Mathematics

Eric Silva

6:00 AM

the $\pi_{n}(thing) = H_{n}(other thing)$

Daminark

Oh yeah lol

Eric Silva

the application you gave sounds like bringing a gun to a knife fight lol

is there any other interesting space which is $SP(something)$

ALannister

They put one of ours in the hospital, we put one of theirs in the MORGUE!

It's the Chicago way.

anon

@Daminark I mean after you take the limit it wouldn't matter, but for finite n it changes what SP^n(X) ends up being right?

ALannister

Thanks @anon

Daminark

6:07 AM

It will, and I'm thinking if anything you don't want them to be distinct

Since otherwise you may not get the nice inclusion I mentioned above. If you have a point in $SP^n(X)$ for which one of the coordinates is already the basepoint, how are you gonna fit that in?

@Eric I don't know anything else offhand

But I'm only just starting, originally I was gonna do this paper in the break but Peter was like "Actually how about we put a deadline?"

Wait how do you otherwise prove that $K(\mathbb{Z},2) = \mathbb{CP}^{\infty}$?

Eric Silva

follows immediately from LES of a fibration

I really would phrase that as saying $\mathbb{C}P^{\infty}$ is a $K(\mathbb{Z},2)$ and not an equality

Perturbative

@EricSilva One of the topologists in my math department said using homology to prove invariance of domain was like bringing a nuclear bomb to a thumb war

Eric Silva

wait why

I don't know how to do it without homology

Perturbative

Me too

Eric Silva

i feel like proving it another way is almost kind of dishonest

because you probably have to disguise algebraic topology as other shit to do it

Daminark

6:18 AM

Kek

What's invariance of domain?

Perturbative

Invariance of domain

Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space Rn. It states: If U is an open subset of Rn and f : U → Rn is an injective continuous map, then V = f(U) is open and f is a homeomorphism between U and V. The theorem and its proof are due to L. E. J. Brouwer, published in 1912. The proof uses tools of algebraic topology, notably the Brouwer fixed point theorem. == Notes == The conclusion of the theorem can equivalently be formulated as: "f is an open map". Normally, to check that f is a homeomorphism, one would have to verify that both f and its inverse...

@Daminark You use it to show that a topological manifold $M$ cannot be both of dimension $n$ and $m$ for $n \neq m$ if I recall correctly

Eric Silva

tbh idr a use other than showing that manifolds have a well defined dimension

Perturbative

@EricSilva But's that's a pretty big use of the theorem itself

Eric Silva

sure i guess

Daminark

Wait why does this give you that? Do you show that all open sets in $\mathbb{R}^n$ are homeomorphic and then given two different dimensions find sets which aren't or something?

Eric Silva

6:22 AM

"all open sets in $\mathbb{R}^n$ are homeomorphic"- Daminark

5

Daminark

Frick

I just realized that was stupid

Perturbative

Nah in the wiki article it says - "An important consequence of the domain invariance theorem is that $\mathbb{R}^n$ cannot be homeomorphic to $\mathbb{R}^m$ if $m \neq n$. Indeed, no non-empty open subset of $\mathbb{R}^n$ can be homeomorphic to any open subset of $\mathbb{R}^m$ in this case."

Daminark

I mean it's better than all compact sets being homeomorphic, which happened (jokingly) last year

I mean it says that but like... how do you prove it from invariance of domain?

Perturbative

So like you'd probably reach a contradiction that $M$ isn't homeomorphic to itself when you descend into Euclidean space or something like that

Eric Silva

do it man

it's an exercise

Daminark

6:27 AM

I guess you could try to go from large to small, then push it back into large

Like

$n > m$ and $f:U\to V$ where $U\subset \mathbb{R}^n$ and $V\subset \mathbb{R}^m$

But then push that back into $\mathbb{R}^n$, at which point invariance of domain gives you a homeomorphism

Which is probably a load of horseshit

Because you need an open ball to live inside a smaller subspace

Eric Silva

this is essentially the proof

Daminark

Sick

Lol I wonder why none of us ever noticed it and made a fuss about this in Neves' class

Like "w8 w8 w8 what's stopping you from having a bunch of dimensions tho?"

I guess it felt pretty intuitive that this couldn't happen but like, pedants abound, surely at least someone was like :thonk:

Eric Silva

bc it's a dumb unenlightening technicality

and as neves has said "proofs are overrated"

Daminark

That's a very Neves thing to say

Kevin Driscoll

@Perturbative I frequently like to refer to it as 'nuking mosquitos'

Daminark

6:38 AM

I've said once "nuke a potato from orbit"

Perturbative

My brain autocorrects Neves to Jeeves

I imagine asking Neves a question to be more fun than AskJeeves tho

Daminark

AskJeeves never puts on that strange but somehow nice music Neves listens to

Eric Silva

he has decent taste

Daminark

See now I wonder what Schlag listens to, if anything

Eric Silva

classical music a lot

lmao i think im actually pretty familiar with the musical tastes of my potential letter writers

Mozibur Ullah

6:48 AM

That sounds like an advantage...

Kevin Driscoll

When I see Neves, I think of Devin Nunez

Or maybe its Devin nunes rather

Perturbative

@Daminark Here's a simpler proof. Suppose $M$ is both a $n$ and $m$-dimensional manifold for $n \neq m$. Pick a point $p \in M$, there exists two charts $(U, \phi)$ and $(V \varphi)$ around $p$ for which $\phi$ maps $U$ homeomorphically onto $\widetilde{U} \subseteq \mathbb{R}^n$ and $\varphi$ maps $V$ homeomorphically onto $\hat{V} \subseteq \mathbb{R}^m$., by definition of $M$ both being a $n$ and $m$-dimensional manifold

Put $W = U \cap V$, we have that $W$ is open and $\phi|_{W} : W \to \widetilde{W} \subseteq \mathbb{R}^n$ and $\varphi|_{W} : W \to \hat{W} \subseteq \mathbb{R}^m$. Both of which are homeomorphisms being the restrictions of homeomprhisms. Since $W \cong \widetilde{W}$ and $W \cong \hat{W}$, we have $\widetilde{W} \cong \hat{W}$, contradicting invariance of domain.

$\cong$ just means homeomorphic there

Daminark

@Eric that's no surprise in hindsight

I'd crack up if he was into like, techno or whatever

Eric Silva

he literally wanders the halls of eckhart playing vivaldi sometimes

Daminark

LOL

(Dunno what Vivaldi is but like LOL)

Eric Silva

6:58 AM

r u srs

one of the most famous composers of all time

Mozibur Ullah

The four seasons...!

Daminark

I'm like, really out of touch with music

Eric Silva

he did a lot of violin stuff, schlag is a violinist, schlag likes vivaldi

Perturbative

I thought Vivaldi was like a web browser

Eric Silva

have you never seen him walking around playing violin?

Mozibur Ullah

6:59 AM

I quite like Holst the planets.

Daminark

I have never seen him walking around, no, but I've seen the picture

Mozibur Ullah

lol ;).

Eric Silva

i like holst too

Daminark

I've almost never seen him at all

Eric Silva

i guess he was around more for my bootcamp than yours

Mozibur Ullah

7:00 AM

Also Nyman and phillip glass.

Eric Silva

so we got to know him more

Daminark

Like, I saw him in class, office hours, that one time when he came to the bootcamp, and like, twice otherwise

Mozibur Ullah

I don't know if they're classed as classical classical.

Eric Silva

philip glass lived in my dorm hallway when he was in college

Daminark

Soug and I are best mates but like, I have rarely interacted with Schlag

Eric Silva

7:01 AM

philip glass is super not classical lol

Mozibur Ullah

@eric silva: seriously?

Eric Silva

yeah my house invites him for a celebration every year but he never comes

it's sad

Mozibur Ullah

Well, he ain't blues, or rock, or rave, or trance - that just leaves classical by elimination.

Eric Silva

clearly there are only 5 genres

Mozibur Ullah

@EricSilva: Fames got to his head, like.

Eric Silva

7:02 AM

i honestly dont like glass that much

it's good studying music i think

Mozibur Ullah

@EricSilva: there's only five fingers on one hand - I'm a poor mathematician.

Eric Silva

but i dont really get into it

which i guess is probably why i think it's good studying music

Kevin Driscoll

Hot take: classical music is kind of bad

Eric Silva

nah it's ok

Mozibur Ullah

I probably listened to Glass at the right time in my life - lake late teens.

Eric Silva

7:03 AM

it's not my favorite but it aint bad

Kevin Driscoll

Its not bad music, but I really dont like it

Mozibur Ullah

what do your own tastes run to?

Eric Silva

i dont really listen to it that often unless im learning to play something

but i will say it's like a lot more fun to play than like pop kinda stuff

Mozibur Ullah

Sounds like art, I don't like art unless I'm trying to draw or paint something.

I mean I don't like looking at art unless...

Kevin Driscoll

I'm a jazz man myself

Eric Silva

7:07 AM

i was literally listening to jazz rn

and I rarely listen to jazz these days

Mozibur Ullah

I've never been a fan of jazz...I've tried listening to it on occasion to get into it.

Probably not listening to the right stuff.

Kevin Driscoll

It can be like coffee or beer or scothch. At first it seems real dumb. but if you just tell yourself its amazing for a while, you start to believe it and notice the nuances

Eric Silva

i kind of unironically liked scotch the first time I had it

Mozibur Ullah

I'll try that trick next time ;).

My brother told me he had a mystic vision the first time he drank whisky.

Kevin Driscoll

I mostly stick to single-malt Irish right now

Mozibur Ullah

7:12 AM

neat?

Kevin Driscoll

of course

Mozibur Ullah

I drink it with something like coke or juice - I'm a late developer.

Eric Silva

I think you should just drink stuff how you enjoy it tbh

Kevin Driscoll

When Im out I will drink a whisky and coke or a whisky sour

But thats realtively cheap whisky

Eric Silva

I don't really drink other than on special occasions

Kevin Driscoll

7:15 AM

I wont mix single-malt stuff because there's just no point, IMO. When you mix it with a greater volume of coke, you cant tell the difference between a $20 bournbon and a $200 bourbon

So why waster you money?

Eric Silva

so usually when i do drink it's some really nice bourbon i keep on my shelf

Mozibur Ullah

I drink cheap stuff, like I said I'm a poor mathematician!

Kevin Driscoll

someone gave me a mimosa once, and I later found out it had Dom Perignon in it and I was flabbergasted

that was a long time ago though

Mozibur Ullah

I thought mimosa was a flower?

Eric Silva

isnt it orange juice and champagne