point-set-topology

[abstract saffron]

Nah, for Nobel you need to have it making a non-trivial verifiable prediction

[tiny ridge]

all good but i think these folks went too deep into string theory and now everything will be experimentally verified 10 years later

[abstract saffron]

They hate theoretical stuff

[tiny ridge]

oh 10 years past? 10 more

and so on

you know what the string theorists say when CERN returns emptyhanded? they say the particles that were supposed to be created were created, just in another dimension

lol

[abstract saffron]

No shit

[tiny ridge]

coz they have 22 extra dimensions

literally this is the excuse

also they say were not using enough energy

youd need to build a galactic particle collider to verify what theyre talking about

and even then particles can go into an extra dimension lmao

[abstract saffron]

They burn electricity from 3 countries, wdym not enough energy?

Bruh these people, they want a Dyson sphere or what?

[patent quarry]

??

[abstract saffron]

Noble prize is only for experimentally verified results

[patent quarry]

did you even read what i said?

[abstract saffron]

I did

[patent quarry]

are you agreeing with me?

[abstract saffron]

You can build all models you want, but if they don't make experiments, good luck with that

[patent quarry]

???

[abstract saffron]

Yes, I am, altho I don't think I understood your wording

[tiny ridge]

i just love the idea that hep.ex people are sitting in CERN, working day and night in front of kilotons of machines, colliding particles together to bang into one of these 22 extra dimensions

big troll by big chief witten

[abstract saffron]

When all goes south, they can still win a Fields medal 🤷

[tiny ridge]

tbh string theory is underfunded these days

mostly because talking heads like brian greene and michio kaku has been trolling the televisions since 3 decades saying this will lead to the grand unification of gravity and qm and nothing happened lol

the big thing now is stat phys

or quantum information

which are both scam in their own right, just with a lot of buzzwords so you wont know

i like math because its obviously scammy

purest of all ponzi schemes

[urban zinc]

Hey I'm taking a class in that next semester!

So excited

[abstract saffron]

✨ quantum ✨ computing ✨

What a scam

[tiny ridge]

huh, khovanov homology detects the unknot

[abstract saffron]

Yes?

and Floer detects the genuses of the knots, doesn't mean it's computable tho

[tiny ridge]

isnt Khovanov super computable

i thought that was Manolescu-Sarkar

[abstract saffron]

I only know it is for a given diagram

[tiny ridge]

yeah thats what i meant

[abstract saffron]

Yeah, but sometimes we can't rely on diagrams, no?

😄

Alexander polynomial gives bound on the unknotting number, but we still can't compute it efficiently

Unknotting number of 10_21 is still open iirc

[tiny ridge]

huh

[tiny ridge]

is there a geometric proof of this or is it skein manipulation

[abstract saffron]

there should be a geometric proof, no?

Taking alexander polynomial as the determinant of seifert matrix

Then the degree trivially gives bound on the unknotting number

Seifert surface is really OP in this case

[tiny ridge]

Ah good

very nice

[abstract saffron]

It's only a mess because the bound depends on the diagram, and we don't know what diagram gives best bound

[tiny ridge]

makes sense

[abstract saffron]

But for alternating knots, Seifert's algorithm does give best bound iirc

Wait, I thought you studied knot theory?

Cus holy hell, you know much more than I do

[tiny ridge]

Nah, just know bits and pieces that I needed to learn other things.

freudian slip

[abstract saffron]

basically me 😄

[tiny ridge]

Sort of makes sense

[abstract saffron]

I don't recall the proof tho 😄

[tiny ridge]

must be some combinatorics. i was thinking, alternating knots, for the most part, is hyperbolizable

the minimal area seifert surface should be the minimal genus gadget

and seiferts algorithm feels like producing a minimal surface

just a feeling, really

[abstract saffron]

Ugh, I don't think so, idk

it was only recently that we constructed seifert surface with differential geometry iirc

My intuition is that for alternating knots, it just happens that seifert's algo actually makes the minimum number of twisting edges to connect the discs

Then the rest follows at once

[tiny ridge]

That also sounds right

probably much closer to an actual proof

There’s a way to “cubulate” an alternating knot, let me see if I can find an image

The union of the planar regions and the square pieces at the crossings is the Seifert surface produced by Seiferts algo, I think

[abstract saffron]

I have deliberated avoided touching satelites and complements because I don't wanna touch hyperbolic stuff, lol

[tiny ridge]

super useful I think but field sorta dead

[abstract saffron]

Well, yes, but you could have done it much simpler 😄

[tiny ridge]

What I’m trying to see is if one can see that this is minimal genus from the structure of the cubulation

the cubulation above is very “special”

[abstract saffron]

which paper is this?

[tiny ridge]

its a book

from riches to RAAGs by Dani Wise

[tiny ridge]

@abstract saffron Hah, OK, this is funny. Suppose the alternating link was actually fibered ie theres a fibering by Seifert surfaces. Then this is true by a well-known theorem of Thurston.

We circle back to where we started from

[main briar]

can anyone tell me if there is a compelling formula that uses the divergence theorem to relate the surface integral of a solid not in a vector field to its volume?

[abstract saffron]

sounds like a #diff-geo-diff-top thing

[tiny ridge]

(The theorem states that if you have a taut foliation by surfaces on a 3-manifold with boundary, everywhere transverse to the boundary, then the leaves are minimal genus)

[abstract saffron]

I'll google myself what a foliation is

[tiny ridge]

Very easy, its like a generalization of a fiber bundle. Any fiber bundle gives an example of a foliation

Taut just means theres a closed curve transverse to all the leaves

[abstract saffron]

me, not knowing what is a fibre or a bundle

[tiny ridge]

Oh ok sorry, didnt realize.

[abstract saffron]

mind you, I've done only very basic diff geo

it's OK. I recall it's something resembling a product of something else

[tiny ridge]

A foliation of a 3-manifold M by surfaces is like writing M as a union of (a continuum many) surfaces, such that locally it looks like R^3 written as union of continuum many R^2’s stacked on top of each other

[abstract saffron]

Sure, why not? 😄

[tiny ridge]

Taut means there’s a closed curve which is transverse to all the leaves of the foliation

https://upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Reebfoliation-ring-2d-2.svg/300px-Reebfoliation-ring-2d-2.svg.png

[abstract saffron]

wait, this is not trivial as I thought it was

[tiny ridge]

This is not taut

(the red boundary circles are leaves also)

[abstract saffron]

why is something seemingly simple can be this deep?

Well, ig it goes for everything in math

[tiny ridge]

Thats cool right?

[abstract saffron]

That's really deep, ngl

[tiny ridge]

Well, but I’m not sure how to do it for an arbitrary alternating knot

Fibered alternating knots are a subclass of alternating knots.

[abstract saffron]

I feel like there must be another way 😄

it cannot be this deep

[tiny ridge]

Most likely, theres multiple ways

Usually you can hammer things in knot theory with post-Thurston technology

Thats why hes so famous lol

[abstract saffron]

I am happy that papers in knot theory are from 20th century

I'm so done with stuff whose authors are all dead

But this doesn't make knot theory any easier

[tiny ridge]

True!

[nimble portal]

Can someone give me a physical description for how to construct a Klein bottle

[hidden crag]

you take a very small bottle

[nimble portal]

Like for a torus I would say "take a rectangle and glue two opposing edges together so it looks like a cylinder, then pull the circular ends together so they meet"

[umbral panther]

No
https://ymsc.tsinghua.edu.cn/__local/3/3F/62/1293A0A1D1BE2BC710A48385251_F41A5764_34E7A6.pdf

[nimble portal]

and...?

[hidden crag]

it was a joke

klein = small in german

[nimble portal]

Oh

LOL

I guess I can sort of see how

So do a similar construction to the torus

[urban zinc]

The Klein bottle is not embeddable in 3D

So you can't really give a physical description without the paper passing through itself

[nimble portal]

Except instead of pulling the two circular ends of the cylinder together, move one end so it's concentric with the other (but not overlapping), then "pull out" more material from that end until it meets the other end

[nimble portal]

Ya

[urban zinc]

https://i.pinimg.com/originals/80/b6/19/80b6199992ea7200f74d2111c27a0bc4.gif

[nimble portal]

Oh

Similar to what I had

That's interesting

Why is that a meaningful shape? Lol

[urban zinc]

Do you know the classification theorem for surfaces?

[nimble portal]

Or do I just think it's meaningful because I see it in popmath

I don't

Are you talking about the Euler characteristic formula thing

V + T + E = 2 or whatever it is

[tiny ridge]

Yeah I forgot Jones discovered it before Witten pointed out the relationship with gauge theory. Someone already pointed that before I think

[urban zinc]

Okay essentially there are four different ways to connect together the ends of a square

The Klein bottle is one of them

[tiny ridge]

I don’t really know the quantum algebra picture at all

[urban zinc]

it turns out that any (connected and compact) surface is homeomorphic to the sphere, the connected sum of tori, or the connected sum of real projective planes; the klein bottle is the connected sum of two real projective planes

[nimble portal]

wtf...

[urban zinc]

So essentially connected compact surfaces are like Z

the sphere is 0, a torus is 1, connected sum of two tori is 2, RP is -1, klein bottle is -2, etc

I think

[nimble portal]

Is there a weaker version for paracompact surfaces?

[urban zinc]

I might be messing up the details

[urban zinc]

Dunno

[tiny ridge]

Except (Klein bottle) # RP^2 = (torus) # RP^2, but -3 is not 0!

Truth hurts!!

[urban zinc]

Oh it's a commutative monoid with the relation that RP # Klein bottle = RP # torus

I lied

[nimble portal]

uh

So say I'm trying to prove surjectivity of an injection

I need to show for an injection f: X -> Y that for every y in Y there exists an x in X such that f(x) = y

If I find a general expression for x given y (and show that this x always exists in the domain) and show that f(x) = y

Is that enough to say that my formula for x is the inverse of f?

[tiny ridge]

Pretty much

[nimble portal]

Like this

patiently waiting to get flamed for using word

Here was the problem

[tiny ridge]

Its OK honestly

Anything more seems like more work for little gain

[nimble portal]

I agree

Computing sigma^(-1) circ sigma looks like a nightmare so this was my way of copping out

[tiny ridge]

You basically did compute it, just indirectly

[nimble portal]

Ya

[tiny ridge]

By finding the accurate expression during proof of surjectivity

[nimble portal]

It's kinda cute how I can feel myself growing as a mathematician little by little

I'm coming up with counterexamples to my informal intuition much quicker

And I'm getting better at putting my ideas into words

[tiny ridge]

Sometimes, if I feel like it, I find the expression for the unique preimage in rough, then consolidate it as a map g : Y -> X and then show f o g = id_Y and g o f = id_X. This shows surjectivity and injectivity both in one go. But this can be a little painful

Unnecessary, even

[nimble portal]

Yeah it's computing sigma^(-1)(sigma) that's painful

[tiny ridge]

Ya

So I like what you did here better

[nimble portal]

If g and f are inverses then is their composition an odd function?

[tiny ridge]

Odd or even is an adjective you use for functions from R to R, yes?

[nimble portal]

Oh

Does the domain have to be R

[tiny ridge]

Because, what would f(-x) mean otherwise?

[nimble portal]

You just need an underlying structure where inverses are defined

[tiny ridge]

Fair enough

[nimble portal]

Holy shit I just corrected someone wtf

Sorry 💀

[tiny ridge]

In that case, f o g is the identity map

Is the identity map odd or even, given such a structure?

[nimble portal]

Odd

[tiny ridge]

So there you have it

[nimble portal]

So this is totally valid?

sigma tilde (x) = -sigma(-x) btw

[tiny ridge]

If sigma(-t) = -sigma(t), I see no issue

[nimble portal]

What

[tiny ridge]

You’re using that in the third equality, it seems

[nimble portal]

I was using the fact that sigma & sigma^-1 are inverses

[tiny ridge]

How does the third equality follow from there?

[nimble portal]

So sigma(-sigma^(-1)(x)) = -x

[tiny ridge]

That doesnt follow from sigma and sigma^-1 being inverses

[nimble portal]

Why not? Sigma(sigma^(-1)(x)) = id

So sigma(-sigma^(-1)(x)) = -id

[tiny ridge]

Careful.

[nimble portal]

That's why I was asking this

[tiny ridge]

sigma(sigma^-1(x)) = x, yes

Replace x by -x

[nimble portal]

That's what I was worried about

[tiny ridge]

You get sigma(sigma^-1(-x)) = -x

Why can you pull out a minus sign in the inner parentheses?

[nimble portal]

Only if sigma^(-1)(x) is odd

[tiny ridge]

Right, so its not about the composition being odd.

Which it always is, being identity

[nimble portal]

It's about whether the composed function is odd

[tiny ridge]

True

[nimble portal]

I don't think this is odd

[tiny ridge]

Nope

[nimble portal]

Oh and also sigma isn't odd anyways

Darn it lul

Too lazy to compute

By any two charts does it means every any chart needs to be smoothly compatible with any other chart in the atlas, or that there exists at least two charts that are smoothly compatible?

The latter condition doesn't make much sense to me so I feel like it's the first

Also don't you need to check smooth compatibility in both directions?

If psi circ phi^(-1) is a diffeomorphism does that automatically imply phi circ psi^(-1) is too?

Ahhhh hehehe

[urban zinc]

USE LATEX

[nimble portal]

ERIC MAY I DM?

[urban zinc]

Yeah

[urban zinc]

About what lol

[nimble portal]

To check my solution to the next part of the problem

[urban zinc]

Just post it here

[nimble portal]

But I think can do here idk why I asked DM

Yeah

[urban zinc]

Haha

[nimble portal]

Okay so for the first part (hte computation) I did this

Now for the second part, to show it defines a smooth structure, we need to show that the atlas is smooth & maximal

[urban zinc]

Hm

I don't think you should get id

[nimble portal]

o

well darn that ruins my next steps

[urban zinc]

Lemme check

[nimble portal]

Hm geometrically I feel like you should get -1 actually

[urban zinc]

yeah sigma-tilde is sigma(-x) not -sigma(-x)

[nimble portal]

wat

[urban zinc]

at least it is in my edition?

[urban zinc]

LOL okay

I guess he changed it

[nimble portal]

Well using the version I have

I'm p sure my computation is correct

So for the next part to show it defines a smooth structure we need to check that the atlas is smooth + maximal

[urban zinc]

okay yea it looks fine

[urban zinc]

it's not maximal

[nimble portal]

Oh sorry that's the smooth structure

[urban zinc]

but any smooth atlas is contained in a maximal atlas

[nimble portal]

Yes

I just need to show smoothness

[urban zinc]

yea

but id is clearly smooth

[nimble portal]

Yep

[urban zinc]

And obviously every point on the sphere is contained in one of those charts

[nimble portal]

Normally I'd have to check the smooth compatibility both directions right

[urban zinc]

So you're done for that part

[urban zinc]

Wdym normally?

[nimble portal]

Like I need to sigma tilde circ sigma^(-1) and sigma circ (sigma tilde)^(-1) are diffeomorphisms (which just amounts to showing they're smooth)

But by the second thing he writes ("Alternatively...")

I can show that they're smoothly compatible just by verifying one direction is smooth & injective w nonsingular Jacobian

Which it is since id is smooth and Jacobian is identity matrix which has det 1

[urban zinc]

yeah if one is a diffeomorphism, then the other one is a diffeomorphism by definition

in this case, id is clearly a diffeomorphism (smooth with smooth inverse)

[nimble portal]

What did you get for the composition in your edition?

I think it should just be -(id) right?

[urban zinc]

Yeah

[nimble portal]

I rlly don’t want to do any more problems

Technically still have one left

[urban zinc]

Which one haha

[nimble portal]

Just bc I’ve spent so much time on these (though I expect future problems will take less time) that I’m ready to move on LOL

[urban zinc]

Btw my diff geo reading group is starting soon, we're reading ch 1 and doing the problems by next week lol

[nimble portal]

This problem

It looks interesting enough

Just ready to move onto learning new things instead of this 😭

It's my only problem on manifolds w boundary...feels like it's important to do just to make sure I know how to work with manifolds w boundaries

[urban zinc]

Yeah sure why not

[nimble portal]

but again

me lazy

[urban zinc]

Lol

I don't think I did this one

[nimble portal]

[urban zinc]

But you should at least know how to do it

[nimble portal]

I can see it in my head at least

Yeah

[urban zinc]

Once it's trivial, you can stop :^)

[nimble portal]

LMFAO

That's all of math

[violet summit]

Not so sure about proving how (x,y) -> xy is continuous

For (x,y) -> x+y, I showed the preimage of open intervals is open (by showing all points in the preimage are interior), and therefore it is continuous, but I cant find a good way to show all points in the preimage are interior with this one

[gritty widget]

use epsilon-delta

then all you need to do is use the triangle inequality

[nimble portal]

My brain is so fried

[nimble portal]

Just thinking abt this rq

The std smooth structure on B^n is the one where we split it into an upper half & lower half and so on?

I assume for the smooth structure on cl(B^n) we’d need two interior charts with upper & lower half domains then one boundary chart

[unreal stratus]

B^n is already closed

[nimble portal]

eh?

[novel acorn]

the problem says that B^n is the closed unit ball

[nimble portal]

Oh I cut off

the snip

oop

B^n is open unit ball

B^n with bar over it is closed

Here's a better crop

[novel acorn]

[urban zinc]

[nimble portal]

wat

[urban zinc]

Lol that was just funny

[nimble portal]

As far as showing that each point in S^(n-1) is bd and B^n is int

I just need to show S^(n-1) subset bd(B\bar^n) right?

[urban zinc]

yeah upper half and lower half, left half and right half, etc

[nimble portal]

And likewise B^n subset Int(B\bar^n)

[urban zinc]

so like 2n charts

[nimble portal]

2n + 2

[urban zinc]

huh why 2n+2

[nimble portal]

...I don't remember

[urban zinc]

oh wait this is wrong it doesn't cover the origin lol I was thinking of just the sphere

[nimble portal]

I can't find anything on what the std smooth structure on the ball is

Just sphere

[urban zinc]

standard smooth structure on the open unit ball is just one cover

(it's already a subset of R^n)

[nimble portal]

Oh I see

Why can't you say the same for the sphere then?

It

s a subset of R^(n+1)

[nimble portal]

You mean the chart (R^n, Id) right?

(Atlas consisting of that one chart)

So ig std smooth structure on B^n is generated by {(B^n, Id_(B^n))} lol

[urban zinc]

Yes

[nimble portal]

Where Id_(B^n) is the restriction of Id_(R^n) to B^n (Do I need to specify that since I'm talking about a submanifold?)

[urban zinc]

You can just write id

It's unambiguous

[arctic relic]

The issue with S^n having the same atlas is that S^n is not homeomorphic to R^(n+1)

[nimble portal]

MMMMMMMMMMMMMM

GOOD POINT

I'm such an idiot how did I miss that lol

Thanks Chern!

[urban zinc]

you might as well show the sphere is equal to the manifold boundary of the closed ball

[nimble portal]

yeah but I'm lazy

._.

[urban zinc]

more explicitly you need to give charts so that the boundary points get mapped to the x-axis (in at least one chart)

and the manifold gets mapped to the closed upper half plane

[nimble portal]

it doesn't say boundary points are points on the sphere, just that points on the sphere are boundary points, so tehcnically that's just S^(n-1) subset bd(M) right?

[urban zinc]

feather

😭

[nimble portal]

what

._.

[abstract saffron]

TIL I rediscovered concordance in knot theory 😄

[tiny ridge]

Useful

[nimble portal]

What's wrong with my proof for the boundary point? It seems too simple

[tiny ridge]

Something I don’t know the answer to: If K is a knot, it is easy to see K # Kbar is slice (ie concordant to the unknot). Can one explicitly show it is ribbon?

[nimble portal]

It's a boundary point if it's in the domain of a boundary chart on M that sends the point right to the boundary of upper half space

Ahh I don't think phi is injective

What else?

[abstract saffron]

Remember the proof that it is slice? You connect K with its mirror, and see that it is actually a disc B_2 intersecting itself.

Well, the way it intersects itself is in ribbon way 😄

if I'm not mistaken

[tiny ridge]

I don’t follow. The disk is in one dimension up

Its in D^4

The way I prove it is slice is by taking K x I in D^3 x I, then removing a little strip around {pt} x I

[abstract saffron]

I know, it's B_2, but intersecting itself. The trick to make it not intersecting itself (and thus slice) is to take small neighborhood around intersections, and lift them up to B_4

[tiny ridge]

I don’t understand your picture, I think

Your proof of sliceness must be different from mine

[nimble portal]

I think this is fixed if we just make the domain a subset of S^(n-1) right? As then if all the first x^(n-1) coordinates are the same that guarantees the x^nth coordinates must be the same too

[tiny ridge]

The slice disk is automatically embedded in D^4 in my construction

You seem to be constructing some explicit ribbon disk in S^3. How?

[abstract saffron]

Euhm... I guess you can find it in here ? https://www.youtube.com/watch?v=2pmpE1P1xF0

I don't have a blackboard with me to draw, sadly

[tiny ridge]

Time stamp?

[abstract saffron]

Around minute 30

But he didn't show it's ribbon though. That I found separately.

I can be wrong

[tiny ridge]

That disk does not seem ribbon to me

Yes it lifts to D^4 but that doesnt mean its ribbon

Its basically the same proof that I gave but projected to R^3

[abstract saffron]

Hmm, somehow in my head it folds in ribbon way. I'll have to check that rigorously

[tiny ridge]

Try this example

If you run the algorithm the lecture presents, you’ll get non ribbon singularities in the disk, near the crossings

Hm, I am wrong actually

This is very much ribbon

So your claim is if you take K, put Kbar on top of it, and then drop segments from K to Kbar, you’ll get a ribbon disk

After deleting a little piece

Just have to check at the crossings, which works out because the crossings are mirror images of each other

Interesting!

[nimble portal]

I see some other issues but I think I know how to fix them

I need a homeomorphism that only makes the last coordinate zero if the point is in S^(n-1)

Which I think is where their hint comes in

[abstract saffron]

So.... I was right accidentally?

😄

[tiny ridge]

Literally I think, not accidentally

[abstract saffron]

My intuition for topology is strong (or so I hope), but I'm always afraid that this kind of intuitive reasoning will one day bite me in the ass.

[urban zinc]

our Hatcher reading group had our first meeting today!

why is algebraic topology so hard 😭

[ornate berry]

It's mostly that Hatcher is a crappy book xoxox

[abstract saffron]

Should have tried Munkres

[tiny ridge]

Strong disagree

[coarse night]

Hatcher proofs are hard to read

[novel acorn]

it's easier than it seems!

[thorny agate]

I'm stuck on 9

I want to use 6 but haven't figured out a good way to construct such an open set

[novel acorn]

what are those symbols

[thorny agate]

uhhhh just problem markers

[novel acorn]

ah ok

[thorny agate]

diamond means "harder" and the other one means "this result will be used later in the text"

[coarse night]

You can give a nice proof is you assume one of the closed sets in compact

Otherwise it involved

I suggest you check out Munkres for the proof

[coarse night]

Let me know if want this

[thorny agate]

can't assume that

[coarse night]

Yeah ik, just a special case if you’re interested

[thorny agate]

I'll try that on my own later

and then come back if I can't figure that out

I don't think I can use problem 6 to do this

so imma try something more direct and construct open sets lol

[abstract saffron]

Can't you use Hausdorff distance?

[coarse night]

The metric you mean? You need one of them to be compact. That’s exactly what i was hinting at

[abstract saffron]

I don't see how 😄

What can go wrong ?

[coarse night]

See what, unless I misunderstood Hausdroff dist

[abstract saffron]

Oh no, nevermind

I didn't see why one of them must be compact

[coarse night]

You meant use d(x,A) right?

Or something else

[abstract saffron]

I meant d(A, B)

[coarse night]

Yeah that can be 0 even if they are disjoint closed

Take 1/x and x axis for example

[abstract saffron]

Right

[abstract saffron]

That's why I said nevermind

😄

[coarse night]

Lol chronology

[abstract saffron]

But why does it fail? R^2 is still metric and normal, hmmm

Oh, you can't use a global parametrisation for the open sets. The open sets in this case must be constructed more locally

[coarse night]

There’s a modified one that works tho

[abstract saffron]

That should give a hint on how to solve this exercise

[coarse night]

Explicit construction of Urshyon for metric spaces

||d(x, A)/(d(x, A)+d(x,B))||

Believe this is the right one

[thorny agate]

oh what

man that's what I was gonna use

[abstract saffron]

Topology moment

[coarse night]

I should spoiler mine then

[thorny agate]

but yea 1/x and x-axis are perfect example

damn it

[coarse night]

That’s exactly I was hinting before lol

[abstract saffron]

I think I found a good construction. It's pretty weird, but it's good

But I don't know that Urshyon construction thing-y. Maybe I rediscovered something

[coarse night]

Idk what it’s called

Urshyon normality criterion

[thorny agate]

what is this Urshyon thing you're talking about

<@&268886789983436800> plz

[coarse night]

Joined

[abstract saffron]

Uryshon's lemma?

[coarse night]

Yeah

Whatever its called

[thorny agate]

oh interesting I've never seen this lemma

[abstract saffron]

I've seen it once, but totally forgotten it

Oh yes, I just made another re-discovery.

[coarse night]

Then look up Tietze

[abstract saffron]

Second time for today already

[coarse night]

Really helpful btw

[thorny agate]

interesting

I'll look these both up

[novel acorn]

that's because most intro topology books don't talk abt the separation axioms as much as they should

[thorny agate]

I'm looking at Bredon rn

there's a small section on it (but tbh general topology is just 1 chapter of the text)

[coarse night]

lol bredon for general topology

[thorny agate]

I just wanted something small and quick

only learning topology to then learn some basic manifold stuff to prep for a manifolds class I'm taking next sem

that's it

[abstract saffron]

Lmaoooo

[thorny agate]

I would have just taken topology next sem but the prof is ass

[abstract saffron]

That some basic manifold stuff will require all topology

Trust me on this

[thorny agate]

yea

it's ambitious

but like tbh I just need enough topology to skate by

and then be able to fill in the gaps later

[abstract saffron]

I mean, I can see myself doing it, because I am doing something similar

[thorny agate]

goal isn't to learn all of topology, just enough to be able to plug gaps as I need down the line

[coarse night]

Read Lee ITP instead

[abstract saffron]

Nooooo, not Lee

[coarse night]

Topological

I like that one

[thorny agate]

and it's not like this manifolds class is required so worst case I've self studied some basic topology

not a bad outcome

[abstract saffron]

feather can attest why not Lee

[thorny agate]

Lee ITP?

[coarse night]

Not talking about ism

[gritty widget]

it's ok ryu if you say it a few more times your point will be clear

[coarse night]

[abstract saffron]

Bredon is ok if you want a quick reminder and move on to Alg Topo,

[coarse night]

hAtChEr

[abstract saffron]

But it's not so great if it's your first time. He proved both IVT in Chapter 2.

Might as well go with May at that point

[coarse night]

Gl

[thorny agate]

Like I said this is the goal

And someone else recommended Bredon and so far it's been nice

So I'mma stick with it than enter textbook choice paralysis

[abstract saffron]

I guess Bredon is OK, if you only do first 2 chaps. But be sure to do a ton of exercises

[coarse night]

ISM appendix is enough

[thorny agate]

I'm doing basically every exercise

And proving everything left to the reader

[abstract saffron]

The book is still hell dense tho

Lmao, Sard's theorem in 2 pages

[thorny agate]

It's so dense

[abstract saffron]

Sternberg took half the chapter 1 to discuss Sard before moving on to Whitney's embedding

[thorny agate]

I'm moving quite slowly

[abstract saffron]

Bredon is like, digestible Lang

[white oxide]

In absolute terms or relatively speaking?

[abstract saffron]

Wdym?

[honest narwhal]

Yeah Bredon is p good

[tiny ridge]

just do Kirk&Davis, More Concise by Peter May and Hirsch for differential topology

go hardcore mode

thats the only alternative if you dont like Hatcher

[abstract saffron]

Everyone always cite Nash's PhD thesis, but TIL Piccirillo's PhD thesis is also quite impressive

What a chad

[tiny ridge]

what did Nash do in his PhD thesis again

he did too many things

[abstract saffron]

Game theory

[honest narwhal]

My PhD thesis is quite literally too good to be true

[abstract saffron]

Non-cooperative game theory

[tiny ridge]

ah ok so thats the least impressive of all of Nashs works

[abstract saffron]

Well, apparently he got Nobel prize for it

[tiny ridge]

meh

[honest narwhal]

It was revolutionary in econ but as far as I'm aware, mathematically simple

Contrast to say, Nash embedding

[abstract saffron]

She overheard an open problem, locked herself up, and cracked it within a week. The whole PhD thesis is 33 pages, with main result is like, 9 pages.

[quiet thorn]

is that in a good way or a bad way

[abstract saffron]

Aced the PhD in 14 months, and got a tenure-track at MIT right after

[tiny ridge]

open to interpretation

[abstract saffron]

Econ ppl learned how to bs better in math after that

[tiny ridge]

not a day goes by that i don’t see a path integral dogwhistle in the papers i read

[abstract saffron]

Are you in PDE department or something?

[tiny ridge]

nah

thats too hardcore for me

[abstract saffron]

I guess you must work closely with mathematical physics or something. And most of them focus on PDE: Navier-Stoke, Einstein's field, Dirac, Schrodinger, Yang-Mills.

Or at least that's how I perceive

[tiny ridge]

i like elliptic pdes, which means stuff like Laplacian + (nonlinear terms) = 0

they appear in geometry

Dirac, Yang Mills, are examples

[abstract saffron]

This kind of writing slows down progress of science. I'd spend days on filtering out the junk from the gold.

[tiny ridge]

welcome to academia

aint nobody has time to write good

write fast, publish

[abstract saffron]

It's not even that hard to write decently. Just write how you understand it, e.g. what is the motivation, why does it make sense

You can't convince me this is how they came up with the thing in the first place

[tiny ridge]

i think this is fine

subjects get exponentially complicated as you get closer to the cutting edge

and theres less and less expositories

i think writing is still subpar these days even within those constraints but eh

people have to eat and get a job, so they write fast and publish as soon as they have something

most papers will be read by a handful of experts so theres very few incentives to write better

not many journals which emphasize on exposition

what im reading is one of the better ones imho

[abstract saffron]

This helps no one 😄

Now I know why Master theses are mostly reviewing and introducing topics.

This fast-paced approach is so error-prone. I fear the same Italian school of gemeotry's catastrophy can happen again.

[tiny ridge]

on the contrary the italians were very good expositors

they didnt care about correctness, only examples and test cases

[abstract saffron]

That's not how math works 😄

[tiny ridge]

maybe not to you, but to many people it does

everyone has a different style

but math is a complex social activity and theres no one way of doing it

[abstract saffron]

If only everyone were Milnor. That guy writes so well

I read his papers much easier than reading textbooks on the same topic

[tiny ridge]

hes also a very good experimental mathematician, in the sense that he knew how to integrate a lot of interesting examples with a lot of correct proofs

and he was willing to write

[abstract saffron]

Exactly, he proves stuff

[tiny ridge]

but thats not why hes famous

[abstract saffron]

And he writes very clearly how he thinks

[tiny ridge]

everyone proves stuff

[tiny ridge]

i wouldnt say hes writing down his thought process

hes writing down some extremely streamlined and slick thing

thought processes are messy

[abstract saffron]

I didn't say that he writes down his thoughts, but you can easily follow his footsteps.

Contrary to others for whom I take hours to see what they said to be "trivial"

[tiny ridge]

i will say though that sometimes milnor is too slick

its not clear why something should be true but you can follow his proof

at least i felt that way, for example, with his exotic 7-sphere paper

[abstract saffron]

I think thought processes should deserve more lime light

[tiny ridge]

you have company then

https://mathoverflow.net/questions/38639/thinking-and-explaining

[abstract saffron]

You don't need to write down the whole thought processes leading to the result, but at least you can write down what you've tried and why they didn't work.

People may find ways to make use of those failed attempts. Ideas are far more important than results.

[tiny ridge]

i agree, thats why Stallings famously wrote down how he didnt prove Poincare conjecture

https://www.degruyter.com/document/doi/10.1515/9781400882076-011/html?lang=en

[abstract saffron]

Even Abel-Ruffini theorem was based on both Abel's and Ruffini's failed attempt iirc

Had Ruffini not published that failed proof, God know how long it'd take

[tiny ridge]

it was much better in the olden days because mostly people wrote like the experimentalists

the bar of precision wasnt as bad as it is now

arnol’d talks about this problem in great length, according to him this happened because mathematics separated itself in some fundamental way from other scientific fields

eg physics

[abstract saffron]

Math folks pride themselves for their derivable truth. Other branches of science need experiments to verify.

[tiny ridge]

although, one must remember, eg, i think Kummer discarded Grassmann’s work — where he invented the exterior algebra — as not very rigorous

actually he even defined what a vector space is

[abstract saffron]

Even Fourier's original work was at points discarded. The notion of uniform continuity didn't exist back then.

It was a different time.

But I agree, writing is a lot of work. If I have to sit down and write a text book on undergrad math, I'm pretty certain that I'll do better than textbooks I have read, but it's just too much work to explain all my intuition and thought processes.

[abstract saffron]

So... that's what a PhD life looks like?

[tiny ridge]

i hope not

[abstract saffron]

Waita minute... You're not a prof, are you?

So you're not a grad student, but deal with advanced stuff. Either you're a prof, or Ramanujan.

[tiny ridge]

im a grad student

i was just saying i hope its not a typical thing for every math grad to run into path integrals every day

[abstract saffron]

Particle physicsists

Actually no, iirc Feynman diagram helps make it easier

I remember reading Emanuel Derman's autobiography, he mentioned this detail

[tiny ridge]

Feynman diagrams are Feyn

its when you start integrating over all Feynman diagrams that you should start questioning your life choices

[urban zinc]

Is this statement true for any topological space? I can't figure out how to prove it

A function f defined on the union of two closed sets K_1 \cup K_2 in a topological space X is continuous if it is continuous when restricted to each of the closed sets separately.

[tiny ridge]

Take R, K1 = [0, infty) and K2 = (-infty, 0]

What do you think?

[urban zinc]

That would seem to be true using the normal epsilon-delta definition of continuity on R

Unless I'm missing something

[tiny ridge]

Correct

Now do it using the preimage of open sets definition

[urban zinc]

Hmmm

[unreal stratus]

Also like this is often known as the pasting/gluing lemma

[urban zinc]

Oh I've heard of that before

[unreal stratus]

It's useful for constructing functions lol like you can often just appeal to this to prove stuff is cts if in doubt xd

[tiny ridge]

Try it in this simplest case, then convince yourself that it generalizes.

[unreal stratus]

(Also like, note the same thing is true for open sets, in fact including infinite collections, whilst for closed it doesn't necessary hold in the infinite case)

those are other good exercises to try lol

[urban zinc]

With open sets it makes sense to me

[unreal stratus]

Yeah

[urban zinc]

So (U) is an open set in the image of (f), and we want to prove that (f^{-1}(U)) is open.

We know that there exist open (Y_1), (Y_2) such that (f^{-1}(U) \cap K_1 = Y_1 \cap K_1) and similarly with (K_2) and (Y_2). Then,
[f^{-1}(U) = ((Y_1\cap K_2^C) \cup (Y_1\cap Y_2) \cup (Y_2\cap K_1^C)) \cap (K_1\cup K_2),]
so (f^{-1}(U)) is open.

[gentle osprey]

boolean_satisfiERIC

[urban zinc]

This is my guess for what should work, I haven't worked out that long equation though

Okay yes I think this should work

[tiny ridge]

Pretty much

[urban zinc]

yay, ty! :)

[tiny ridge]

You can use a little trick to do it easier

Preimage of open sets is open iff preimage of closed sets is closed

[urban zinc]

Ohh

[forest glen]

where can i learn this math

[urban zinc]

What level are you at right now?

[forest glen]

calculus

:/

i mostly played video games my whole life now i am becoming interested in the field of academia

i dont know what order or what classes to do though

[urban zinc]

That's fine, personally I found that learning proofs through following an analysis textbook was useful for learning general topology

[next crystal]

our lord and savior munkres

[urban zinc]

Munkres's topology is good but it might be a little unmotivated if you don't at least know the theory for the topology of R in my opinion

I also liked the introduction to Lee's Introduction to Topological Manifolds for a review of intro topology :)

[forest glen]

I dont know any of this

like my brain is a blank slate

[next crystal]

do you know proofs

[forest glen]

right now I am learning multivariable calc but that has nothing to do with this it looks like

[forest glen]

very basic

[urban zinc]

there are a lot of relations!

I'd recommend picking up an intro analysis book (there are a lot, like william wade's, bernd schröder's, abbott's, etc)

[forest glen]

do you have a link?

[next crystal]

i think even after going through the first couple sections of an intro analysis book you should be fine for topology

[urban zinc]

Yeah

[next crystal]

just getting through the topology of R section

[urban zinc]

Learning the intermediate value theorem and extreme value theorem would be nice too

[forest glen]

is this not calculus

[urban zinc]

It is calculus!

It shows you how topology can help you do calculus

[forest glen]

oh wow

😮

wonderfu

l

[next crystal]

the cool part is these theorems are more topological than they are analytic and when you learn it in calc its a special case of the topology version

the structure of R is "nicer than it needs to be" to allow for results like these

[urban zinc]

This seems to be an option for one mathematical analysis textbook that's freely available online http://www.trillia.com/zakon-analysisI.html

There are tons though

[forest glen]

okay i will read this

in combination with the multivariable course i am doing rn

[urban zinc]

I think I first learned from a youtube series that a prof uploaded during the pandemic

[next crystal]

abbott is pretty nice for intro analysis as well

[forest glen]

are you guys doing this for fun or for a course?

[urban zinc]

https://www.youtube.com/watch?v=njzIR53eJOs&list=PLLFpXNanTP9WGfbjxR5kCMXQgol4bGehz this series is pretty slow but good for the basics

[next crystal]

i took an intro topology course last semester

[urban zinc]

I guess sort of a combination of both, I think it's fun and I'm not taking a course specifically on topology right now but it's content that it would be helpful to know for future courses anyways

There are also interesting topological methods for data analysis

Which I'd like to learn about

[forest glen]

hmm yeah

[novel acorn]

clearly we do not have to teach kids real analysis as it's all just info you can learn abt through metric spaces

[urban zinc]

heine borel

[next crystal]

this was me after covid

[urban zinc]

There's honestly nothing wrong with having fun tbh

If you wanna learn math that's great though :)

[novel acorn]

During Covid I gamed like fucking 12h a day that made me a husk of a man so I started learning (chemistry -> Physics -> math)

[forest glen]

yeah i played lots of chess and league of legends

[forest glen]

🙂

[urban zinc]

Okay 12h is a lot 😭

[next crystal]

at least chess is kinda using your brain

[forest glen]

i might have done more

[forest glen]

honestly i felt like i used my brain in league more LOL

[urban zinc]

Don't obsess too much about learning math either, doing anything for 12h/day can lead to burnout p fast

[novel acorn]

bruh if I get ahold of a new game I ain't letting go like an orangutan doesn't let go of his banana

[next crystal]

i agree with this when i first started getting into it i spent an unhealthy amount of time on it

[forest glen]

LOL speaking of chess

[next crystal]

now that im in college i dont do shit and need to spend more time doing it lmao

[urban zinc]

To be clear, I've never read that textbook, I just found it for free online so no quality assurances lol, seemed to cover the topics that were helpful though judging by the table of contents

[forest glen]

cant hurt to read it

its either that or im playing league

lol

[urban zinc]

It'd probably be good to go up to chapter 4 (function limits, continuity, compact sets, connected sets, sequences of functions) before delving into more general topology

chapter 5 (differentiation and antidifferentiation) isn't really as relevant to a first course in topology

[next crystal]

this good too http://ndl.ethernet.edu.et/bitstream/123456789/88631/1/2015_Book_UnderstandingAnalysis.pdf

[urban zinc]

I agree that book is nice and gentle!

[next crystal]

link looks so sus

[urban zinc]

I liked it

[forest glen]

once i finish reading those chapters i will ask for more advice on topology then

[urban zinc]

If you have any questions about any of the textbook content feel free to ask in #real-complex-analysis :)

[novel acorn]

A book that'll make you sweat is Pugh's book

[forest glen]

thank you 🙏

[urban zinc]

reading this up to ch 4 would also be super helpful for learning topology ^^^

it covers more or less the same ideas

so if one doesn't work for you, you can just pick up the other one haha

[forest glen]

kk

bet

[tiny ridge]

@forest glen Want to learn topology? Ignore everything else and just read https://pi.math.cornell.edu/~hatcher/Top/Topdownloads.html

These 50 pages will make you an expert in topology

Bookworms hate this trick!

They’ll come at you with their Munkreses, their Follands and their Engelkings, even. Tell them, stop! But politely.

[forest glen]

Hmm?

[tiny ridge]

Yes, you too can now have access to a royal road to topology

It’s shocking! And the big government won’t tell you about it. For good reason!

[nimble portal]

I love Hatcher

But I can’t read his alg top book

Idk why it has great visuals I just don’t like the way he explains stuff

[tiny ridge]

his point set notes is much more easily accessible

[forest glen]

@tiny ridge do i need prior knowledge for anything

[civic verge]

I share with you this book to study topo at Elon's level.

http://mathonline.wikidot.com/

I also share a link that I found a long time ago, I think they had sent it too but here I pass it to you too, go to the topo section, there you will find the complete course, the exercises you can look for them in the bibliography of any book.

[quasi steppe]

What do you suggest for algebraic topology?

[coarse night]

hatcher again lol

[unique wolf]

I recommend beckham myer harvard notes, very clear.

[urban zinc]

so I understand why intuitively r is continuous, but how would you actually prove it? would you just write down a formula for it

[coarse night]

Try solving

[urban zinc]

so
[r(x) = x + c(x)\left(\frac{x-h(x)}{|x-h(x)|}\right)]
where (c(x)) is such that (|r(x)|=1) but I'm not sure how to find an explicit formula for (c(x)), the distance between (x) and the circle in that direction

[gentle osprey]

boolean_satisfiERIC

[next crystal]

hol up i found a stackexchange post about this a couple months ago when my class went over this

[coarse night]

You can try solving with x,y coordinates

[urban zinc]

Okay

[next crystal]

cuz my prof just breezed over it too saying its not worth it

[urban zinc]

So really it's just finding the intersection between a line and a circle

Which shouldn't be hard but it's annoying

[next crystal]

https://math.stackexchange.com/questions/4540516/proof-of-brouwers-fixed-point-theorem-in-hatcher

[urban zinc]

Oh god

[next crystal]

have fun

[coarse night]

it’s a pain, you have to get uses to this kinds of handwaveing otherwise hatcher (or any AT except may) is going to be infuriating

[urban zinc]

That's sad, I kinda want to fully understand what's going on

Maybe I'll just have to get really good at spelling out all the details

[next crystal]

it seems like in topology they handwave a lot of the analysis type arguments

since its not really the point ig

[coarse night]

God why would anyone do this

[next crystal]

i wonder how long it took that guy to come up with that answer lmao

[urban zinc]

I'm guessing not terribly long if they're a mathematician, very long if they're a student

[coarse night]

No the point is it’s pointless

[urban zinc]

I want to know how to do it!

[coarse night]

There’s more pain on the way then

[urban zinc]

I don't mind

At least not right now

[next crystal]

its okay ur a masochist and i accept you

[urban zinc]

I get that it's not the main point

[coarse night]

I’ve seen an explicit proof of S1 diffeo to unit square

With explicit functions

[urban zinc]

But I want to know

[coarse night]

No I’m not saying you shouldn’t

If you enjoy details then sure

[urban zinc]

Oh this Cauchy Schwarz argument is rather clever, I wouldn't have come up with it

I guess I just need practice with Cauchy Schwarz

[nimble portal]

Can anyone give me a hint IN A MOMENT AFTER I EXPLAIN WHAT IM THINKING to this part of the problem

[next crystal]

no i think i will give you it before you explain

[urban zinc]

It seems obvious to use in retrospect

[nimble portal]

Ok so wtf is the chart supposed to be? Is it pi circ sigma^(-1) restricted to the ball?? Or something along that line?

Just like

The most vague hint ever

Is all I want

[urban zinc]

Oh I have lots of friends that use that app lol

[nimble portal]

Ya luv GoodNotes

Look my pretty vector field

1s

[coarse night]

Even I use this lol

[urban zinc]

I think you just need 2n charts, one for every hemisphere?

[nimble portal]

Wait I gotta find it

[urban zinc]

I could be wrong

Wait no you only need two

[nimble portal]

I don’t think the charts are the same construction as for the sphere

[urban zinc]

One excluding the north pole and one excluding the south pole, no?

[nimble portal]

Yes

[urban zinc]

Idk what I'm talking about

[nimble portal]

No I’m pretty sure you’re right

And I think I see how it’s supposed to come together too

I’m pretty sure you use their hint to generate a chart that makes the last coordinate zero if the point is on the boundary and nonzero otherwise

Or something along those lines

Because then that covers your local Euclideanness (fuck that word I’m calling it Euclidity) checkbox and it clearly shows that points in S^(n-1) are bd pts and points in B^n are int pts

[urban zinc]

here's a pretty bad drawing of how I imagine it for the 1-ball

[coarse night]

Try this maybe

[urban zinc]

wait I did the stereographic projection wrong, flip it across the x-axis lol

mb

[coarse night]

Idk explicit for higher dim but you can do in 2D using mobius transformations

[languid patrol]

I am going to make a bold suggestion

[nimble portal]

What am I looking at?

[urban zinc]

I think using the given hint via stereographic projection is prob better

[languid patrol]

Why don’t you use three charts

[coarse night]

Or just linear interpolate

[nimble portal]

One for the boundary and two for the hemispheres right?

[coarse night]

There’s a hint lol

[languid patrol]

One chart is the ball of radius < 1-epsilon

[coarse night]

Oh I see what the hint is saying

[urban zinc]

I feel like you need only two charts, one omitting one pole and one omitting the antipodal pole

But maybe there's something wrong with my logic

[languid patrol]

I think that’s fine

[urban zinc]

Lemme try to do a drawing of how it would work for the 2-ball but my drawing skills are weak

[languid patrol]

Just trying to suggest something different since she didn’t like the hint

[nimble portal]

No I do like it and I’m so close to seeing how it plays into the proof I feel

Just don’t fully understand how to apply it

[coarse night]

There’ll probably be some level set theorem in the later chapters of the book

[nimble portal]

Wait a second

[urban zinc]

Oh wait I think you do need more than two charts

Or else the projection isn't injective

[nimble portal]

Agh I wish my dad hadn’t broken my laptop zzz it’s so much harder to do math

No WORD to type on

[urban zinc]

STOP WITH MS WORD

[nimble portal]

JUST FOR THIS LAST PROBLEM

THEN I TRANSFER HOUSES TO OVERLEAF

[coarse night]

Spherical inversion

[nimble portal]

ON MY MOMMA

[gritty widget]

[urban zinc]

So I think you can cover every point except the origin by 2n charts with the map they suggest

And then just cover the origin with a ball of radius between 0 and 1

[nimble portal]

He spilled some wine 💀 no abuse here!

[urban zinc]

Actually wait yeah that works

[urban zinc]

Piggybacking off this idea

I think that's the easiest way to do it, following the hint

And then it's not too hard to show everything is smooth

[urban zinc]

so you map each hemisphere (not including the base of the hemisphere) to the upper half plane like this

[nimble portal]

My brain stopped processing information

Feel free to type but I don’t want too much of a hint

That’s all

[urban zinc]

I feel like you just need 2n of these + something to cover the origin and you're good

[nimble portal]

What’s so special about the origin?

[urban zinc]

It's not covered by any hemisphere (since we're not including the base of the hemisphere, to keep things open)

[nimble portal]

mathematicians design an interesting problem that doesn’t have 12471293712983 cancer little counterexamples/annoying constraints that make the proof involve any actual thought challenge

[urban zinc]

Tbf covering the origin is easy

[urban zinc]

The details help you learn

🥺

[nimble portal]

Unfortunately

I think I’m just tired of this kind of problem LMAO

[urban zinc]

LOL

[nimble portal]

I really enjoyed the second one

[urban zinc]

Ngl I've stopped doing the exercises, I just wait until you ask so we can do them together

it's funner

[nimble portal]

Because they gave me the freaking chart and I didn’t have to go through silly p-

AWWWWWWWWWW

UR ACTUALLY SO CUTE OMFG

fuck you now I have to do this problem hehehehe ok tomorrow we BALL then hopefully we move onto actual differential geometry not just verifying boring conditions 🤓

[gritty widget]

[urban zinc]

TTerra

[urban zinc]

I'm still trying to read this post which verifies that r(x) in this diagram is continuous

😭

I think I'm getting it though at least!

[nimble portal]

Oh one thing I never understood properly was when Lee says “omit the coordinate” (and denotes it with a hat over the coordinate), what exactly does that mean? Make that coordinate 0? Or just literally remove it and move the coordinates of everything after one dimension down?

[urban zinc]

the latter

[nimble portal]

Cool

Perfect

[urban zinc]

Ohhh I get it, you just solve the quadratic formula

Okay I think I just need practice doing these manipulations

But they're fairly straightforward, I just got somewhat lost

Brouwer's fixed point theorem is so neat

[coarse night]

hmm

try looking up his original proof

[urban zinc]

I'm not ready

I only know the 2-dimensional disc case so far 😭

[nimble portal]

Doesn’t it show up a lot in diffnl eqns

[urban zinc]

I think so

I now know how to prove S^n is simply connected if n≥2 :D

[nimble portal]

is it hard?

[urban zinc]

It's not too bad

[nimble portal]

what’s simply connected vs connected again? LOL

[urban zinc]

simply connected = path-connected and all loops are contractible to a point

corollary of this is that R^2 is not homeomorphic to R^n for n≠2 🥺

[nimble portal]

uh

don’t you get that from like

two vector spaces are iso iff they have same dim

[shadow charm]

This doesn’t tell you anything about whether they are topologically iso

Ie homeo

[nimble portal]

homeomorphisms are kinda like isomorphisms for top spaces

[shadow charm]

Showing R^n not iso to R^m is actually kinda non trivial

Only proofs I know use homology or degree

[nimble portal]

homology seems boring ngl…don’t shoot me topologists..

[shadow charm]

Homology is very fun, and you can prove a lot of nice unexpected things with it

[urban zinc]

You can do it by comparing tangent spaces I think?

Treating R^m and R^n as smooth manifolds

[shadow charm]

Well the tangent spaces are just themselves

So it’s a circular argument

[urban zinc]

Yeah but the tangent spaces are vector spaces

And R^m is not isomorphic to R^n as a vector space

[shadow charm]

Oh fair enough

[urban zinc]

It's a neat argument

[shadow charm]

But now this only tells you about diffeomorphisms no?

[urban zinc]

oh I think so

that's a good point

[nimble portal]

I know next to nothing about it besides classifying the “holes” of spaces algebraically

[shadow charm]

I don’t know much diff geo so not sure but yeah I doubt it works for any homeo

[urban zinc]

Yeah I think it only works for C^∞ diffeomorphisms

[nimble portal]

kinda like fundamental group stuff and homotopies right?

[urban zinc]

also why is r1(X) only a subset of A here and not just equal?

isn't r1(A) = A

[shadow charm]

Depends how you define def retracts

I think Hatcher only takes subsets

[nimble portal]

literally how do you read hatcher this typesetting is giving me eye strokes

and I type on word ffs 😭

[urban zinc]

Oh I see

[shadow charm]

And calls it a strong deformation retract when r_t(X)=A

[urban zinc]

Ahh

[shadow charm]

Ah wait

[urban zinc]

ty!

[shadow charm]

No cause he says r_t|A = id so never mind

[urban zinc]

😭 maybe he's just weird

I mean it's... true I guess

[shadow charm]

Yeah not sure actually

[urban zinc]

just kinda weird to write it that way

[shadow charm]

Yeah

[coarse night]

or is it

[coarse night]

Jk

[tiny ridge]

Precompose with a Blaschke transformation of the disk taking 0 to h(x).

Or conjugate with it, rather.

Then r becomes the map D^2 -> S^1 given by sending x to the radial projection from 0 of x onto the circle

Which is obviously continuous. The Blaschke transform is continuous on its parameters

Conclude

[coarse night]

Cool proof

[long remnant]

Hey guys, I'd love to chat with a transdisciplinary, meta / holistic mathematician about a problem I'm trying to solve.

It has to do with sweeping the entirety of an N dimensional state space with a single variable, or finding a mapping from an N dimensional space to an X dimensional space (without any data, not at all talking about dimensionality reduction techniques like PCA or t-SNE or UMAP that require data)

The problem is quite hard to articulate without a wall of text, so would rather chat

[nocturne basalt]

I think hatcher uses subset instead of subseteq

[left merlin]

How do i draw diagrams of subsets of order topologies

For example, the ordered square, I^2, where I = [0, 1]

[urban zinc]

isn't it just equal regardless?

[urban zinc]

Have you looked into space filling curves

[long remnant]

Yeah, this solves my many to one dimension problem but not N to X

So I want to compress 10K to 1K to 100 to 10 to 1

[urban zinc]

wait what's a blaschke transform? can you send a link, nothing showed up when I looked it up for some reason

[long remnant]

There are hilbert space filling surfaces, someone shared a link of that here

But I want an extension of that

that generalizes beyond a static number of dimensions

[tiny ridge]

Its the Mobius transformation (z - a)/(1 - abar * z). It sends the unit disk to itself, and maps the point a to the origin.

|a| < 1

Written in complex coordinate z

[urban zinc]

ohhh okay

[nocturne basalt]

yes in this case its equal

[urban zinc]

that makes sense, ty! that's a cool proof

I like that

[urban zinc]

okay so the point here is that homotopy equivalence is weaker than isomorphism of pointed topological spaces because you only require the compositions to be homotopic to id, not equal to id?

can someone give an example of two pointed topological spaces that are homotopy equivalent but not isomorphic? struggling to wrap my head around this definition

[plain raven]

Let X be the real number line with distinguished basepoint 0
Let Y be the singleton space {0} with distinguished basepoint 0
Then X and Y are homotopy equivalent, but not isomorphic.

[urban zinc]

ahh okay so to prove that, we'd show there's a homotopy between the identity function on R and the constant zero function on R

given by F(s,t) = st for 0≤t≤1?

[plain raven]

Yes.

[urban zinc]

ty so much!!

[urban zinc]

Question: Show that composition of paths satisfies the following cancellation property: If (f_0g_0 \simeq f_1g_1) and (g_0\simeq g_1), then (f_0\simeq f_1).

So here are we assuming that (f_0(1)=f_1(1)=g_0(0)=g_1(0))?

[abstract saffron]

Huh, this goes from the fact that R is contractible, no?

That's literally the definition

That the identity is homotopic to a constant function

[quick bough]

i want to prove the universal coefficient theorem, but i’ll need to cover all the basics of algtop for this (i want to write like a small thesis on this), do you guys have any good literature on this?

[abstract saffron]

Universal coeff theorem as in homology?

[quick bough]

uhh, i can choose, i heard there's also the universal coeff theorem for cohomology

whats the difference

[abstract saffron]

You invert the arrows

[quick bough]

[abstract saffron]

No but, it's a standard theorem. Any textbooks will cover this piece.

[quick bough]

i honestly dont have any textbooks for algtop haha

do you have any that you can suggest?

[urban zinc]

Yeah, isn't this the proof that R is contractible though