Spring has ergodic and GMT
#point-set-topology
sleek thicket
I kind of want to take GMT but there are also aspects of toro's course style I dislike
If it is less assheaded I will probably take it
sleek thicket
Ehh kind of both
sleek thicket
TA is the biggest issue
Well like
It is an intro lol
I don't have an issue with that part
elder yew
HAHA
sleek thicket
You'll start to answer her question
elder yew
That's funny
sleek thicket
And then if your proof diverges from the thing she has in mind
elder yew
That's real funny
sleek thicket
She immediately pounces
It would be funny if I weren't taking the class lol
And also
elder yew
For second semester real, we had to prepare a 55 minute lecture
sleek thicket
There's a participation grade
elder yew
There was one girl that did hers on Fourier Series and poisson kernel
sleek thicket
ooh cool
elder yew
In the middle of the presentation, she wrote down a fourier series
prof asked what the coefficients were
She couldn't say what they were
sleek thicket
oh my
elder yew
Yeah...
So it was a shipwreck
I get up out of the room for ten minutes cuz it's just pure pain watching
I come back
Prof. is lecturing
sleek thicket
:///
elder yew
Even stranger - she straight up stalked me
sleek thicket
bruh
elder yew
She'd follow me around campus, made jokes about seeing me naked
sweet wing
elder yew
yA all 100% true
sleek thicket
I have a less weird lecturing story
elder yew
Ok. She also accused a prof. of sexual assault and told me about it
It was 100% not sexual assault
She's fucking psycho
sleek thicket
oh my
elder yew
anyways go ahead sham
sleek thicket
so chmonkey and I did this algebra study group together freshman year
good times
There was no prof
A student lectured each week
And like, it was tough and people weren't great, but that's okay
Chmonkey got a bit of a short straw tho
He got the week which was all about the structure of the symmetric group and An
And I don't think we'd even defined An before
so he had a ton of material to cover
his lecture went over time by like, 2.5
I left to get lunch (after he had passed time) and when I came back he was still going
chmonkey is not to my knowledge a psycho stalker
elder yew
I can go on about this woman
There's several stories
involving cheating on test
she once told me I reminded her of her father
Oh, we got to the tits group in knot theory
She said "Omg it's like a tit! That's so funny!!"
gritty widget
elder yew
Professor went bright red
sleek thicket
You just don't say it in class
LMAO
Why are you danning ttera the word tits is funny
elder yew
Oh someone had that kanye west meme in their presentation interrupting Tay Swizzle
sleek thicket
Bruh
sleek thicket
Oh I think you've told this one before
sweet wing
She said "Omg it's like a tit! That's so funny!!"
literally twitter_irl
how is she not kicked out
elder yew
The only thing good about leaving Long Beach is getting away from her
She's about to be. You only get 2 tries on the quals
sweet wing
wait you cant get expelled for literally acting like a psycho irl?
elder yew
She posts on FB about being able to hear the voice of God
and it telling her to do things
I am not making any of this up
sweet wing
actual psycho lol
elder yew
Yeah and she wanted to bang me so bad. She followed me to the bathroom once
She knew I had a wife. Everytime she tried to talk to me, I just talked about my wife
sweet wing
elder yew
The entire time
sweet wing
wtf
elder yew
To give her a hint
sweet wing
do you not have anyone to report to?
and like
m8 theres a insane person in your school
sweet wing
oh
elder yew
Yeah, I just feel bad for her. She's probably schizo
Anyways that's enough of that story LOL
elder yew
Btw, she accused a professor of sexually assaulting her
(to me at least)
For trying to convince her not to drop the course she was failing
Cuz he thought she could pull through
sweet wing
oh gosh
sleek thicket
There is someone in a similar situation at uw
sweet wing
ok i would go on a rant how people should stop being so sensitive but wtv
sleek thicket
Not as severe
It's not clear to me whether they were actually ever in the program
But they currently aren't
They take a bunch of high level courses but don't really know what's going on, ask questions that sound correct unless you know what they're actually talking about in an attempt to seem smart
and like, clearly have some mental health stuff going on
idk it's a weird setup
elder yew
Yeah, a lot of these people at one point were able to do good mathematics - but then their minds snap
sleek thicket
very disruptive but it seems like the unofficial dept policy is to be neutral
elder yew
(Go on)
sleek thicket
because like, they're in a bad state and it is not getting to be better by telling them to fuck off
¯\_(ツ)_/¯
elder yew
That's unfortunate
sleek thicket
Yeah
elder yew
I always feel sad when I hear stories like that sham
sleek thicket
I think they went off their meds at sons point last year
sweet wing
when you're having wild thoughts
elder yew
One of my close friends at UCLA had a full-schizo break down. Hospitalized for 2 weeks. Had to go on full medication
He was taking grad AG, Complex Analysis, etc.
On track to be a good mathematician
Now he can't do any math
sweet wing
Now he can't do any math
wait why?
sleek thicket
One day in junior year of high school I woke up to a wall of text from my cs teacher
elder yew
He said it doesn't process the same way anymore. Like he couldn't
elder yew
Do basic algebra
sweet wing
He said it doesn't process the same way anymore. Like he couldn't
oh no
sleek thicket
Like, really unsettling
anyways I avoided school
Other friends text me later
she is fully manic during school
sweet wing
oh no
sleek thicket
was incredibly inappropriate towards a student who she'd had it out for for years
like
it was messed up
I didn't go to my hs for a week because I was in a dual enrollment program and had no classes there
Because I had no idea what was going on
elder yew
[I know 4 people who completely burned out at UCLA and basically quit mathematics for life after going there - which is why I tell ppl not to go there]
That's fucked sham
sleek thicket
anyways I was the only student to get that text
She sent it on the staff school wide mailing list too
but it was really unsettling how like, I was the one student
elder yew
oh jesus fk
sleek thicket
idk
sweet wing
[I know 4 people who completely burned out at UCLA and basically quit mathematic...
is the place there actually that hectic?
sleek thicket
She ended up leaving that school a year after but for possibly unrelated reasons
School was cutting down the cs program, she moved inside the district to a new school
elder yew
UCLA profs don't give a fuck about you. It has a huge math department and everyone is good at math there, so if you're not a superstar they just don't care about you. You won't get any opportunities, any letters of rec, and they'll prevent you from taking the courses you need to take to get into a good grad school
sleek thicket
yike
sweet wing
omg
elder yew
My friend got the 9th highest grade in honors algebra out of 40
sweet wing
ok that is pretty bad
sleek thicket
That makes sense
elder yew
Prof said he couldn't write him a letter of rec for an REU
sleek thicket
wild
sleek thicket
Eh, I mean the uw isn't like that
elder yew
Yeah, from what I've heard UW is a healthy environment
sleek thicket
Like my honors analysis prof would write you a letter after 1 quarter
It's weird because like
elder yew
I had night terrors for the first time in my life after 1 year at UCLA
sleek thicket
I hear worse things from everywhere else
But also people here still say the department is super un supportive
elder yew
There are courses where no matter how much you study for, you will not be prepared for the exam
sleek thicket
idk I don't want to like, universalize my experience
but I feel like it's a pretty healthy culture
elder yew
I've heard it from other ppl to shammy
You'll get a great education at UCLA, you won't realize it till you leave
and go somewhere else to do math
sleek thicket
oof
sweet wing
yikes
elder yew
But it burns you out fast ~ like that analysis midterm was significantly easier than our undergrad stein and shakarchi real analysis midterm
sleek thicket
my analysis midterm?
elder yew
Just cuz profs don't give a fuck
sleek thicket
yikes
elder yew
Yeah the measure one
sweet wing
wait what are the qns like i have no reference xd
elder yew
They'll have averages in the 30s-40s
sleek thicket
Oh yeah I got 8/10 for figuring out everything except the last implication
Which was the hard part
elder yew
Then they rank you by your classmates. If you're not top 5 consistently you don't get anything
sleek thicket
And the mean was 6.9
sweet wing
Then they rank you by your classmates. If you're not top 5 consistently you don'...
painfully competitive ouch
sleek thicket
Yike
elder yew
My friend started at MIT math PhD after undergrad at UCLA
He said MIT was very refreshing
IDK what happened to the school
sleek thicket
Wild
sleek thicket
I really like math, but academic math culture can be really painful and bad
Like
Sometimes I wonder if professors actually think of their students as people
lol
elder yew
I'm lucky I went to CSULB for my MS. Absolutely fantastic professors that made me fall back in love with math
sleek thicket
That's great
elder yew
I hope I get into UW lol
sleek thicket
I'm rooting for you!
I need to think about where I'm applying for phds
It's coming up
sleek thicket
I'm not sure, sorry
elder yew
Ah it's k
sleek thicket
She's not like cold at all
Like
I think she would be more receptive than most
But I don't know the protocol
sleek thicket
Ahh I feel I would then
elder yew
along with a few others
My advice for grad applications sham is fuck ranking
Find someone who you want to work with that is reasonably friendly
sleek thicket
this only widens the search space!!
elder yew
haha
elder yew
You don't like GT or RG
sleek thicket
Too many pictures/pictoral arguments
elder yew
You like AG
sleek thicket
Well, I don't like RG because I'm currently taking a class on it
And I like AG because I'm no longer taking a class on it
elder yew
LOL
sleek thicket
I think I'm interested in both, they're just both pretty painful
elder yew
My GT prof. gave picture arguments for intro
but in topics courses we went into heavy details
You might like GT more if it's detail oriented?
sleek thicket
I mean I haven't really given it a fair shot, one online reu in covid times isn't really representative
And we didn't really do knot theory, it was mostly category theory and noncommutative algebra (I didn't like the knot theory papers tho)
elder yew
I'm thinking of doing an online intro to GT here on the discord
sweet wing
wdym gt has pictorial arguments
sleek thicket
oh neat
elder yew
I'd do it over summer ~ nobody seems to know much about knots or 3 manifolds
@sweet wing Geometric topologists are notorious for writing
sweet wing
oh
sleek thicket
wait what did you think gt was
elder yew
"Take a properly embedded surface that bounds a ball with an essential simple closed curve that intersects this other thing transversally. By gluing the boundary of a disk to this and capping it off, then via an isotopy we may assume that we've fully compressed this manifold w.r.t. this curve"
And ur like wot
sweet wing
wtf
elder yew
and it takes like 20 minutes of drawing pictures
To even begin to understand what it means
sleek thicket
So like, the things I am so far interested in are algebra \cap geometry/topology
Like ofc this includes AG and AT
But also just like, vector bundles
sweet wing
OHHH ok yes i remember the geometric picture pain
sweet wing
i completely skipped those and went to formal definitions
sleek thicket
I'm planning to apply to big departments
elder yew
Like I'm working on understanding one of my profs papers that's literally "Restrict this to two bounded components and one unbounded component, and play compression games with isotopy and a spine to get the thing you want"
sweet wing
consider some geodesic homologous to xxx ...
sleek thicket
That have people do specific things I think I'll like
elder yew
@sleek thicket word to the wise, USC gives 1 year paid time off from TA duty to finish a dissertation
sweet wing
consider some geodesic homologous to xxx ...
i cant visualise anything lol so i jus use these to actually write out explicit examples and screw any visualisation
elder yew
For all students
sweet wing
woa cool
sleek thicket
I genuinely enjoy TAing
elder yew
IDK if USC has ppl u like
sleek thicket
One of the worst parts of Covid-school for me is not TAing
elder yew
I enjoy it too, but it's nice to have options like that
sleek thicket
true true
sweet wing
teaching people is so exhausting
sweet wing
do you slowly develop the stamina for taing lol if not theres no way i can survive teaching a few times a week
elder yew
But yeah cat, GT is a fun subject to learn about
sweet wing
bai gn
elder yew
cuz it uses a different part of your brain than your used to
sweet wing
But yeah cat, GT is a fun subject to learn about
yee am jus trying to brush up my AT rn cuz im always tripping over at stuff
sleek thicket
@sweet wing for me it's not like an energy suck
like
It takes energy but also gives it
elder yew
It gives me energy when I TA
sweet wing
oh wow
sleek thicket
Yeah exactly
elder yew
I feel refreshed
sweet wing
interesting
sleek thicket
Right
elder yew
Like if I do a lot in one day
sleek thicket
Like sometimes it can feel draining or like too much work
elder yew
Yeah I'm exhausted
tough imp
idk it's a weird setup
I think I know who you’re talking about. Were they living in the building?
exotic root
What am I missing here for $X = V(xy-1), \bar{X} = V(xy-z^2)$? Doesn’t seem like $\pi(X)$ is closed in the affine line?
Obviously the right hand side is closed because projective space is complete.
X should project down to A^1 minus the origin which is not closed.
gentle ospreyBOT
Lejoon:
exotic root
Duh (0:0:1) is in the closure...
Why is my brain so bad rn
Wait I actually know the answer
sleek thicket
I figured out what went wrong with my inequality!!
I applied AM-GM to two nonpositive reals
elder yew
Is there any theorem regarding these types of bounds
That you can just use to nuke it?
gentle ospreyBOT
shamrock:
sleek thicket
Where $h$ is a symmetric negative semidefinite bilinear form
gentle ospreyBOT
shamrock:
sleek thicket
wait
This is a fucked up version of cauchy Schwarz lmao
If you multiply h by -1 you get a positive semidefinite symmetric bilinear form
But you don't change the two sides of that inequality
we might have h(v, v) = 0 for v nonzero but h is like, almost an inner product
hmmm
So like
The proof of Cauchy Schwarz I know involves dividing by stuff
Which might be zero here
So let $h$ be a symmetric bilinear form, either positive or negative semidefinite. If $h(a, a) \neq 0$ or $h(b, b) \neq 0$ then $h(a, b)^2 \leq h(a, a) h(b, b)$
This is proved the same way as cauchy swcharz
gentle ospreyBOT
shamrock:
sleek thicket
If both of these vanish then $h(a, b) = \frac{1}{2} h(a+b, a+b)$, but I see no reason why $h(a+b, a+b) = 0$
gentle ospreyBOT
shamrock:
nimble jolt
@sleek thicket applying to both a+b and a-b you get h(a,b) must have both signs and is hence zero.
sleek thicket
ahh that makes sense
I got that h(a+b, a+b) = 2 h(a, b) and h(a-b, a-b) = -2 h(a, b) but I didn't think about what that implies for the signs
nimble jolt
And in the real case, the discriminant proof of CS goes through fine for h not def.
sleek thicket
Wait why?
oh "not def" as in "only semidef"?
Because that's the proof I had used for the case where h(a, a) ≠ 0 or h(b, b) ≠ 0
nimble jolt
Yeah, although I suppose vanishing leading coeff might be annoying.
sleek thicket
Actually maybe I misunderstand what you mean by discriminant proof
What proof are you thinking of?
nimble jolt
h(x+ty,x+ty) >= 0 for all t. Expand as quadratic in t (non vanishing leading coeff used here), discriminant of poly must be =< 0.
sleek thicket
Oh I see, the proof I know is to evaluate that quadratic at t = -h(x, y)/h(y, y)
and that gives that the discriminant is <= 0
nimble jolt
Ah right.
Right but yeah this discriminant proof will work for all cases where h(y,y) is not zero, and the remaining case is trivial, as a linear poly cannot have fixed sign, so h(x,y) must be 0.
sleek thicket
Sick
Oh wait
In the case where h(a, a) = h(b, b) = 0, were looking at the linear poly h(a+tb, a+tb) = 2 t h(a, b)
Evaluating at t=1 and t=-1 gives the same proof as before
neat
nimble jolt
Yep
elder yew
is h alternating?
sleek thicket
no, symmetric
sleek thicket
@elder yew the thing I needed to prove/proved was that if $h$ is a negative or positive semidefinite symmetric bilinear form then $h(a,b)^2 \leq h(a,a) h(b,b)$
gentle ospreyBOT
shamrock:
sleek thicket
so like an upgraded version of cauchy schwarz
in my case $h$ is the scalar second fundamental form of a hypersurface
gentle ospreyBOT
shamrock:
nimble flower
how true is this answer? https://math.stackexchange.com/questions/296170/isotopy-and-homotopy?noredirect=1&lq=1
An isotopy is a deformation that involves only bending. It must be one-to-one and onto at every step. In this way, any two handlebodies of equal genus are isotopic.
specifically the "only bending" part
actually.. nvm, im interested in ambient isoptopy, which is a different thing
cold vine
I had to prove for X and it's suspension that for reduced homology Hn+1(SX) ~= Hn(X), and I just created Mayer-Vietoris seq for upper and lower cones (or more precisely open nbds of these cones to satisfy that the union of their interiors is SX, which strong deformation retract onto the cones) and now I get the isomorphism straight from the sequence since Hn(CX)+Hn(CX') = 0 for all n. But we had a tip in our assignment that told to use both M-V and excision. Where would I need excision here, am I missing something?
uncut surge
I think that sounds fine, maybe there's another way using a long exact sequence in relative homology, but yours sounds good
uncut surge
yeah i don't think you do
fading vale
i dont think you do but i assume thats what they were referring to?
uncut surge
with mv
fading vale
mhm
if you were solving it with LES for rel hom you could do like
a neighborhood around the cone point and then deformation retract (red. susp(X), CX') to i guess uhhh
CX, X
by excising a neighborhood around the cone point contained in the lower cone and then retracting to CX, X
but then you dont need mayer vietoris sooooo
cold vine
so yeah this M-V should be ebough then?
fading vale
yea i think so
cold vine
ok nice. but yeah that LES way seems pretty nice w excision this M-V feels like cheating lol 😅
cold vine
it is it is
cold vine
its a class
fading vale
nice
my AT class is so fucking slow
we're literally still proving van kampen and we havent even touched covering theory
cold vine
we just had almost two lectures of proving the excision axiom 😅
fading vale
that proof is awful
elder yew
Most AT classes go very slowly
fading vale
it makes me sad tho cuz no time to cover pi_n in second sem
we literally spent all of last lecture introducing group presentations like
we're running out of time for the semester? we havent even touched covering spaces we spent more time on point set than actual AT?
elder yew
There's a lot of subtlety in presentations
fading vale
yea but we have like 3 lectures left i think lol
elder yew
That sucks, was point set a pre req
cold vine
yeah we also had another full lecture of proving homotopy invariance of homology a while back. But no more that long proofs so I guess the last weeks go a bit quicker xD
fading vale
its a grad class moonbears so its supposed to be i think??
i feel like if the prof wants to do stuff like this it makes more sense to assign it as readings
especially the point set stuff
idk it just seems rly unnecessary to spend so many weeks covering point set material most of which isnt even rly used
esp for a grad class where i feel u can expect it to go ok if ur assigning readings and stuff
he could have just assigned hatchers point set notes and it would have been like. fine lol
cold vine
what point set stuff do you have? i have a class in homotopy in the spring
fading vale
like personal materials :?
cold vine
yup
fading vale
if u want a quick coverage hatchers point set notes are good
if you want something more comprehensive chapter 1 of bredons geometry and topology does quite a bit
elder yew
Munkres start at chapter 2
fading vale
u can do munkres if u want but like
ngl u dont really see that much point set in intro AT lol
well idk i havent learned anything about pi_n in general yet
tight agate
I had to prove for X and it's suspension that for reduced homology Hn+1(SX) ~= H...
If you're doing this with cellular homology I think you can show that suspending just shifts the cellular chain complex
tight agate
Suspension of an n cell is an n+1 cell, and it is a left adjoint in the homotopy category so it's not bad
oh okay
fading vale
i feel like i still dont understand cellular homology well tbh i need to relearn it at some point
elder yew
I think it's better to go through things slowly and carefully
cold vine
but yeah that seems interesting 🤔
elder yew
Than it is to race through
fading vale
do you mean re: point set moonbears?
elder yew
Not point set
fading vale
oh
elder yew
That has no place in an AT class
fading vale
yea
its silly
i wouldnt rly mind if it was just going thru the material slowish and in depth but
we are literally about to run out of time in the semester without touching covering theory
because we spent so much time on point set we arent even using
elder yew
Yeah, but I'm sure the prof has their reasons
fading vale
yea im not rly sure why
cold vine
but are the reasons good 😄
elder yew
You could ask!
fading vale
i dont attend this school im just auditing but my impression is that nyu is more analysis and applied heavy
so maybe the grad students just havent been exposed to algebra or point set before?
fading vale
whom
elder yew
Ya could elaborate a little in your texts
fading vale
like he said a few times that he was unhappy he was spending so much time on point set
elder yew
Oh hell yeah
Those are the best kinds of classes
AT feels a lot like book keeping to me
All the technicalities to make more interesting things work out
fading vale
hn
elder yew
My topology prof is teaching knots again next term
since everyone who took it in 2019 graduated
everyone take knots
tight agate
hmm I think there's going to be a knots class next quarter over here too
elder yew
where is dat
tight agate
ucla
tight agate
235 or 236 i think
elder yew
There's only a few ppl who can teach that
tight agate
yup
tight agate
there's also going to be a class on fukaya categories
tight agate
nope
tight agate
when did you graduate?
elder yew
I highly recommend him
tight agate
prolly will do that
tight agate
I did sit in on a couple of classes
elder yew
He's really good
elder yew
How is it?
tight agate
taught by kefeng liu
elder yew
Mine was very easy
elder yew
Sounds about right
If you want knots, you can cross enroll to CSULB ~ Ryan Blair is teaching it
He's a monster of knot theory
(yA probably won't tho, that's ok)
Are you ug or grad at LA
elder yew
That's what I thought and then Blair showed me the way
Super cool topic
Unfortunate LA doesn't have many knot theorists
USC has a few
tight agate
I'm pretty sure it is super cool, but there's a lot of super cool stuff
tight agate
did not do a and b
elder yew
I just did A, after I finished that I graduated
tight agate
was it on riemannian stuff?
elder yew
Yeah, we just did RG
Covariant derivatives, brackets, vector fields
Sections, curvature etc.
tight agate
yeah I was hoping we'd do that
elder yew
Some Algebraic Topology stuff was used for a while
tight agate
but we're doing complex geometry
elder yew
Oh RIP
tight agate
which is also dope
we did do some fun stuff
chern-weil theory
deformations
hodge decomp
hard lefschetz
elder yew
@gritty widget I opened up little spivak
The proofs to go from general stokes to the specific cases are pretty short
gritty widget
do carmo what the fuck is wrong with you
it should be a crime to put this into books and make people learn it
elder yew
lol
gritty widget
rauch comparison theorem better be worth it
there was all this really nice topology
y'know
algebraic topology and whatnot
fading vale
thats disgusting
gritty widget
now there's just this shit
gritty widget
i do want to read hatcher at some point
gritty widget
one of the theorems in RG pulled out some stuff about like
deck transformations
and universal covers
fading vale
hi sham!
fading vale
how many more do you have left?
gritty widget
3
fading vale
oh scary
gritty widget
probably one for ring theory and two for field/galois
fading vale
wait monkaS are you on the quarter system
fading vale
lol
gritty widget
there is a september-december semester and a january-april semester, this class goes on for both of em
2 week break in between
sleek thicket
Ahh so it's across a whole year?
sleek thicket
Less insane
fading vale
hurb
gritty widget
so like i have no idea how im doing
gritty widget
i thought i was good and then that midterm trashed me
sleek thicket
We gave everyone 3.8+
fading vale
what was it on?
gritty widget
the one i had today?
fading vale
mhm
sleek thicket
And gave up on grading the backlog at the end
gritty widget
group action and sylow stuff
fading vale
yucky
sleek thicket
Pog
fading vale
sylow is not pog
sleek thicket
Sylow is extremely pog
gritty widget
yeah i clearly did not review the group action stuff enough 😔
fading vale
twitter has deepened the ruin
gritty widget
the statement of this rg theorem is half a page long
fading vale
also im reviewing 3.1 in hatcher its very cool
fading vale
shamrock listen to knives in the coffee
gritty widget
it's so fucking long
everything recently in this class has had like
2-3 page proofs
cold vine
So I'm calculating the homology of torus minus a point but i cant figure this out. this is probably MV but i cant get the right divisions for an isomorphism
woeful oasis
The torus minus a point retracts onto the wedge of two circles
cold vine
aahhh nice I didn't think of that. thanks!
woeful oasis
yay!
uncut surge
i just had the thought that when you're building that retraction of the torus minus a point, you're literally tearing up that hole
i felt the need to write this down
uncut surge
i also don't know what to say really
woeful oasis
I think that in general setting up specific complexes is computationally tough for me, much easier to build on known things instead of constructing stuff every time
fading vale
this is your brain on the mathematics discord server topology-and-geometry channel
woeful oasis
if I only had a brain...
uncut surge
sometimes i have these very intrusive but very visual math thoughts that just don't let me go
fading vale
i just imagine it as like
poking a hole in the square and then
pushing ur fingers into the hole and pulling it open
fading vale
its like that scene from evangelion when the angels core is torn open
woeful oasis
If they won't let go, I think important to put it on pen & paper (not just type). tactile connections help
uncut surge
jesus christ that is even worse
fading vale
yes i do math yes i am depressed we exist
uncut surge
another example is when you consider the quotient of $\mathbb{R}^2$ by the equivalence relation $x \sim t \cdot x$ for all $t > 0$, the space is a circle plus a point for zero
gentle ospreyBOT
Lartomato:
woeful oasis
Torus - pt retract, more like torture scene on Clockwork Orange. 3rd eye pried open!
uncut surge
and if you consider the topology of this space, you cannot disconnect the zero point from any of the points in the circle; i think it's non-hausdorff
woeful oasis
non-sober?
uncut surge
hmm i don't know soberness
uncut surge
no my default metaphor was a fat sweaty man in an overfilled bus, who touches everyone else
and no matter how much you try, you can't get away from him
fading vale
thats not better!!!
woeful oasis
fat sweaty homotopy
fading vale
arguably it is worse!!!!
uncut surge
but it's a visual which stuck with me forever
woeful oasis
i just learned it, so yay!
woeful oasis
If they'd stop making it illegal to be me, there'd be no problem
dusk heron
I have heard that the Ricci tensor on a Riemannian manifold should intuitively be viewed as the "Laplacian" of the metric, even though the actual Laplacian of the metric is obviously zero. Is there any further explanation for this?
The first part, that is: that the Ricci tensor should intuitively be seen as the Laplacian of the metric.
cold vine
Ok now I'm completely stuck 😅 I have 0<k<n and I'm supposed to calculate homology groups of R^n\R^k. I can calculate for Hp(R^3\R^2)=0 which I believe I can show easily for all n=k+1 and Hp(R^n\R^0)=Hn(S^1) I believe for all n since we can retract all the dimensions down to 1, but I don't see how these help with the general case?
cold vine
Ah, I guess it's actually H(R^3/R^0)=H(S^2), H(R^3\R^1) = H(R^2\R^1) = H(S^1) and H(R^3\R^2)=H(S^0)
we dont have cellular homology😅
cold vine
ok ^^ cheers
Ok I have an idea: H(R^n\R^0)=H(S^n-1) for all n and now if I prove H(R^n\R^k) = H(R^{n-k}\R^0)=H(S^{n-k-1}) which would seem to be the case for the low numbers and I think is just done with a projection as the strong deformation retraction. Then I would actually have all the cases.
fading vale
how do you plan on getting that first isomorphism?
if you could use kunneth formulas
cold vine
Doesn't it follow since we can see R^n\R^0 = R^n-{*} and with linear homotopy shrink the space to the boundary of the point? and now S^n-1 is a homotopy equivalence of the space?
cold vine
ahh right xD
fading vale
im not rly sure how you would get this tbh
cuz R^n \ R^k = (R^{n-k} \ R^0) x R^k right
cold vine
I think we can project the space to the subspace orthogonal to the R^k? sry hard to explain. Like similarly as we do in the case of Hn(R^3\R^1)=Hn(R^2\R^0)
hmm
fading vale
i think this might not be true sorry
cold vine
sad T_T
fading vale
u havent seen kunneth formulas yet but they give a way to calculate the homology/cohomology of products and at first glance it kinda doesnt look like this holds :c
ok mostly done
need to figure smth out re tor functor
cold vine
well this is complicated then. I just though we could reduce the dimensions one by one like H(R5\R3)=H(R4\R2)=H(R3\R1)=H(R2\R0)=H(S1) by doing at each step linear homotopy parallel to the subspace. But if it doesn't work it doesn't work 😄
Ok nice ^^
cold vine
lets hope you did 😄
fading vale
this is scary
nvm i think this is just cursed
i didnt fuck it up but it doesnt tell us anything
sad!
cold vine
ahah ok
fading vale
this is actually depressing
wait double no it might imply that all the homology is trivial which makse no sense
ummmmm
Hm.
cold vine
fading vale
i dont like this
cold vine
Kunneth pls
fading vale
i am being cyberbullied
ok R^n \ R^k definitely is equal to (R^{n-k} \ R^0) x R^k right??
im not just being super silly
oh hm wait
cold vine
I actually can't see how they'd be the same
fading vale
maybe it just reduces to
cold vine
but i still think my homotopy equivalence works but i might be silly there xD
fading vale
uh well topologically it factors thru right
the product factors through the set difference
cold vine
cant say😅
cold vine
ok
fading vale
ur right H_p(R^{n-k} \ R^0) is iso to R_p(R^n \ R^k)
im not sure how ud show the htpy
but ur intuition was right
@tough imp ur my personal Tor computer
tough imp
Shamrock is better at it
cold vine
ok great ^^
tough imp
I only calculate Tor via LES in homology
fading vale
shamrock can also be my tor computation machine
tough imp
Shamrock is willing to literally take homology of a flat resolution
fading vale
oh theres a Tor LES?
fading vale
woke
tough imp
???
tough imp
yeah so if you have 0 -> A -> B -> C -> 0
you get one like
-> Tor_2(C,M) -> Tor_1(A,M) -> Tor_1(B,M) -> Tor_1(C,M) -> A (x) M -> B (x) M -> C (x) M -> 0
so if you want to calculate Tor_1(C,M) if you can find a SES
0 -> A -> B -> C -> 0 with Tor_1(B,M) = 0
that LES says Tor_1(C,M) is the kernel of A (x) M -> B (x) M
tough imp
this is how you show like Tor_1(A/(x),M) is the x torsion of M
fading vale
ive only seen the construction explicitly with Ext before ig but even then hatcher didnt rly talk much abt computing it
he was just like
tough imp
I mean the other way is to take a flat / projective / free resolution of something
remove a term
fading vale
"ext is 0 if its free, it commutes with direct sums, Ext(Z_n; G) = G/nG"
fading vale
"ok good enough"
tough imp
and that's not very pog
cold vine
so my intuition was the homotopy by taking R^{n-1} orthogonal to R^k and just using parallell lines w.r.t. R^k to shrink R^k to R^k-1 (subspace of that orthogonal space) and R^n to the R^{n-1}. Still might be too bad of an explanation 😅
fading vale
i kind of see it but im not sure im convinced tbh
quaint agate
Hey everyone
cold vine
ok ^^
fading vale
but maybe thats just poor intuition atm
and i mean ur computation does work so its probably right?
cold vine
yeah I don't know ^^well ill do it like this and check the model solution unless i get an epiphany
fading vale
yea thats fair
cold vine
but thanks for help all of ya ^^
cold vine
haha that feels nice at least ^^
quaint agate
Can anyone give their take on my notation for link diagrams
I’m trying to see if it seems feasible to use
fading vale
hello 
fading vale
fading vale
hurb
fading vale
this goes in #geometry-and-trigonometry
sweet wing
moth is a bot in this channel
fading vale
then go to a questions channel
sweet wing
we dont do this geometry here lmao
sacred kayak
sorry!
fading vale
issok
sweet wing
read the history
fading vale
ari i am not a bot!
i just have notifs set up so i see
every msg sent in this channel
sweet wing
every msg sent in this channel
simp tbh
sweet wing
its only bad when ppl are diff geoing
i will ask diff geo soon lol
fading vale
sad!
gritty widget
👀
too bad moth
this and all of the channels are my DG complaining channels
or simping channels
sweet wing
wait tt u have any info on like moduli spaces of hyperbolic info
tough imp
||TTerra say it's after they learn multi variable calculus||
gritty widget
what does that even mean
sweet wing
mm
coral pivot
fading vale
hello john
fading vale
: (
sweet wing
wait also theres a thing i have been thinking for some time but havent had a answer
given the fundamental group G of a hyperbolic manifold how do you calculate volume
gritty widget
integrate the volume form duh
sweet wing
my book just says lol volume and doesnt actually give how to conpute volume wtf
that only rlly works for simple cases where you know where to integrate tho?
sweet wing
finding the region to integrate actually needs big brain imagining skills
sweet wing
just visualize
help me
fading vale
moving around irl to mimic the movement i was trying to imagine
until i didnt have to anymore
it sounds rly cringe but like
gritty widget
idk knowing you you can probably visualise whatever dimensional manifold you want 
sweet wing
im not that chad to visualize things not in 3D
gritty widget
compared to us euclidean beings
fading vale
visualizing things not in 3d is easy just embed
it will self intersect but i dont see why thats a problem
coral pivot
ari then just paramatezie using equations 4 head
sweet wing
what a chad
fading vale
the way i visualize the 4 sphere is as like
a 3d paperclip
lmao
idk how to describe it
u just have to thonk abt how to build the CW complex from S^2
and do that
gritty widget
ari it seems like there are a few things on MSE about the volume of hyperbolic manifolds idk if those posts will help
fading vale
and then blur
sweet wing
you see the poincare ball model sucks when you have a explicit fundamental group
sweet wing
ari it seems like there are a few things on MSE about the volume of hyperbolic m...
oo havent checked those out yet ill poke at those thanks!
gritty widget
ill share
fading vale
woke
sweet wing
you see the poincare ball model sucks when you have a explicit fundamental group
while the poincare half plane model is slightly easier to understand what π_1 does but even harder to visualize
gritty widget
maybe related to what you're doing
sweet wing
yes i got lol i dont actually do this reply
sweet wing
maybe related to what you're doing
im just playing with random stuff lol but looks cool thanks!
fading vale
ya lmao
sweet wing
by playing around with stuff
fading vale
i remember now that was based
fading vale
ari plz
sweet wing
number theorist at heart