understand the worry about people who will always give too much away and solve the problem for the student

honestly i learn more form teaching than asking questions

learning is collaborative.

all i do in my math class is help other student

but it seems like theres got to be a better solution

because im still in algebra 2 :(

I agree that learning is collaborative, math is collaborative, mathematicians work together often. But I understand that he wants us to become better mathematicians, so I can see why he is doing what he's doing. He's a very good teacher.

I learn maths best by discussion

I do as well, for sure.

@MeowMix it must be very exciting to be in HS though.

not very

it's a new chapter dood

meh

Hope everyone has been well during my absence! :)

so meow mix you like linux now? :)

Always have.

Also, they are the same

Can I see a context where they're shown as different? I may be wrong

I was so confused in highschool

my teachers always made them seem so different

Oh, they're the same

they made us describe k in some way I dont really understand

something by a factor of 1/k

oh man it's just the "explain it to me like Im not studying in this field" short hand. The "easy" tricks always seem so complicated to me, and in reverse the technical things that other people find complicated is simple to me

now I need to study the a(k(x-d))+c formula works and why it always works to display the transformative properties o.o

ohh right I think I understood it before, it's because x is being offset by d so whatever value x -d is computing to could be viewed as an actual x and the rest of the values can be viewed as just really messing with the graph but still why cant a different value be just that, interesting choice we made

oh is x-d for example in a quadratic function inside the x? before the exponent? or is the entire thing inside the x it is isnt it.

@Blue Now I am

Summon Balarka and he appears

is he a pokemon?

Balarkamon

Daminark use quick tackle!

uses quick tackle

@Blue Hm, I haven't seen that before. So if $f$ is differentiable, then $f(c + h) - f(c) = hf'(c) + g(h)$ where $\lim_{h \to 0} g(h)/h = 0$.

balarkamon dodges, and daminark gets stunned as he bashes head first into the wall D:

Daminark return! T.T

I want $\phi(h)^{Ah} = f'(c) + g(h)/h$? This looks weird.

Ow

You did great Daminark we'll get em next time!

I haven't been here for months yet render math jax is still my only bookmark

@Blue Notice that in particular we want $\lim_{h \to 0} h \phi(h)^{Ah} = f'(c)$.

What was the context of the appearance of this?

@MeowMix my teacher said "students commonly make mistakes where for a function that looks like this $f(2x)$ they think the $k$ value is actually $1/2$. I just thought to myself what a face palm, and whose fault is that confusion? why are we using such confusing 'indirect' explanations

@Blue Which looks suspicious because $\phi(h)^{Ah}$ is an expression where $\phi(h) \to 0$ and $Ah \to 0$ as $h \to 0$, so it's a "0^0 indeterminate"; those usually converge to a certain value rather than heading off to infinity with order $1/h$, which it needs to be here for $\lim_{h \to 0} h\phi(h)^{Ah}$ to exist.

FUNCTIONS ARE SO FUN NOW THAT I GET WHAT ALL THE TRANSFORMATIONS MEANS THANK YOU @MeowMix

oh okay so they used horizontal compression and whatever because the $k$ lives inside the $x$ in $f(x)$ and therefore does a direct transformation on $x$

so there must be a distinction between $a$ and $k$

I'm learning :)

@Daminark how is k

k is kek

(i don't know what i'm on about, time to sleep)

lol

Yo @Balarka

oy

@Balarka so to verify that the tangent bundle of a manifold is a vector bundle

Would it be that you find a local diffeo to some open set in R^n?

The idea being that this would locally be an isomorphism of tangent bundles of the open ball and the neighborhood of the point in the manifold

But then tangent bundle of the open ball is trivial

@Daminark Right, exactly.

Sick

Pick charts on your manifold that constitute an atlas. That's a trivializing atlas for the tangent bundle.

Okay so here's a non-trivial example of a vector bundle, that's cool

The easiest non-trivial example is of course the Moebius strip, but yeah

So you know that $TS^2$ is nontrivial right?

I know that this is true but I don't know how to prove it

I meant trivial not officially as much as dumb, and in principle that non-trivial tangent bundles are a thing

@Daminark Take that as an exercise :)

Also understand what direct sum of $TS^2$ with a trivial line bundle on $S^2$ is.

Good story, melon

One potentially sneaky way to do this is to try hairy ball

mhm

Since if it were trivial, I think the map $x\mapsto (x,(0,1))$ would be a vector field

Excellent!

Not only a vector field, but...?

Nowhere vanishing vector field?

Exaaactly.

That breaks hairy ball theorem, like you said.

That feels trolly as shit but I'll roll with it

heh

Also, we know a manifold is metrizable because we can embed it in $\mathbb{R}^n$ and slap on the Riemannian metric we inherit from the ambient space, right?

@Daminark You mean metrizable in the Riemannian sense? Exactly.

Alternatively you can "construct a Riemannian metric by hand".

I mean also that Riemannian metric makes the manifold a metric space, that's what I had in mind more

Actually I don't know the proof of Whitney, how does that go?

That's true, but that it admits a metric in the usual sense is easy by Urysohn metrization theorem

I dunno what that is

Ah, it says every Hausdorff second countable and regular space is (homeomorphic to) a metric space.

I see

@Daminark Technically that you can embed any manifold in $\Bbb R^N$ for some $N$ is weaker than Whitney; Whitney says you can embed it in $\Bbb R^{2n+1}$ where $n$ is the dimension of the manifold.

But let's see.

Suppose $M$ is a compact $n$-manifold for simplicity's sake.

I think we saw in GP how to reduce the dimension of the embedding in the compact case to 2n+1

I think?

Right.

It's a clever "secant variety" type argument

Anyway I'm more concerned with the fact that it can be embedded at all

That's what I'm trying to tell you; let's see if I succeed.

Suppose $\{U_1, \cdots, U_m\}$ is an open cover of $M$ by charts.

Take a partition of unity $\rho_\alpha$ corresponding to $U_\alpha$ (recall that means $\rho_\alpha$ are real functions on $M$ that have compact support inside $U_\alpha$ and $\sum \rho_\alpha = 1$ at each point)

@Dodsy You can read the Feynman Lectures for free online at feynmanlectures.info.

IIRC the candidate for the embedding is going to be $f : M \to (\Bbb R^n)^m \times \Bbb R$ given by $f(x) = (\rho_1(x), \rho_2(x), \cdots, \rho_m(x), h(x))$ for some function $h : M \to \Bbb R$ that I don't recall.

Can I figure out what $h$ looks like? Hm.

Ah, as of now, if $x$ and $y$ belong to the same set of charts and inside the support = 1 region, then the $h$-free components of $f(x)$ and $f(y)$ would agree.

So it's not an injection yet.

@Daminark Oh, ok, I got it. Every chart $U_\alpha$ comes with a chart map $\varphi_\alpha : U_\alpha \to \Bbb R^n$. Take $f : M \to (\Bbb R^n \times \Bbb R^n)^m$ by $f(x) = ((\varphi_1(x), \rho_1(x)), \cdots, (\varphi_m(x), \rho_m(x))$.

Oh I think that works

Oh wait hmm

The $\varphi$ components distinguish points on the same chart; the $\rho_i$ components distinguish points on different charts, being the idea.

I think it's easily checkable that $f$ is an immersion.

Like, over each chart it's literally nothing but inclusion of a graph inside the product of domain x codomain; those are immersions. Locally immersion means globally immersion.

Finally injective immersion of compact manifolds are embeddings.

Aight, this checks out

It's a cute trick

Nothing but taking the graphs of the chart maps over each chart, and gluing it all by partition of unity by passing to higher dimension

@Daminark You can prove using similar ideas that any vector bundle $E/M$ embeds fiberwise in the trivial bundle $M \times \Bbb R^N$ for high enough $N$ using similar techniques. Eg here

(That implies any rank $k$ v.b. $E/M$ embeds in $M \times \Bbb R^\infty$, which of course gives rise to the map $M \to \text{Gr}(k, \infty)$ by sending each point $x \in M$ to the inclusion of $\Bbb R^k \hookrightarrow \Bbb R^n$ that's happening over $x$)

Nice

I meant $\Bbb R^k \hookrightarrow \Bbb R^\infty$ but yeah

Lol, yeah I mentally autocorrected

thumbs

man you should read the first chapter of vbkt

it's so beautiful

I'll check it out

Also I may consider working a bit from Bredon. Partially to get some point-set done, maybe learn a less flimsy difftop

Problem is, it's dauntingly long

ah, I have never tried that book

yeah

why don't you read Hatcher's point set notes lol

it's quick and dirty

math.cornell.edu/~hatcher/Top/TopNotes.pdf

I think one of the reasons I've been avoiding Hatcher in general is that I'm a bit eh on having the theorems and proofs buried in exposition

have you tried to read any nontrivial hatcher so far

or are you basing your opinions on other's words

I mean it's entirely possible that you spent a nontrivial time reading a nontrivial bit of Hatcher and it didn't resonate with you

The only thing I've been taking from others was the worry that he didn't write things precisely. I haven't gotten too far, I just started reading and I was like "Okay this isn't a book I can scan"

well if you freak out after first two pages then you'd have hard time resonating with any book whatsoever

So I won't say I spent a non-trivial amount of time. Probably about 5-10 pages of point-set, less time in AT

ah you read the point-set notes, Ok

that's not his best work imo

I imagine not, it seems to be one of the "Get this out of the way" things, and the subject matter is a bit less interesting

Is there a Riemannian submanifold of $\mathbb{R}^3$ which is locally symmetric, but not a globally symmetric space?

Read ch 1 from VBKT and tell me afterwards if you love it or hate it. That should settle the question once and for all

because it's very representative of his style

The first 30 pages aka

Alright, will do. But I'll say point-set was okay, what really made me meh was chapter 0 is his AT book

I struggled with ch 0 in AT

Like I didn't get far before I was just like m8 w0t?

Anyway I'll give vector bundles a chance

Cool

Hi can someone give me simple idea to find this bound:

1/2^i \sum_{j=0}^{i} |2j-i|\ \binom{i}{j} \leq d \sqrt{i}

d is fixed constant

when i = 10, I found that the summation take the values: 0, 2, 4, 6, 8, 10. So i can instead of writing |2j-i| in summation I write i-2j

Is it Ok!

@MaryStar I think this is "permutation with repetition". So, use it for all, then take out the 3 choose 10.

doing this "when i = 10, I found that the summation take the values: 0, 2, 4, 6, 8, 10. So i can instead of writing |2j-i| in summation I write i-2j" is not possible

Is it $\frac{10!}{2!\cdot 2!\cdot 1!\cdot 1!\cdot 4!}-\binom{10}{3}$ ? @YOUSEFY