point-set-topology

[gritty widget]

can it hurry up and come out already

[flint cove]

time to transfer

[tight agate]

he's retiring

[tight agate]

I'm probably going to buy a hard copy as soon as it comes out

[tight agate]

read ch 23 and 24 from a concise course in alg top while you're waiting

[flint cove]

Yeah, that's kinda what was my starting point :>

[tight agate]

pretty great starting point

[flint cove]

Ah, didn't come far enough to notice „a glimpse at the general theory“

until now it just seemed very much restricted to VBs

[tight agate]

VBs are dope tho

[flint cove]

I mean I still have to say that I am a bit surprised that the functoriality of B is not very easy to find. Things like this question https://math.stackexchange.com/q/1458470/91103 make it seem like this was common knowledge, and nobody in the comments asked about that construction.

[tight agate]

this is the problem with alg top

people don't write shit down

[marsh forge]

ive found a lot of things in alg top have an issue where like

i cannot google them for the life of me

but i can dig them up in textbooks

[flint cove]

In that case I'm glad that I haven't set a deadline for my BS thesis yet, lol

[marsh forge]

or long manuscripts

[tight agate]

what's your thesis on?

[sleek thicket]

yeah unfortunately it's from an unreleased book, sorry

[flint cove]

characteristic classes and in particular the chern character

[sleek thicket]

oh he did ask me if I could help him edit it/keep looking at draft copies after the class ends though

That was kind of cool

[tight agate]

from which POV

[flint cove]

I don't know which POVs there are, but my main guideline is Hatcher

[tight agate]

there's also the super nice diff geo POV

[flint cove]

Ah, yes, with the curvature form stuff

I remember

[tight agate]

TAKE BACK THE SULLY

[marsh forge]

i will sully diffgeo however i please

[sleek thicket]

I think that's what my class ia doing right night

But I got very lost

Turns out not having a textbook+awful diff geo computations is bad

[marsh forge]

you can remove the first summand

[tight agate]

chern-weil theory is super nice

[flint cove]

that diffgeo view actually the stretch goal, i.e. to give a description both ways and show that they're equivalent
But to give a complete intro do the algtop view will be enough work on its own I think

[tight agate]

read huybrechts

[tight agate]

noice

[flint cove]

Most texts I've read so far do some intro-level talk and at the hard parts do stuff like „we can prove that with spectral sequences“ and you're just like WTF

[tight agate]

there's another POV via lambda rings and stuff

and you do everything via the splitting principle

which is super nice

[flint cove]

What are lambda rings?

[tight agate]

rings + operations which behave like exterior power operations

[marsh forge]

this is the woke part

[tight agate]

so for each natural number n, you have a map \Lambda^n: R ---> R

[flint cove]

Oh, the big lambda.

[tight agate]

and then these Lambda^n satisfy pretty much all the relations that exterior powers satisfy

yeah and once you have this you can define Adams operations and stuff

[flint cove]

I've not quite gotten to internalize the splitting principle, but it seemed like embedding H(bundle, A) into H(Σline bundles, A) via the pullback of the flag bundle, which is inherently nicer

What counts as an „Operation“?

[tight agate]

?

Adams operation?

[flint cove]

I've heard of things like Adams operation and „cohomology operations“ in general, but have no Idea what these are

Oh, it's an NT between cohomology functors

nice

[tight agate]

adams operations are a bunch of operations on K-theory

but these can be defined on any lambda ring

the k-th adams operation acts as the k-th power operation on line bundles

and this determines them

by using the splitting principle

but yeah there's an explicit definition using newton polys and stuff

[flint cove]

I think I'll keep that in the back of my head for a few weeks when I've understood the splitting principle and the necessary polynomial relations a bit better, but it sounds interesting.

[tight agate]

I need to read more about this stuff too

once you have the lambda ring formalism set up there are a whole bunch of weird filtrations you get on the ring

and I have no intuition for them

[shut moat]

@marsh forge

An understanding of exact sequences and very basic homological algebra would also be good.
if I have zero clue about both of these things, can I still get something out of your talk?

[marsh forge]

Hm

the issue is that the computations would be largely nonsense

and i think the meat of the talk is actually like, how you go through the computations

rather than the results being that interesting

[shut moat]

ah, rip

[marsh forge]

I mean it wont be that long and i wont get mad if you leave in the middle

so no reason not to just show up if youre interested

[sleek thicket]

wait where actually is #events

I can't see it in the sidebar

[tight agate]

introduction

[sleek thicket]

ah okay found it

dang looks like we have quorum

[flint cove]

Well the lazy way would be to just write #events and then click on the link 😄

[sleek thicket]

You found my secret plan all along :P

[gritty widget]

max's talk

finally some good content onthis server

[marsh forge]

i hope to inspire

other people to give talks

[sleek thicket]

My posts are good

[marsh forge]

in fact i will make mine intentionally bad

to set the bar low

all part of my plan

[chrome dew]

I'm inspired, I too will give an intentionally bad talk

[gritty widget]

I'm inspired, I too will give an intentionally bad talk

[flint cove]

“This talk intentionally left bad”

[marsh forge]

maybe i should learn how to use inkscape for this

i want to draw out all the SS stuff beforehand

so people dont want to kill themselves trying to keep up w the indicies

[gritty widget]

just wanna get my definition correct. Is it true that the closure of (GnH) is: R - (GnH)
Where G,H are subsets of R

[river granite]

no, that sounds like the complement of G\cap H if both were subsets of R

[gritty widget]

oh oops

i forgot to mention yeah they are

because im looking at this solution here, and just wondering on this part

[river granite]

gee that's some awful notation for the complement

anyhow seems like more of a set-theoretic question than a topology one

[gritty widget]

hmm i guess it is. well thanks anyway

[jolly ice]

I have a question on some basic topology

Proving every bounded perfect set is seperable

I have a proof but need to confirm can someone help me in off topic voice?

Or 384kbs voice

<@&286206848099549185>

[wind portal]

How does this point radially inward friends. I thought it would be the negative of this vector field?

[gritty widget]

it points inwards because it takes points in the cube S and decreases their magnitude

i.e. makes them go into the origin a little

imagine a box and then a sphere sitting inside, both concentric. if you take a point on the box and then project it to the sphere along the line though the origin, it's going inwards, yeah? (if there's some rigorous definition of "points inwards" that i'm missing that you mean, that isn't just "look where points go and draw arrows," ignore this)

[cloud owl]

but the actual function should have a minus sign in

wait no

phi is just a scalar

oh no wait it isn't

it's a vector divided by a scalar

ok so rn for phi(0, 0, 1), it returns (0, 0, 1)

which points outwards, not inwards?

[wind portal]

That's what I thought but I'm reading @gritty widget explanation

[cloud owl]

wait no

[cloud owl]

what comes out of phi isn't the adjustment

it's the end vector

ok, and (0, 0, 1) is on the sphere so that makes sense

ok and for any other vector it just normalises it

this is just a function that normalises a vector

which gives the vector of a point on the sphere

not the change between the point on the cube and the projection on the cube

ok, right, makes sense

i'm too tired for this lol

[wind portal]

@gritty widget I think I understand

[gritty widget]

[shut moat]

that's lee isn't it :o

[tough hill]

#groups-rings-fields message

[sleek thicket]

I got invited to come to a stable homotopy theory seminar?!

I'm gonna be so lost lol

This thingy https://www.math.purdue.edu/~pvankoug/spectra/index.html

[cedar pebble]

oh sick

[quasi forum]

Okay. So I have questions about this.

1. I found out the sets in 3i) are equivalent, however, I believe I am about to come to the conclusion that the relationship for both 3ii) and 3iii) are supersets. How can this be?

[quasi forum]

nvm, I think I just concluded that they are all equivalent.

[quasi forum]

Hold on, is the third one equivalent? If not, I could use some help piecing it together.

[native raptor]

oh that's awesome!!

[long coyote]

https://math.stackexchange.com/questions/1971562/union-of-connected-subsets-x-y-is-connected-if-overlinex-cap-y-neq-var

why does the accepted answer define a continuous function

if we let $x\in\bar{A}\cap B$, then there exists neighborhood x such that x intersets with A is not empty

[gentle osprey]

亜城木 夢叶

[river granite]

a subset $A$ of a top. space is connected iff every continuous function from $A$ to ${0,1}$ is constant.

[gentle osprey]

derivada.schwarziana

[zealous fossil]

hi

[sleek thicket]

hello!

[zealous fossil]

stuck on this one

i think this is the right channel for it, I believe

[sleek thicket]

This is more appropriate for #geometry-and-trigonometry imo

[zealous fossil]

really i find it to be advanced

[zealous fossil]

or maybe too hard for that

[sleek thicket]

It's not really about difficulty, more subject matter

[zealous fossil]

ah ok i will post it in there, nonetheless do you think u could guide me to the solution?

[sleek thicket]

sure, I'll think about it

[sacred gust]

wait is this an amc problem

[zealous fossil]

no its a cayley proglem

problem*

[sacred gust]

oh

[zealous fossil]

thanks boss

[coral gale]

is this close to being right

i have a feeling it's really wrong

G is a lie group here

[gritty widget]

i don't think $\varphi_p$ is the restriction of the projection $G \times M \rightarrow G$ is it?

[gentle osprey]

8da | dumbass

[gritty widget]

oh, unless you mean something like, there is a commutative square involving $G \times {p}, Orb(p), G\times M, M$, and we wish to show that the map $G \times {p} \rightarrow Orb(p)$ (inverse of $\varphi_p$) is smooth, given that the other maps are smooth

[gentle osprey]

8da | dumbass

[coral gale]

@gritty widget aaaa i'm so confused

[sleek thicket]

ive decided partitions of unity are bad actually

huh

that is very cute

brb stealing this and passing it off as my own for clout

[tight agate]

petthecat

[sleek thicket]

https://twitter.com/grassmannian/status/1366638203340398592?s=20 no

Hi I’m Brendan (mean) (@grassmannian)

Just realized "partitions of unity" is an oxymoron 🤯

Twitter · Mar 2, 2021, 6:35 AM

[shut moat]

https://tenor.com/view/smh-kanyewest-gif-4544077

[sleek thicket]

anyways I need to think really hard about why a certain cover admits a partition of unity

and do not see it

bad

oh okay

https://twitter.com/grassmannian/status/1366639037038030849?s=20

Hi I’m Brendan (mean) (@grassmannian)

okay i fucked up yall I stole this from someone else. I am ashamed and will do better in the future. Thank you "Ultrapyknotic set" for pointing this out to me, you are the best of us

Twitter · Mar 2, 2021, 6:38 AM

okay so homework time

[tight agate]

lol

[sleek thicket]

i thought about doing a notes app apology

but that's too much effort for 10:39pm

here E -> M is a rank k numerable vector bundle

[tight agate]

10:39pm implies PST

doxxed

[sleek thicket]

,ti shamrock

[gentle osprey]

The current time for shamrock is 10:39 PM (PST) on Mon, 01/03/2021.

[marsh forge]

why do ppl write like that

spacing is your friend

[sleek thicket]

lol

i am

lazy

Fram

[marsh forge]

its too much effort to even proof read

[sleek thicket]

you don't know fram?

[marsh forge]

if it looks like that

[sleek thicket]

she's an cool guy

I refuse to use new lines, they are for cowards

[tight agate]

where is she fram

[sleek thicket]

it's a pretty trivial neighborhood

[marsh forge]

i use [] like

oh no

[gentle osprey]

Carly Rae Maxsen

[marsh forge]

\[\]

[sleek thicket]

oh no

[gentle osprey]

Carly Rae Maxsen
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[sleek thicket]

oh no

[marsh forge]

every 3 sentences

[sleek thicket]

this is not more readable max

[marsh forge]

it is!

[sleek thicket]

you've made the bot angry

so

less readable

[sleek thicket]

wtf is this

[marsh forge]

it thinks im display mathing

discord killed the backslashes

on the first try

[sleek thicket]

yes i understand lol I'm joking that the accidental triggering of the bot makes your messages unreadable

[marsh forge]

oh

[sleek thicket]

ergo it's bad to use \[ \]

ugh

how do you not do that???

[marsh forge]

time for Latex Conventions Discourse

[sleek thicket]

lmao

alright im going back to

[marsh forge]

if you use double dollarsa

[sleek thicket]

thinking real hard

about

[marsh forge]

instead of brackets

[sleek thicket]

why this admits a partition of unity

[marsh forge]

i will literally kill you

[sleek thicket]

why would I use double dollars

i have no need for newlines

jk jk i of course use them

[shut moat]

the slash sign is so much effort to reach tho

[marsh forge]

sham if ur REU paper looks like that

im gonna make peter kick u out

[sleek thicket]

lmfao

it's gonna look worse

I promise

[marsh forge]

pls

[sleek thicket]

this has been very unhelpful thank you everybody

i am going back to my homework

well you did point out the typo

I removed fram

i do not think a partition of unity exists

gonna need to like

think harder about this

pain

me rn being stuck

[marsh forge]

have u considered just boldly asserting it

[sleek thicket]

no, because I think it is likely false

I only boldly assert things I've already checked

[marsh forge]

all the more reason

[sleek thicket]

okay soi guess gentop question

supppose X admits a cover which admits a partition of unity

what does this tell me about X

or about other covers of X

oh no this is very boring

because I can just cover by X

hnnnnng

so a bundle then

with a numerable trivializing cover

[near crypt]

So I’ve got this question but I’m gonna give what I know first: a cube orthogonally projected onto a plane from the right angle is essentially a regular hexagon, which can then be divided into three rhombi that represent the three sides.

I discovered I can model an octahedron from the same angle by connecting the midpoints of the faces (because they’re duals) or just by drawing an equilateral triangle “inscribed” (not sure if that’s the right word) in the hexagon.

My question is, how would I go about drawing other Platonic solids from the same angle? As far as I know a tetrahedron is a quadrilateral made of two similar triangles, one the mirror of the other, connected. But I don’t know the exact angle measures or the ratios of sides to each other, both of which are important to know.

Here’s a picture of what I’m talking about:

Pixel art is my medium

Also I did figure out an icosahedron would be really similar to the octahedron, just the triangle in the middle would be shaped differently

[coral gale]

does this work

[gritty widget]

i was thinking something like this diagram
$\begin{tikzcd}
G\cong G \times {p} \arrow{r}{i} \arrow[swap]{d}{\varphi_p^{-1}} & G \times M \arrow{d}{\rhd} \
Orb(p) \arrow{r}{i}& M
\end{tikzcd}$
and presumably that the fact that all other maps besides $\varphi_p^{-1}$ is smooth implies that $\varphi_p^{-1}$ is smooth as well. but i can't say i am good enough at manifolds to know off hand if this is true/how to show it

what is the p map in your diagram? oh, you are saying it is projection

[gentle osprey]

8da | dumbass

[coral gale]

m

i think that's essentially the same as my triangle

[gritty widget]

yeah i agree

[coral gale]

but yea same i have no idea if this is justified/how to justify it

oh and yes whoops p is my projection a bit overloaded mb

[gritty widget]

it looks like lee's ISM page 166, proposition 7.26 has some relevant discussion

actually it looks like it is basically this problem lol

[quasi forum]

I have a question: why are cl(A) and cl(int(A)) Not necessarily equivalent?

[gritty widget]

what if A is a closed set with empty interior

[quasi forum]

So like a set of isolated points?

[gritty widget]

that works

it doesn't have to be a set of isolated points though, e.g. R x 0 in R^2

it's closed in R^2 so it's its own closure, and its interior is empty so its interior's closure is empty

[quasi forum]

Ohhhh, I see! That makes a lot of sense.

Thank you. I was struggling with that intuotion. I have one more question

I am not really sure what notions of boundary we can use to solve 3iii)

[median glade]

try closure - interior

[quasi forum]

The thing that throws me off is i) and ii) are equivalent. But iii) is not. I can't wrap my head around the intuition for that

[quasi forum]

Can you explain what's going on here?

[median glade]

the square is the product of two line segments

the boundary of square is the black square

but the boundary of each line segment is just the two red dots

[quasi forum]

Sorry, I was studying while I was thinking about this, but I think I understand what you are saying.

So bd(A)xbd(B) would simply just be those 4 dots, where bd(AxB) would be the border of the entire square.

This also doesn't break apart the equalities from i) and ii)

[median glade]

exactly

[quasi forum]

How is this?

I mean it gets the job done, but would you say I'm missing any important steps?

[gritty widget]

this picture is kinda blurry, can you type it in @gentle osprey ?

[quasi forum]

Let me just get a better picture. LaTeX will take a bit too long right now

Better?

[gritty widget]

Can someone help with sections on a bundle over a torus? Consider the trivial bundle T^2\times C^2, i want to show that a section of this bundle can be given as two smooth functions on T^2

So i know that a section is a map s:T^2 to T^2\times C^2 such that s(p) lies in the fibre at p

It seems this should be $\eta_i\in C^{\infty}(T^2,\mathbb C)$ but I thought $C^{\infty}(M)$ always meant real functions

[gentle osprey]

lime_soup

[gritty widget]

Hi, we just started an intro to diff geometry course, and we're parametrizing curves a lot

So I'm wondering, do I need to memorize all of their parametrizations?

Or just the basic ones like ellipse, hyperbola, parabola...?

you'll probably end up working with them enough that you'll remember them

no point in sitting down and trying to memorize various parametrizations of common shapes

okay thanks

[brittle beacon]

Hello! This is probably a stupid question but isn't the final topology (https://en.wikipedia.org/wiki/Final_topology) always the discrete topology, since any function from the discrete topology should be continuous? Like if $f:X\to Y$ where X is equipped with the discrete topology and Y is equipped with some other topology then if U is an open subset in Y, we have that $f^{-1}(U)\in P(X)$ so it is open hence f is continuous.

[bitter yoke]

It seems that you're reading it backwards? The maps are from Y to X

[brittle beacon]

OHH

I see, thanks!

[gritty widget]

nice tex

[brittle beacon]

lol

[bitter yoke]

its cause of the underscore in the link

latex needs underscores to be in math mode

[brittle beacon]

I see, idk how I would fix it though

[bitter yoke]

\url{https://en.wikipedia.org/wiki/Final_topology} $f^{-1}$

[gentle osprey]

Zopherus
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[bitter yoke]

yea you need the package rip

[gentle osprey]

slimvesus
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[gritty widget]

\text{https://en.wikipedia.org/wiki/Final_topology}} $ f^{-1} $

\text{https://en.wikipedia.org/wiki/Final_topology}} $f^{-1}$

[gentle osprey]

Vanhousen
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[gritty widget]

lol

[bitter yoke]

think its just discord formatting being weird

[gritty widget]

Am I the only one that likes to pronounce topology with a hard g?

I guess I am

how hard of a g are we talking?

As hard of a g as I am 😎

[cloud owl]

yes, you are alone and you should feel bad about not being more like me

[gritty widget]

Can someone help me understand a connection in this context?
As a map from $\Gamma(E)\rightarrow \Omega^1(E)$

[gentle osprey]

lime_soup

[gritty widget]

So I have a trivial bundle over the torus E=T^2\times \mathbb C^2

Since this is a trivial bundle we can think of any section as s=(s_1, s_2) where s_i are smooth functions from T^2 to C

I want to see that D which maps from sections on E to one forms on E by Ds=D(s_1,s_2)=(ds_1,ds_2) is a connection

but i

not even sure how to see concretely that d is a connection on the trivial bundle

[tough imp]

You say topolo-ghee?

[gritty widget]

I like to say it this way

I am admittedly shy to say it this way in front of other people though

[tough imp]

D:

[median glade]

Are you German?

Pretty sure it's like "Topologie" with a hard G in German

[gritty widget]

in the definition of a conncetion we have this fs

clearly it doesn't mean composition because that doesnt work

I also assume it doesn't mean multiplying

so how does a smooth function act on a section

Oh lol no, I didn't know that, interesting though. I just have a weird inclination towards pronouncing things weird

[bitter yoke]

@gritty widget I might be misunderstanding, but

at every point on p on M, you get a real number f(p)

so like f \nabla(s) at p is just f(p) \nabla(s)(p)

[gritty widget]

ah yes the sections are module over the smooth functions

thank you

[bitter yoke]

right

[gritty widget]

now the other think, lets say have a trival bundle, the the exterior derivative should be a connection on this

on the trivial bundle our smooth section s will just be a smooth function from M to R

so f*s is again a smooth fuction

so we can indeed act on this by the exterior derivative

where does the vector field come into play?

do we just evaluate the form dfs on the vector field?

[digital peak]

how do I extend a smooth function from a closed embedded submanifold to a neighborhood thereof?

[gritty widget]

smells like partitions of unity

one of your open sets should be ambient manifold setminus closed submanifold ( iirc the "closed" condition just ensures you can choose the neighborhood to which you extend to be the entire ambient space. you don't need to assume closed to extend to just a neighbourhood of the submanifold.)

[sleek thicket]

Partitions of unity reduces you to a closed set in R^n

Oh or even a closed line segment by taking rank theorem charts

Which is very nice

[gritty widget]

Is anyone good with bundle stuff?

[sleek thicket]

If it's about the stuff you're doing above I think I'm learning it right now too

But am not comfy enough to give definitive answers

[gritty widget]

Can i try ask anyway even for rubber duck purposes?

[sleek thicket]

Sure

[gritty widget]

I have a trivial bundle over torus, T^2 \times \mathbb C^2. We will call this C^2

[sleek thicket]

yup

[gritty widget]

I have a nice map from the torus into two by two matrices with complex entries defined by

[sleek thicket]

sugar are f, g, h ?

[gritty widget]

oh and here are the f g h

[gritty widget]

But what if my manifold is R, and my SUV manifold is everything except the origin. Then smooth functions on the SUV manifold cannot generally be extended

[sleek thicket]

And are θ, φ points on the circle?

[gritty widget]

yeah view the torus as S1 \times S1

[sleek thicket]

The result in general is that they can be extended to an open set

[gritty widget]

its not hard to show p^2=p

but I have no idea why this gives a map from the bundle C^2 to C^2

Ah I see, the neighborhood here being the whole suvbmanifold itself

[sinful pecan]

matrix multiplication?

[gritty widget]

Sorry what do you mean by that alias?

C^2 is T^2\times \mathbb C^2

[sinful pecan]

elements of C^2 as 2x1 row vectors

you have a map for every point, which acts fibrewise

[gritty widget]

If you could introduce a social ranking system here I wold transfer a large amount of my score thank you

if we*

[sinful pecan]

np

[gritty widget]

im assuming this map acts fibrewise because p^2=p

but how do we correspond a point on the torus to that matrix?

[digital peak]

the question is how to extend it locally at all

I figure if I can extend to an open set, I can extend to the entire manifold by partition of unity

but how do I do that?

[sleek thicket]

choose rank theorem charts

so you look like R^n in R^(n+k)

[digital peak]

what

[sleek thicket]

what do you know about embedded submanifolds?

[gritty widget]

We are already embedded, so k should be 0 right?

[digital peak]

the definition

[sleek thicket]

no 8da, S^2 is embedded in R^3

[digital peak]

that's about it

[sleek thicket]

ah, that's trickier

[sinful pecan]

lime_soup what do you mean correspond a point? The mapping P is a correspondence

[gritty widget]

Ah, maybe I am getting my definitions mixed up, I thought embedded meant codimension 0

[digital peak]

frankly we hardly defined what df is

the definition was on the order of "given a chart on here and a chart on there this composite map has full rank jacobian" or something

[gritty widget]

What i mean is, p is a map from T^2 to Mat(2,\mathbb C)
we want to use this to define a map P on the bundles so we send (theta, phi) , (z,w) to (?,?)(p(theta,phi)*(z,w))

[sleek thicket]

so what's your definition of embedded submanifold? Topological subspace with the structure of a smooth manifold where the inclusion is an immersion?

[gritty widget]

does my question make sense (pretend (z,w) is a column vector?

i don't know what we do with the basepoints in our map

[sinful pecan]

i see, i just read it as (theta, phi) , (z,w) to (theta, phi)(p(theta,phi)*(z,w))

[gritty widget]

oh okay

[digital peak]

immersion via jacobian on charts, a submanifold is an image of an injective immersion, an embedding is when the submanifold is homeo to the domain

[sleek thicket]

cool

[gritty widget]

I thought there was some way to maybe take the columns in the matrix and turn them into complex numbers and then somehow relate this to a base point

but yeah it it makes more sense that is just the identity onteh base poits

in the base*

[sleek thicket]

Okay so we can wlog to be in R^n

[digital peak]

"sure"

[sleek thicket]

There are nice ways to characterize embedded submanifolds here

But I guess you don't know those

I'm just trying to avoid reproving a bunch of stuff that would simplify this

and failing

[digital peak]

I mean modulo locality and charts, we can talk about an embedding R^k -> R^m

that much is evident

[sleek thicket]

Yup

[digital peak]

showing that "slice coordinates" exist seems nontrivial

[sleek thicket]

Yeah, it is

[gritty widget]

how would we prove that the bundle map between $F(TM)\to M$ and $F(TM)\to F(TM)/GL_n(\bR)$ is an iso

[sleek thicket]

But it's also the nicest way I can think of to do this...

Maybe I'm being a coward though

[digital peak]

especially since it's like not true if we lose homeo

[sleek thicket]

well sort of. You can still solve the local problem

[digital peak]

not in the points where homeo is broken

[sleek thicket]

I mean that if S <= M then there's a cover of S by S-open subsets {Ui} where Ui is embedded in M

[gritty widget]

are you sure you have phrased that properly spinsicle?

[sleek thicket]

but then these may fail to be open in M

[gentle osprey]

spinsicle

[sleek thicket]

I'll think about this mniip but it seems very irritating

No promises I can figure it out

[gritty widget]

sorry there we go

[sleek thicket]

I think you need some kind of slice coordinates thing

By which I mean

You need to apply to inverse/implicit function theorem at some point

I just don't see any way to get local control of an embedding otherwise

[gritty widget]

why can't mnlip just use a partition of unity?

[sleek thicket]

that takes you to the local problem

how do you solve the local problem?

[gritty widget]

So we have some f:V to R?

[sleek thicket]

yup

Or even f : R^n -> R

And an embedding F : R^n -> R^m

[gritty widget]

and just so I understand the problem, we have this f:V to R, and we want to extend this to f':M to R where f and f' agree on V?

[sleek thicket]

no

we have some embedded submanifold N of M

and a smooth function f : N -> R

And we want to find an open set U of M containing N on which there is a smooth extension g : U -> R of f

extending to all of M may not be possible

Consider 1/x on N = R\{0} and M = R

[gritty widget]

oh okay

[digital peak]

right

the entire problem asks about Im(f) closed

at which point you readily apply partitions of unity

but you have to extend f to U first

[sleek thicket]

It's easy if you know things about embedded submanifolds

[digital peak]

might as well know the answer

[sleek thicket]

I didn't mean that as a diss?

You don't have a lot of tools for working with embedded submanifolds

Mniip I guess my advice is to try and use the IFT

It's the only way to get reasonable info out of something being an embedding

but like, all the ways I can think to apply it end up proving slice coordinates exist

Idk

[digital peak]

I'll sleep on it

thanks

[gritty widget]

If i have a map from a manifold into some nxn matrices, is the differential just done entry wise?

For example if I have this map p :T^2 to 2x2 matrices, and here f,g,h:S^1 to \mathbb C

[sleek thicket]

Sort of yeah

Think about it as a map into R^n²

And the differential of a map X -> M × N is just given by the direct sum of the differential of X -> M and X -> N, identifying T_(p, q) M×N with T_pM (+) T_qN

[gritty widget]

Ty

[obtuse meteor]

wow some of the stuff you get for free groups

from covering space theory

is really really cool

[tight agate]

nielsen-schreier?

[obtuse meteor]

the better version (tm)

which is like a corrolary / computational thing

if you have a map F -> G where F is free

and G is some group

then thinking about the kernel of this map often allows you to construct the universal cover graph explicitly

and grab generators for the kernel

explicitly

[near crypt]

Well I’m dumb, the tetrahedron turned out to be two 45-45-90 triangles and so it’s just a square

Weird how that works

Thank you for the answers both of you though

[rough obsidian]

Hello, can anyone tell me the difference between a basis and a subbasis of a topological space? Thank you!

[shut moat]

Bases generate topologies via arbitrary unions, subbases generate topologies by arbitrary unions + finite intersections

[rough obsidian]

Thanks a lot, may I ask what do you mean with finite intersections? I understand the fact that they generate topologies via arbitrary unions

[shut moat]

as in, just taking unions is not enough to generate the whole topology with a subbasis

you need to take unions of intersections of the elements of the subbasis

another way to think about it is if you add all the intersections to the subbasis, it becomes a basis

[rough obsidian]

Okay, but if you intersect elements of a subbasis you get a "smaller" element, then how can it become a basis?

[shut moat]

For a collection of subsets to be considered a topology, it must be closed under arbitrary unions and finite intersections

A subbasis on its own may not necessarily satisfy the latter condition

By using intersections, you guarantee that this works

[rough obsidian]

oh oh

I get it

Thanks a lot!

[shut moat]

np!

[gritty widget]

Does anyone have an idea for this? I'm trying to show that if $\phi: V\rightarrow V$ is a map of symplectic vector spaces (not nesccairly linear). Then
The graph of $\phi$ is a lagrangian subspace of $V\times V$ if and only if $\phi$ is a linear symplectic map

[gentle osprey]

lime_soup

[gritty widget]

symplectic map means phi pulls back the symp form on V to itself, yeah?

yes

there is this very slick proof which even works in the case of symplectic manifolds

but I'd like to be able to do this concretely as I feel there is something good to understand from doing it concretly

also its actually not clear to me why this proof works

it's worded a little strangely imo

let me attempt to clarify it

[gritty widget]

at least in the context of this

(if it holds for a vector space it ought to hold for manifolds by doing everything fiberwise; this is me being lazy)

let's say $\phi\colon V \to V$ is a linear symplectic map. the linearity part says $\Gamma_\phi$ is a subspace of $V \times V$, and since this subspace is isomorphic to $V$ via the map $\gamma \colon v \mapsto (v,\phi(v))$, you get that its of dimension $\frac{1}{2}\dim V \times V = \dim V$. let's write the symplectic form on $V \times V$ as $\Omega = \pi_1^\omega-\pi_2^\omega$. now, we wanna check that $i^\Omega=0$, where $i \colon \Gamma_\phi \hookrightarrow V \times V$ is the inclusion, yeah? it's enough to check that $\gamma^(i^\Omega) = 0$. if we write this out, we get $$ \gamma^i^\Omega = (\pi_1 \circ i \circ \gamma)^\omega - (\pi_2 \circ i \circ \gamma)^\omega = \omega - \phi^\omega = 0, $$ showing that $\Gamma_\phi$ is a lagrangian subspace of $V \times V$.

[gentle osprey]

(T*Terra, dqⁱ ∧ dpᵢ)

[gritty widget]

i'm trying to be really explicit about the forms and pullbacks involved

ooh

i am

silly

if $L$ is lagrangian, then $L^\perp=L$

also the converse follows from the computation at the end

[gentle osprey]

lime_soup

[gritty widget]

so L is lagrangian iff the form is identitly zero on L

ah i probably should hvae asked what your definition of lagrangian is lol but it seems like you got it

Thank you very much though

no problem

Is the dog in your profile picture the same dogs that were in Shamrocks previous picture

just for max clarity: the definition of "lagrangian" i had in mind is: sub(space/manifold) of dimension half that of the ambient space onto which the symplectic form of the ambient space/manifold restricts to 0

[gritty widget]

it is not

it's the beastars wolf

also

If V is a symplectic vector space

then the only lagrangian subspace is 0?

But a presymplectic vector space can have lots of lagrangian subspace?

If V is a symplectic vector space
then the only lagrangian subspace is 0?
i am not sure about this, although to be completely honest i couldn't give you an example right off the top of my head (lee's smooth manifolds book 1000% has something on this)

he does all the usual symplectic linear algebra stuff and also tells you how to find symplectic/isotropic/coisotropic/lagrangian subspaces, in terms of coordinates where the symplectic form looks like the standard one. maybe you can look there?

oh well i guess if you take the usual symplectic form on R^2 you get a lagrangian subspace by looking at an axis LOL

[tight agate]

x_1, x_2, ..., x_n is a lagrangian subspace

following the usual naming convention

lagrangian subspaces need to have dim n (if the vector space has dim 2n)

[gritty widget]

what brofibration is saying is exactly the thing in lee i was thinking of

[tight agate]

noice

[gritty widget]

yeah there's an exercise in lee that tells you exactly how to construct nice subspaces of symplectic v spaces using a symp basis

one of them is a lagrangian subspace

[gritty widget]

Okay thanks

That was a good mistake to make

[tight agate]

petthecat

tfw R^2n w/ standard simp form is the only simp mfld I know

[gritty widget]

[tight agate]

[gritty widget]

S^2

[tight agate]

yes

cotangent bundle

and every projective variety

is simp

right?

yeah they're kaehler

[gritty widget]

not sure what a projective variety is but i guess so

[tight agate]

it's a variety

and a closed subscheme of P^n

the kaehler structure can be given by the restriction of the fubini-study metric

i think

and once you have the almost complex structure and a riemannian metric

you can recover the simp form

[gritty widget]

Does anyone know what this means? I thought a classification would be any subspace spanned by x_i’s in the standard basis

But then

This is given as the classification for coisotropic

And it doesn’t really seem to tell us much about coisotropic spaces?

[gritty widget]

Okay so i assume the idea is to show if W,W' are isotropic there is and automorphism as above that maps W to W'

I know we can do this for Lagrangian subspaces

as for W Lagrangian, W\times W^* is iso to V

[nimble jolt]

@tight agate that's not a bad symp mfld to know, given that locally it's the only one! 🙂

[gritty widget]

your brain on microlocal analysis

[tight agate]

broser trick

[gritty widget]

i believe it is spelled simp 🙂

[fading vale]

Let S be the pseudocircle

[gritty widget]

[shut moat]

tf is a psuedocircle

wtff

[gritty widget]

[tight agate]

lol

@fading vale https://math.uchicago.edu/~may/FINITE/FINITEBOOK/FINITEBOOKCollatedDraft.pdf

[fading vale]

[gritty widget]

[tight agate]

[obtuse meteor]

finite sets that look like cw complexes up to algebraic topology

really scare me

what's like

the explicit generator of the fundamental group for this dude

ah you can determine this

with groupoid Van Kampen

what happens if you enforce hausdorffness

[gritty widget]

but all topological spaces are hausdorff

[obtuse meteor]

🧠

[sweet wing]

hi i personally dont identify as absolutely flat rings

too flat for me

[little hemlock]

im having trouble seeing how this computation ive bracketed in blue works

[wanton marsh]

the first line is the LHS of 3.12 wedged with ek1...ek(n-r)and the second line is the RHS of 3.12 wedged with ek1...ek(n-r)

[gritty widget]

a bilinear map is positive definite if f(x,x)>0 with equality only when x=0?

I'm actually finding it very hard to find a definition for it but this seems like the only thing that would make sense

as in requiring f(x,y)>0 would not make sense since f(x,-x)=-f(x,x)>0 implies f(x,x)<0

[gritty widget]

good ol' nlab

[gritty widget]

i really like these meta arguements on what a definition should be

assuming the definition is good it cannot mean such and such

[shut moat]

are there any fun exercises applying point set top (besides manifolds)

topology hw has been a bit grindy and dry lol

[marsh forge]

go through an algtop book and explain why all the technical conditions are necessary

and then tell me

[obtuse meteor]

lol

[shut moat]

[obtuse meteor]

technical conditions scary

give me the technical conditions for when a CW complex product agrees with the traditional product

[marsh forge]

after some research

apparently the answer

depends on CH

which is crazy

but if either X,Y is locally finite or if both are locally countable

it works out

[obtuse meteor]

I hate this I hate this I hate this,,,,

do you have a link???

like holy shit that's so fucked

poll: should I introduce homotopies to my real analysis students next week at the discussion section office hour?

(the context of the discussion section is to give them exposure to other parts of math)

(I've been doing topology bc I'm a nerd)

[fading vale]

lol faye are u a TA

its not a bad idea but at the same time it might be better to go broad and cover math from lots of different areas than to keep doing topology

up to u tho (and maybe ur students xd)

[marsh forge]

homotopies are boring imo

unless you learn about like

uses of them

but certainly not really self-motivating

id teach them some group theory or smth

[obtuse meteor]

I have been teaching them group theory stuff

topic list so far

https://estrogen.fun/i/ad92.png

We're doing quotient space today

(Some basic definitions includes topological space + continuous + connectedness, which they're also getting some of in class)

they really like the topology stuff so far. I could cover some elementary number theory or combinatorics if we wanted to?

[marsh forge]

Do some computability theory

or some dynamics

rotations of the circle is the example that keeps on giving

or expanding maps of the circle

[obtuse meteor]

idk anything about either of those lol

[marsh forge]

not 2 late 2 learn

[obtuse meteor]

I'd be willing to learn the dynamics stuff

computability theory is

cringe

[marsh forge]

its cool!

The two even intersect

the theory of like ML random points of a computable metric space

is cool stuff

[fading vale]

just randomly write the word ergodic on a combinatorics paper

done

[crimson imp]

Hey, guys, is this set simply connected? I’m trying to define a universal cover and need this condition. As I draw it’s a subset of $R^3$ consisting of one of the axis with a bunch of tangent spheres which doesn’t intersect each other

[gentle osprey]

pmorelli

[marsh forge]

You can prove it is simply connected

I assume you've seen SvK

if you're doing covering theory

[crimson imp]

I’m studying covering theory but what do you mean by SvK?

[marsh forge]

Seifert Van Kampen

[crimson imp]

Ah yeah, I saw that

I never used it but I know the statement

[marsh forge]

interesting

well

it makes computing this pretty straightforward

but to answer your question yes it is simply connected

(and the universal cover of S^2 v S^1 if thats what you are going for)

[crimson imp]

Ok, I’ll try applying it

[crimson imp]

Yes, that’s the base space

Thanks!

[marsh forge]

np

try computing it for just one sphere wedged with the axis

its the same for countably many

but it might be less intimidating

[obtuse meteor]

you can also do it by applying CW complex stuff

CW complexes are homotopy equivalent to quotienting by a subcomplex

so just identify the line

then this is a wedge of spheres

a sphere is always simply connected

and wedge of simply connected is simply connected

this is how I would do it because I avoid SvK like the plague

[marsh forge]

you still need SvK

to prove that second to last statement

SvK is a good theorem

[marsh forge]

you mean contractible subcomplex right

[fading vale]

SvK makes a lot of sense when u think about it in terms of "when do properties of Top get preserved under the fund. groupoid"

before that i always found it kinda unintuitive

[sleek thicket]

pushie pushie

[fading vale]

when it came to intersections anyway

yes

[sleek thicket]

I have a very good memory of my frosh analysis TA trying to explain de rham cohomology in section when we were doing div/grad/curl

he got yelled at by the professor after the prof found out, apparently

[fading vale]

lmao

based

[marsh forge]

bad take

SvK is like

so visually justifiable

you dont need to think about it w category theory at all

Its just like

[fading vale]

it is when the intersection is trivial

[marsh forge]

even nontrivial

[fading vale]

it never was intuitive to me when it was nontrivial

[marsh forge]

well i dont think ive seen a text

that explains it well

but the quotient you take is literally like

okay so you have loops in space 1 and space 2

they can interact freely

but what loops should we identify

if you look at a picture and think hard about it

i think you'd come up w exactly the relations induced by SvK

[fading vale]

i mean

it makes sense

[marsh forge]

like you're basically identifying things that can be made into eachother

by homotoping them through the intersection

[fading vale]

ik i have the visual intuition now

[marsh forge]

?

i dont see how the categorical approach could have given you the visual intuition lol

[fading vale]

it made me understand the whole picture better and when i went back and thought about it more intuitively it clicked

[marsh forge]

okay

i think you could have achieved the same result without the detour

but whatever makes it click ig

(this is a good example of the type of thing a lecturer can do better than textbooks, though)

[fading vale]

¯_(ツ)_/¯

idk seeing the pushout coproduct stuff made it clear to me that we were basically identifying the two copies of the intersection

but i also think maybe hatchers style where it was really long with a lot of exposition kind of obscures the big picture sometimes

[marsh forge]

yeh

hatcher is a bad book

ill write my own someday

[fading vale]

like the relationship between free products in group and unions in Top was vague to me

seeing it categorified made it really obvious that like oh these are the same so it makes sense that under some conditions one would become the other

[marsh forge]

I suppose so

maybe im just reactionarily anti-category-pill

[fading vale]

i dont think its necessary per se but it was good for me

[marsh forge]

but i think sometimes the niceness of categorical language can prevent people from understanding more fundamental basic/concrete stuff

you went back and understood it

so i dont mean you in particular here

[fading vale]

honestly if i hadnt done hatcher i dont think dieck would do anything valuable for me

like ur not wrong that the categorical approach alone i think can be a little unhelpful

but having already seen it from the more visual perspective the categorical approach i guess "organizes" it in a way that i find nice

also sometimes its just very neat when the two seem really different but get the same results

like hatcher doing universal covers very directly by constructing them and going for the unwinding loops perspective vs dieck doing that stuff with a fiber sequence of the htpy groups is neat

[obtuse meteor]

yah lol

[obtuse meteor]

yeah I know

[obtuse meteor]

based

[marsh forge]

this is incredibly important

im not anti-categorical stuff

i just feel like a lot of people disregard the concrete stuff

for the abstract nonsense

[fading vale]

i feel like that would suck

[marsh forge]

and as a result end up being unable to compute basic stuff

[fading vale]

i think dieck is enjoyable for me as a second very different perspective on intro AT

not really as a primary or first one

[obtuse meteor]

I agree that SvK is very visually intuitive

I haven't done groupoid version

haven't needed it

[fading vale]

the groupoid version is pretty much the same honestly

you just dont have to think about basepoints

[obtuse meteor]

doesn't it allow you to do ridiculous things tho

[marsh forge]

no

[obtuse meteor]

like just compute pi1(S^1)

just with SvK

[marsh forge]

idk if thats 'ridiculous'

[fading vale]

it does but like

[obtuse meteor]

it is to me

[marsh forge]

it seems 'pathologically overcomplicated'

[obtuse meteor]

I mean traditional SvK can't do anything like that lol

[marsh forge]

yes its more general

and good for computing particularly tricky fundamental groups i guess

[fading vale]

also faye the version of that proof i saw is like

honestly its not that different from the normal covering defn at least intuitively you still use the exp map

it still involves a morphism from the groupoid of R to the groupoid of S^1

sooooooo

[obtuse meteor]

oh I see

if you have to use the covering theory

it's not ridiculous

[fading vale]

you dont

[obtuse meteor]

well not like the full theory

but at least the fact that it's a covering space?

[fading vale]

but practically speaking the idea behind the proof is the same

no

[obtuse meteor]

ah

hm

[marsh forge]

i think there aren't that many times where the categorification approach

isnt just like

a better organized version of the same thing

there are a few examples

[fading vale]

i can send it but the basic idea is that you define a topological groupoid using the covering map from R -> S^1

so its really not that different xd

[obtuse meteor]

lol

I guess like correct me if I'm misinterpreting

but SvK will tell you that you're like some pushout of groupoids

two of which are an R groupoid

sharing a common 2 point groupoids

and the issue is that like

to compute the pushout

you essentially end up doing the same work?

[marsh forge]

oh man

this would be

a fun REU project for an apprentice

i feel

give like 3-4 proofs of \pi_1 S^1

[obtuse meteor]

lol

[marsh forge]

and compare them

very cute expository that would involve a lot of learning

ill keep it in my back pocket

[fading vale]

ehhhh like basically the idea is you define a topological groupoid G

like this

you end up with this kind of situation

[marsh forge]

jesus

[fading vale]

lmao right

[marsh forge]

that is worse than i expected

honestly

[fading vale]

ur not even done

u have to prove the functor is an equivalence

[obtuse meteor]

ah this is what I expected then

[fading vale]

then you take a basepoint

[obtuse meteor]

you want to give an algebraic description of the pushout somehow

and that's hard

[fading vale]

yea

[obtuse meteor]

(tm)

[fading vale]

its not like awful and you dont have to develop much theory to pull it off but i honestly dont understand why groupoids are so valuable

or why anyone cares that much

the proof of van kampen is like the only thing

[obtuse meteor]

is there a nice theory of presentations of groupoids?

and like

if you have presentations what happens when you take a pushout?

[sleek thicket]

topology and groupoids does a theory of presentations

[obtuse meteor]

this is what makes computing with groups nice

bc like

you can just use presentations for everything you know

and a presentation pops out from SvK

is this proof nicer when you have a theory of presentations?

@sleek thicket here's some meme material--I could technically do two REUs this summer (idk if this is allowed or not, but I could if it's not expressly disallowed)

(this would be helpful bc I need money to pay for college lol)

[sleek thicket]

I don't think that's a good idea but it is a meme

[obtuse meteor]

no it's a terrible idea

namely this is bc UMich doesn't have like

a set timeframe

it has to be 8 contiguous weeks with nothing else going on during those weeks

but SMALL is towards the latter half of the summer,,,

and if I started early,,,

[sleek thicket]

oh yeah u mich got back to my email asking about decisions and told me it'll be like a month until they're finished and I should take u chicago

[obtuse meteor]

nice

get to have Max beat you over the head

with anti-category meme

[marsh forge]

category memes are okay if you can do the concrete things when its time

[fading vale]

still think max should start pulling some strings to sneak me in but whatever

[marsh forge]

theres a great (perhaps apocryphal story)

about a grad student of Peter May's

who was so category pilled

he failed an interview question

that asked him to compute H_*(S^2)

with mayer-vietoris

[fading vale]

thats kind of fucked im pretty sure i could do that

that is way too category pilled

[marsh forge]

if you cant

put down dieck

and go do it

lol

you should already have done it before

[fading vale]

i think i have

maybe not the ring specifically

[marsh forge]

oh yes sorry

i mean its homology

[fading vale]

yea

[marsh forge]

i just mean H_k for all k

[fading vale]

oh

lol

i thought homology ring was a thing

with some wack product

[marsh forge]

if your space is an H-space

you get a product of sorts in homology

[fading vale]

yea i vaguely remember this

[obtuse meteor]

I wonder when we'll see Mayer-Vietoris 🥺

[fading vale]

probably soon faye

i <3 mayer vietoris

homology in general really

[marsh forge]

good

get cohomology pilled

[sleek thicket]

Lee yelled at me on homework for using MV

[fading vale]

i read the cohomology chapter

[marsh forge]

why

[fading vale]

4head

[marsh forge]

MV is good

[sleek thicket]

Instead of just looking at it as a polygonal presentation

[fading vale]

homology and cohomology computations r so fun

[marsh forge]

gross

[sleek thicket]

I had a space I wanted to show was the klein bottle

[marsh forge]

im basically planning to do a phd in them

[fading vale]

based

[sleek thicket]

And it's nice on a cover

and he was like

[fading vale]

ur done w/ apps right?

[sleek thicket]

"literally just track how it glues"

[marsh forge]

oh

that is headass

i thought you meant something else

by polygonal presentation

[sleek thicket]

no lol

[marsh forge]

can you even use H_* to identify the klein bottle

[sleek thicket]

yes

[marsh forge]

oh

manifold

[sleek thicket]

It was easy to see its a compact surface

[marsh forge]

yeah

i see

classification theorems are broken in dim 2

2 strong

[desert bloom]

hello

so I got iii.) wrong

and I kind of understand why

I think I have a fundamental misunderstanding

what exactly are the rules for taking the union of two subspaces?

[fading vale]

wdym "rules"

[desert bloom]

This is the answer I got

how exactly does the topology of the spaces come together exactly?

lemme think how to phrase this

[fading vale]

uhh im pretty sure whoever was grading you is wrong because if this was false then literally every topological space would be connected

in fact the defn of disconnectedness is that there exist such an A, B (disjoint, open, non-empty) whose union is X

[desert bloom]

I don't think its a general case

I think he means that there exist a space such that this is true

[fading vale]

and furthermore ur example is correct

[desert bloom]

Oh that answer isn't mine

that's the mark scheme

[fading vale]

oh lmfao

wait but u said "this is the answer i got"

[desert bloom]

bad phrasing

[fading vale]

what did u say then? true?

[desert bloom]

I know...

yeah

[marsh forge]

3 is false

for sure

lol

[desert bloom]

I lost my train of thought now

whoops

[fading vale]

thank you ultraproduct

you will go down in history as a visionary and a sage

[marsh forge]

Can you phrase your question a little more precisely

do you understand the counterexample?

[desert bloom]

Im starting to get confused myself

who am I

[marsh forge]

Or why the result should be false in general?

[fading vale]

the question is worded poorly i agree

well... eh i think its fine

[desert bloom]

I don't understand anymore

[fading vale]

but i can see why itd be confusing if you havent learned about the disjoint union topology

[desert bloom]

wait what?

[fading vale]

^

[desert bloom]

huh?

I dunno

yeah

why should it be subsets though?

oh

I might just ask him

I just kinda thought that it has subspace topology

and I thought there are specific rules for unioning two topologies

basically