/mg/ - Mathematics General

[Anonymous No.12457035]

Wear a mask edition

talk maths, formerly: >>12441656

[Anonymous No.12457048]

What are the pre-requisites to learn cohomology?

[Anonymous No.12457049]

first for Sobolev spaces

[Anonymous No.12457050]

>>12457035
Cringe edition.

[Anonymous No.12457055]

imagine posting anime pictures on the Internet past the age of 16

[Anonymous No.12457062]

>>12457055
https://en.m.wikipedia.org/wiki/4chan
>4chan is an anonymous English-language imageboard website. Launched by Christopher "moot" Poole in October 2003, the site hosts boards dedicated to a wide variety of topics, from anime and manga to video games, music, literature, fitness, politics, and sports, among others.

[Anonymous No.12457076]

https://youtu.be/pTnEG_WGd2Q

For anyone trying to learn mathematics from start to finish

[Anonymous No.12457128]

I bullshitted my math degree honestly I am a fraud. I wanna be a guy that answers questions on mathexchange.

[Anonymous No.12457155]

>All of mathematics can be described with sets.
Is this true?

[Anonymous No.12457162]

>>12457155
No, where did you hear that? Set theory completely falls apart in some parts of advanced algebra, that's why category theory was created.

[Anonymous No.12457169]

>>12457162
Book of Proof.
Which are some examples of set theory falling apart?

[Anonymous No.12457173]

>>12457162
>Set theory completely falls apart in some parts of advanced algebra
Give examples.

[Anonymous No.12457176]

>>12457162
>you can't describe injectives and projectives in ZFC ergo set theory completely falls apart in some parts of advanced algebra
Please don't give arguments in bad faith like that.

[Anonymous No.12457184]

>>12457048
singular cohomology? just point set topology. get the hatcher PDF.

[Anonymous No.12457188]

>>12457169
Throw it into the trash, Book of Proof is a complete meme, if you want good proof books go read Velleman's How to Prove or A Transition to Advanced Mathematics

[Anonymous No.12457204]

>>12457184
What's a good book on learning topology?

[Anonymous No.12457210]

>>12457184
What's a good book on learning topology? No Munkres please.

[Anonymous No.12457230]

>>12457210
What do you have against Munkres?

[Anonymous No.12457233]

>>12457188
What makes it so bad compared to those?

[Anonymous No.12457243]

>>12457233
It doesn't go in-depht in logic and it leaves out somethings about sets, the exercises are also bad and there are less than in How to Prove It.

[Anonymous No.12457285]

>>12457230
It's too big and it's too boring. I'm thinking of learning from Lee's Topological Manifolds.

[Anonymous No.12457287]

>>12457210
munkres.

[Anonymous No.12457332]

>>12457285
Give me an example of what you think is not boring. Also, how is something being too big a legitimate criticism?

[Anonymous No.12457348]

>>12457285
It's small and fairly easy so should be a quick read.
If you get filtered by Munkres it's literally over for you lmao

[Anonymous No.12457366]

>>12454233
>Don't ask questions, just prove conjecture, and then get exited for new theorem

[Anonymous No.12457385]

>>12457332
Tao's Analysis wasn't boring to me. Aluffi's Algebra was also pretty cool. Munkres has 600+ pages just for Topology...

[Anonymous No.12457451]

>>12457210
Arkhangel’skii and Fedorchuk doesn't motivate anything, proves even less and the entire text reads like a wikipedia entry.
But it's also very short and detailed.

[Anonymous No.12457456]

Never Give Up on Math!

https://youtu.be/EMHtNNP6s0A

[Anonymous No.12457476]

>>12457451
Those books are kinda old though, right? I think I'd rather read something more recent.

[Anonymous No.12457487]

>>12457456
he should buy himself some eyebrows

[Anonymous No.12457493]

>>12457487
two fifty . seven 1 2. 2020

[Anonymous No.12457512]

>>12457456
I like fightan gaems

[Anonymous No.12457556]

Stay safe from the corona, /mg/. Please wear the mask while posting on /mg/. If you think you may have contracted coronavirus, please leave this thread.

>>12457155
here's an incredible fact - all of mathematics can be described using english language

>>12457366
questions like "is PA consistent" is the math equivalent of "what if the world doesn't exist lmao" solipsism in philosophy
sounds deep and interesting when you first hear it at the age of 12, later becomes unbearable

[Anonymous No.12457582]

>>12457556
What is your area?

[Anonymous No.12457584]

>>12457556
>sounds deep and interesting when you first hear it at the age of 12, later becomes unbearable
Why would it stop being deep, and why is it unbearable to you?

[Anonymous No.12457596]

>>12457210
Stephen Willards book is probably good

[Anonymous No.12457625]

>>12457596
Hum, I don't really trust Dover as a publisher though.

[Anonymous No.12457627]

>>12457210
What do you mean by topology? Do you want a thorough, rigorous and technical treatment of the notion of continuity, or do you want Klein bottles and mugs deformed into donuts? If it's the first, Munkres. If it's the second, Guillemin And Pollack.

[Anonymous No.12457630]

>>12457556
You're not as smart as you think you are.

[Anonymous No.12457683]

>>12457625
You’re just looking for any reason to shoot down suggestions fag. I bet you’ve already made up your minds about what book you want to choose. Don’t ask for advice if you won’t take it.

[Anonymous No.12457745]

>>12457556
are you american?

[Anonymous No.12457757]

>>12457683
Not really, you idiots were the ones who couldn't come up with good recommendations. Forget it, I'll use Munkres.

[Anonymous No.12457786]

post more pics of cute topologists

[Anonymous No.12457817]

>>12457786
here go

[Anonymous No.12457823]

>>12457786
Wouldn't this mean Normal spaces are path connected?

[Anonymous No.12457832]

>>12457366
good meme and way too true.

>>12457385
the first 1/4th is set theory and logic. its actually a great coverage but since your scoffing you can skip.

the last half is algebraic topology. I like it, but some prefer to jump to hatcher at this point.

[Anonymous No.12457840]

>>12457823
no. imagine two separate components. define f(A) =1 and f(B)= 0 this is continuous.

[Anonymous No.12457845]

>>12457832
>some prefer to jump to hatcher at this point.
That's exactly what Im gonna do I guess

[Anonymous No.12457846]

one month to study a whole course of ordinary diferential equations (bachelor level), will I make it?

[Anonymous No.12457863]

>>12457840
ok, then what

[Anonymous No.12457868]

>>12457846
Yes!

[Anonymous No.12457870]

>>12457817
wish it were me

[Anonymous No.12457877]

I've taken courses in analysis and measure theory (no textbook) so is there any reason to go through baby Rudin, or should I just jump straight into papa Rudin. Would there be anything I'd be potentially missing from neglecting baby Rudin?

[Anonymous No.12457889]

>>12457210
Borisovich. Great exposition, gives you a good introduction to what topology is like before getting into all the rigor. Also gives nice examples. Russians seem to be outstanding when it comes to topology.

[Anonymous No.12457890]

>>12457817
do you have more anon?

[Anonymous No.12457899]

>>12457890
Only a couple of old Yukari posts and anime girl math book shops

[Anonymous No.12457900]

>>12457863
not path connected, but that function satisfies urysohns lemmas conclusion.

>>12457889
thanks anon. will look

[Anonymous No.12457904]

>>12457863
What do you mean "then what?" Then nothing. You asked a question and he answered it. What more do you want?

[Anonymous No.12457911]

>>12457846
One month? Heh, I did it in one weekend son...

[Anonymous No.12457920]

>>12457786
>>12457817
What do catgirls have to do with Urysohn? Is it cause it sounds like Furry-sohn?

[Anonymous No.12457947]

corona is gay noone cares shit's about as real as the real numbers

[Anonymous No.12457956]

>>12457920

[Anonymous No.12457971]

im doing my masters in physics but i had done a final year of mathematics bachelors that i failed and damn that final math bachelor year was 10 times harder than what im doing right now
math is just not as easy as phys or other stuff for some people, the level of rigor and abstractness is otherworldly
>>12457456
it actually felt right, although a bit sad to give up

[Anonymous No.12458046]

>>12457971
?
Physics is way harder than mathematics. At least at a high level.

[Anonymous No.12458055]

what are some papers that will improve my chess play?

[Anonymous No.12458057]

>>12458046
well not at my masters level
i guess it also depends on how hard you find math or physics
i know some math students think phys is harder but me and many other phys students thinnk math is much much harder

[Anonymous No.12458066]

>>12458055
Chess is just applied graph theory.

[Anonymous No.12458092]

>>12458066

I'm tired of getting obliterated by autists that remember literally every game they ever played or studied and can immediately just recite good lines from memory. There should be some good methods out there so we humans can apply to compete with these superhumans...

[Anonymous No.12458096]

>>12457868
>>12457911
thank you pals, i'm glad to be into this, although it gets me sad sometimes

[Anonymous No.12458118]

Mochte jemand mit mir Schach spielen?

[Anonymous No.12458133]

Wish I had friends to do math with

[Anonymous No.12458135]

>>12458118
Allahu Alkhbar

[Anonymous No.12458138]

>>12458133
Wir sind dein Freund

>>12458135
Ist das ein Ja?

[Anonymous No.12458139]

>>12458133

>that christmas sweater

basé

[Anonymous No.12458141]

>>12457757
>you idiots
says the person that got filtered by munkre

[Anonymous No.12458145]

>>12458118
1. d4
ja wohl

[Anonymous No.12458147]

>>12458145
https://lichess.org/MEfx1e8h

[Anonymous No.12458153]

>>12458147
You're trash.

[Anonymous No.12458156]

>>12457786

Damn they even wrote the correct proof on the board. I am impressed.

See Rudin for the proof.

[Anonymous No.12458157]

>>12458153
I'm literally winning atm. Finish the game coward.

[Anonymous No.12458158]

>>12457210
http://pi.math.cornell.edu/~hatcher/Other/topologybooks.pdf
Is from 2003 but it may useful to you.

[Anonymous No.12458160]

>>12458157
That's because I wasn't even paying attention, you were struggling against an opponent who was literally masturbating to netorare. I just clicked on that link because I thought it was chess resources, not a match.

[Anonymous No.12458163]

>>12458118

Ich habe ein Masterdiploma auf Mathematik und ich bin ein 2k Go spieler. Verzeihung :-}

[Anonymous No.12458166]

>>12458160
Whatever faggot
>pic related me and my gf laughing at the literal cuck

[Anonymous No.12458171]

math bros how do we beat the giga-memory chads in the game of chess ...

[Anonymous No.12458173]

>>12458158
Finally a good answer. Thank you, the first non-idiotic answer I had.

[Anonymous No.12458175]

>>12458171

Can't. Learning chess necessitates to memorize the classical sequences.

[Anonymous No.12458176]

>>12458171
It's not about memory it's about practice and neural nets. You see a geometric structure once, twice, and you see it recur. The only part anyone really memorizes is openers and even thats gay

[Anonymous No.12458186]

Math bros, how do we get good at match match? The physics department keeps beating us on game night.

[Anonymous No.12458188]

>>12457877
no

[Anonymous No.12458189]

>90 posts in
>no math posted yet, literally nothing written in tex
wow impressive

[Anonymous No.12458199]

warum sprechen die schachautisten deutsch

[Anonymous No.12458200]

Prove that the assignation [math]X \rightarrow ~ i_X \omega[/math] defines a surjective linear map [math]\mathcal{H} _{loc} (W) \rightarrow H^1 (W; \mathbb{R})[/math]. Deduce an exact sequence of vector spaces [eqn] 0 \rightarrow \mathcal{H} (W) \rightarrow \mathcal{H} _{loc} (W) \rightarrow H^1 (W; \mathbb{R} ) \rightarrow 0[/eqn]
Where [math](W, \omega)[/math] is a symplectic manifold, [math]\mathcal{H} (W)[/math] is the vector space of Hamiltonian vector fields, [math]\mathcal{H} _{loc} (W)[/math] are locally Hamiltonian vector fields, and the cohomology is de Rham.
Shouldn't take you more than a minute, it is basically just applying the definitions.

[Anonymous No.12458201]

>>12458189
[math]Fuck\\You[/math]

[Anonymous No.12458203]

Chess is also like a story. You can think of sieges, offensives, sneaky bishops, brave horsemen riding. Do chess on psychedelics, it's awesome. And then various mathematical notions of closeness/forwardness - this move moves my piece to the side but one layer forward, this one moves it backwards but to the middle - which is a better "metric"? And what about flanking of course. (Military command).

[Anonymous No.12458206]

>>12458200
Wow, that's some pretty advanced stuff, I'm still just an undergrad

[Anonymous No.12458244]

>>12458200
should take me atleast few years anon

[Anonymous No.12458251]

>>12458203

>psychedelics

Big memes. If you want a reliable PED just take nicotine which is well documented and pretty safe if taken without the carcinogens from cigarettes.

[Anonymous No.12458252]

>>12458189
>implying it's atypical

/mg/ got shit.
It was always only a banter ground but now there's better places for that

[Anonymous No.12458256]

why would you learn TeX if you can just pay a professional to do it

[Anonymous No.12458258]

>>12458251
Psychedelics isn't for the PED effects mainly in this case, all though low doses of it definitely would help (reindeers eat it before hunting). But it's for being able to "feel" the game more deeply and literally fall into it, let it become your whole reality. It makes you feel each piece on the board as living, each structure/layout of the pieces as straight from a medievel battle. Very cool stuff.

[Anonymous No.12458261]

>>12458256
Why would you learn TeX if you can just handwrite and take pictures/pass through a scanner/write on a touchscreen tablet

[Anonymous No.12458285]

>>12458256
I could write TeX for you professionally. What's the pay? I didn't know you could work as a TeX professional.

[Anonymous No.12458287]

>>12458252
At least 30% of posters here are PhDs anon...

[Anonymous No.12458307]

>>12457062
All the anime retards thankfully left years ago. Now 4chan is something worth visiting

[Anonymous No.12458308]

>>12458287
There's absolutely no way

[Anonymous No.12458322]

Is geometry real? I'm convinced it's just a visual representation of vector spaces, like picture books for children.

[Anonymous No.12458338]

>>12458322
Nothing in math is real but geometry is also more than just (((Euclidean))) stuff

[Anonymous No.12458356]

>>12458308
Then you don't come here very often.

[Anonymous No.12458368]

>>12457035
Strongest program for algebraic topology?

[Anonymous No.12458374]

>>12458338
I meant real as in... Is it really a necessary abstraction that can't be properly captured by pure algebra?

If so, can you give some examples of these non-euclidean geometries which transcend algebra?

[Anonymous No.12458394]

>>12458368
Cambridge, Oxford, MIT and Harvard. And this is not only for AT, but for everything practically.

[Anonymous No.12458395]

>>12458287
Mark out one post in this thread that you know was written by someone with a PhD.

[Anonymous No.12458397]

>>12457582
>>12457584
>>12457630
ABSOLUTELY SEETHING FOUNDATIONS BRAINLETS

[Anonymous No.12458403]

>>12458118
speaking a language other than english outside of /int/ is a bannable offense, you’ve been warned.

[Anonymous No.12458406]

>>12458395
My own.

[Anonymous No.12458409]

>>12458395
>>12458200
>>12457556
>>12457451
And there were more last thread.

[Anonymous No.12458416]

>>12458394
how's Princeton compared to those?

[Anonymous No.12458432]

>>12458403
The downvote is my sword.

[Anonymous No.12458449]

>>12458416
Not as good, but should suffice ifyou can't get into any of those 4.

[Anonymous No.12458452]

>American universities
>America
Imagine getting trained in a false nation

[Anonymous No.12458477]

>>12458403
¿Lo es?

[Anonymous No.12458512]

>only his wish to know more mathematics kept him from suicide

[Anonymous No.12458521]

>>12457035
How are continuity, fixed points, and computability related?
Also, are quantum methods just a fancy name for distributed protocols, but done topologically rather than algebraically?

[Anonymous No.12458554]

>>12458374
Geometries provide semantics for the algebras.
Without spatial representation any symbol manipulation is meaningless.

[Anonymous No.12458596]

>be le me
>try phone posting for the first time
>app doesn't support latex
Reminder to post Latex to btfo phone posters and brainlets.

[Anonymous No.12458692]

>>12458512
I'm more of a Yutaka Taniyama kinda guy

[Anonymous No.12458699]

can someone explain to me what the open sets in [math] 2^{\mathbb{N}}[/math] look like with the product topology? This shit doesn't make sense to me. I want to say they are collections of finitely supported functions but idk if I'm right.

[Anonymous No.12458705]

>>12458554
Does maths have to be meaningful? Perhaps geometry is just a form of applied mathematics?

[Anonymous No.12458807]

>>12457210
http://kevinluo.net/books/book_Topology%20-%20James%20R.%20Munkres.pdf
try this one out!

[Anonymous No.12458813]

>be le me
>5got how to la tex
>5got how to maths
anyone got that cool book image again, I got time on my hands for how to learn some more maths

[Anonymous No.12458819]

>>12457971
>masters in physics

[Anonymous No.12458834]

Hello friends! Abstract algebra, algebra lalalalala laaaaa merry christmas

[Anonymous No.12458844]

we need abstract algebra

[Anonymous No.12459460]

>>12457076
>start to finish
Kek.

[Anonymous No.12459473]

>>12457285
Anon if you're wanting to get into smooth manifolds and later differential geometry then Lee is a great book. I'm reading it rn and the material is very clear. I also love how he sprinkles exercises in the middle of the text so it helps your understanding.

It deals with the same material a first course in topology would too, fundamental groups covering maps etc.

[Anonymous No.12459485]

>>12458819
he's european.

[Anonymous No.12459597]

What are some interesting non-measurable sets besides the Vitali set

[Anonymous No.12459600]

>>12459460
mathematic ends at multivariate calc (or you can call it analysis)
beyond that, it devolves into mostly useless crap kept alive so people can churn out papers

[Anonymous No.12459683]

>>12458147
black can resign

[Anonymous No.12459689]

https://lichess.org/EhuJbaWM .

okay lets go niggerfaggots

[Anonymous No.12459816]

>>12457173
The category of rings is too "big" to be a set

[Anonymous No.12459885]

>>12459600
>mathematic ends at multivariate calc
clearly it ends there for you retardo

[Anonymous No.12459892]

>>12459600

[Anonymous No.12459907]

>>12459600

[Anonymous No.12459914]

>finally proved that [math]\sqrt[n]{n} \rightarrow 1[/math]
We're all gonna make it

[Anonymous No.12459915]

>>12459600
I don't know, the book I'm currently reading on manifolds gives me joy because it's fun to work through. Do you think there are meaning in that or would you still consider that useless?

[Anonymous No.12459962]

>>12459914
How did you prove it?

[Anonymous No.12459972]

>>12459962
Showed that its monotone decreasing for n>= 3
Showed that its lower bounded by 1
Therefore its convergent
Assume it converges againt 1+a, a>0, then the sequence [math]\sqrt[n]{(1+a/2)^n}[/math] converges against 1+a/2 but is larger than [math]\sqrt[n]{n}[/math] for sufficiently large n. This is a contradiction since its limit is lower.
Therefore the sequence must converge against 1

[Anonymous No.12459983]

>>12459597
There are none. "Vitali set" is a meme, it literally doesn't exist.

[Anonymous No.12459985]

>>12459972
Cool. Now try proving it by giving sharp explicit bounds of convergence.

[Anonymous No.12459993]

>>12459985
I already did try
I just could not

[Anonymous No.12459997]

>>12459993
What did you try?

[Anonymous No.12460002]

>>12459997
[math]sqrt[n]{n} - 1 < \epsilon [/math] is equivalent to [math]n< (1 + \epsilon)^n[/math] so I mostly tried to change estimate the right side down and use the binomial formula but I could not get it to work

[Anonymous No.12460006]

>>12459972
>against 1+a/2 but is larger than n√n for sufficiently large n
Why?

[Anonymous No.12460009]

>>12460006
because (1+a/2)^n is larger than n for sufficiently large n
I think one should be able to prove this with induction somewhat easily but I havent tried yet

[Anonymous No.12460019]

>>12460002
>and use the binomial formula
Try using
$$
(1+\epsilon)^n \geq 1 + n(n-1) \epsilon^2 /2
$$

[Anonymous No.12460025]

>>12457786
>Go to his Twitter page
>He is drawing full of fucking lewd pics
Based

[Anonymous No.12460029]

>>12460019
god damn that why did I not think of that
[math]n>\frac{1}{\epsilon^2}[/math] solves it
At least I got the right idea I guess

[Anonymous No.12460074]

>>12459816
That's kind of a boring answer. When do you actually need to work with a proper class of rings simultaneously?

[Anonymous No.12460088]

>>12460025
show

[Anonymous No.12460114]

>>12460074
To define projectives and injectives.
See >>12457176

[Anonymous No.12460149]

>>12458699
so 2^N is the space of sequences of 0s and 1s, and a basis for the topology should be given by cylinder sets, i.e. sets where you fix the first n elements of a sequence and allow the rest to be whatever they want. so definitely not compactly supported.
you should think about product topologies in terms of pointwise convergence. if your sequence is outside an open set and it converges pointwise, the limit needs to be outside the open set. well all you can say about pointwise convergence are things about finitely many elements converging. so open sets will be defined by keeping finitely many elements constant (or unions of such sets).

[Anonymous No.12460160]

>>12460114
You can just use the definition in terms of a dual basis.

[Anonymous No.12460240]

>>12460074
the reality is, sometimes you want to say, for example, "there exists a ring such that for every ring bla bla bla"
and it's perfectly understandable english but sometimes you can't write it down in terms of set theory
a normal person won't care
but to some autists it is such a huge problem that they write 1000 pages long books on category theory or they insist we should redo the entire foundations

[Anonymous No.12460247]

>>12460240
I know basic homological algebra. Give me an example of such a property which cannot be expressed by an equivalent condition in terms of only a set-worth of nodules. Better yet, give me such a property which is actually used in an interesting result.

[Anonymous No.12460250]

>>12460247
*modules
lol

[Anonymous No.12460271]

>A connection on a principal G bundle is a lie-algebra valued form thst is G equivariant and send fundamental vector field to their generators
Who the fuck thinks this is acceptable? Isn't Geomtry supposed to bring intuition and clarity? Fuck this

[Anonymous No.12460323]

>>12459473
So it's for beginners right?

[Anonymous No.12460344]

>>12460271
Ist das hier ein komplizierter unintuitiver Satz, nur weil du nicht Deutsch sprichst? Nein.

[Anonymous No.12460349]

>>12460271
What is not clear and intuitive in that sentence?

[Anonymous No.12460363]

https://www.strawpoll.me/42274371

[Anonymous No.12460364]

>>12459485
yes and by the way i was finally explained why americans dont do masters, because your phds's are way longer (ours is 3 years) and we dont have classes in our phds normally wether you do
so in effect, you do our masters during your phd

its just semantics really

[Anonymous No.12460426]

>>12460364
We're actually awarded master's degrees on the way to a PhD in some cases.

[Anonymous No.12460430]

>>12460344
Knowing the definition doesn't mean it is intuitive.
>>12460349
Why the hell do you need such a construction to define a covariant derivative compatible with a group transformation.

[Anonymous No.12460467]

>>12460271
Usually I visualize an isolated connection as the horizontal splitting thing and use that specific definition for visualizing the entire space of connections.

[Anonymous No.12460505]

>>12460364
How long is an american PhD?

[Anonymous No.12460633]

>>12460505
5-6 years

[Anonymous No.12460640]

>>12460633
Why so long? Are americans retarded?

[Anonymous No.12460655]

>>12460640
Mainly what the poster b4 pointed out, they go directly from bachelor to PhD, and attend courses in the first few years (in europe people start immediately with research in their PhD, but do a 2 year masters degree first)

[Anonymous No.12460658]

>>12460430
>Why the hell do you need such a construction to define a covariant derivative compatible with a group transformation.
do you have a better idea?

[Anonymous No.12460773]

>>12460658
Yes

[Anonymous No.12460780]

>>12460773
can you present it?

[Anonymous No.12460783]

>>12460780
So you can steal it? No way fag

[Anonymous No.12460838]

>>12460783
so I can tell you why principal bundles are better

[Anonymous No.12460898]

>>12458092
don't fucking play chess then. Or try fisher chess of blitz

[Anonymous No.12460949]

>>12460838
Why are they better?

[Anonymous No.12460960]

https://www.strawpoll.me/42275036

[Anonymous No.12460968]

>>12460960
reddit

[Anonymous No.12460971]

>>12460949
they're extremely intuitive and easy to understand

[Anonymous No.12460975]

>>12460971
What is the intuition behind a lie algebra valued form?

[Anonymous No.12460986]

Ahem.
Despite making up 20% of posters, people who randomly post in german are disliked by 60% of posters.
>>12460975
Oh no, you're actually retarded.

[Anonymous No.12460991]

>>12460986
>Oh no, you're actually retarded
For no understanding the intuition behind such an abstract concept?

[Anonymous No.12461006]

>>12460991
In this case, yes.
Remember [math]V \otimes W^* \cong Hom(W, V)[/math] from widdle baby multilinear algebra? That's the intuition.

[Anonymous No.12461017]

>>12461006
No you fucking retard, the intuition behind that object defining a connection and parallel transport.

[Anonymous No.12461032]

>>12460975
take a point 'p' in your manifold and a frame at this point. now consider a path starting at 'p' and extend the frame along this path in two different manners. the two moving frames differ by a curve in GL(n) starting at identity. therefore infinitesimally they differ by an element of gl(n) (you just take derivative evaluated at identity). note that we can compare two moving frames like this, but it doesn't make sense to say that a moving frame is constant. it has to be compared against something. connection is a choice of constant moving frames, but it's all expressed infinitesimally.

[Anonymous No.12461057]

>>12461017
Then what's the confusion?
To every tangent vector in your original manifold (a direction of motion) you associate a tangent vector at the group's identity (a direction of motion from the identity).
And then you ask for obvious compatibility conditions.

[Anonymous No.12461082]

Hiya /mg/!

From tomorrow to Friday, there will be the 8th East Asian Conference on Algebraic Topology.
https://sites.google.com/view/eacat8/home

[Anonymous No.12461095]

>>12458200
ω gives isomorphism between 1-forms and vector fields since it's non degenerate
X hamiltonian <=> dH = ω(X,-) <=> ω(X,-) is exact
X locally hamiltonian <=> ω(X,-) locally exact <=> ω(X,-) closed (Poincare lemma)
all follows from this

[Anonymous No.12461096]

Hi /mg/,
I need to find the limit for f(x) when x tends to infinity. But I have difficulties to continue from g(x). I tried expanding the numerator because I had no other idea how to continue. We haven't yet covered derivatives. Any hints/tips for a novice?

[Anonymous No.12461110]

>>12461096
do the same you would if the powers were all natural numbers. factor out the highest power in numerator and denominator

[Anonymous No.12461123]

>>12461096
First of all, don't do that. Notice that you have [math]\sqrt[3]{\frac{a^3}{b}}= \frac{\sqrt[3]{a^3}}{\sqrt[3]{b}} = \frac{a}{\sqrt[3]{b}}[/math], and then run away to /sqt/ before someone starts complaining!

[Anonymous No.12461218]

>>12461110
>>12461123
Much appreciated, thanks

[Anonymous No.12461393]

>>12460247
You're basically asking us to justify category theory since most universal properties are defined in terms of all ovjects. A lot of universal properties are way nicer to work with directly than equivalent definition. For example no one wants to take an actual construction of ring tensors when the universal property directly tells you what you want.
Of course you could still prove everything without category theory but it'd be a lot more painful.

[Anonymous No.12461395]

>>12461095
Jolly good job, lad.

[Anonymous No.12461426]

>>12461395
post more problems (I don't know *that* much about symplectic geometry specifically though)

[Anonymous No.12461508]

>>12461426
In the spirit of plugging in a small gap on the previous problem, prove that if [math]\omega[/math] is a smooth non-degenerate differential form and, for a not necessarily smooth vector field [math]X[/math] we know that [math]i_X \omega[/math] is also smooth, then [math]X[/math] is smooth.

[Anonymous No.12461511]

>>12460655
Interesting, I always found it strange that they skipped masters.

[Anonymous No.12461515]

>>12461508
w, being nondegenerate gives a smooth isomorphism TM with T*M which we use to recover X from a smooth covector field.

[Anonymous No.12461518]

https://www.youtube.com/watch?v=Liqs0k4bXvY

Thoughts on this math genius?

[Anonymous No.12461525]

>>12461515
I didn't say [math]\omega[/math] is a two-form tho.

[Anonymous No.12461526]

>>12461518
If you've actually watched the movie you'd realize that in this scene he's daydreaming and in reality he sucks at maths.

[Anonymous No.12461528]

>>12461525
You said it's a nondegenerate form. Nondegenerate as a concept only makes sense in the context of two-forms.

[Anonymous No.12461539]

>>12461528
Use this definition, then.
For any tangent vector [math]Y[/math] at any point [math]p[/math], [math]i_Y \omega _p = 0[/math] if and only if [math]Y = 0[/math].

[Anonymous No.12461542]

My friend helped me to find a solution. The solution uses the formula for expanding the binomial in the numerator.

(a+b)3 = a3 + 3a2b + 3ab2 + b3

Here a = x^(1/3) and b = x^(1/4). After a lot of work I ended up with 1/2, which seems correct.

[Anonymous No.12461544]

>>12461526
I know.

[Anonymous No.12461549]

>>12461542
Probably obvious for /mg/, but certainly not for me. In step "p(x)" the numerator is multiplied by 1/x which is the same as x^(-1).
in "q(x)" the numerator is converted to the root notation and denominator is also multiplied by 1/x.

[Anonymous No.12461584]

>>12459983
that's all folks: axiom of choice is a meme all thanks to anon and his deep analysis of the foundation of mathematics.

[Anonymous No.12461599]

>>12461584
this but unironically

[Anonymous No.12461602]

>>12461082
I wish I could be on that level to understand the conferences. Feels bad being a brainlet.

[Anonymous No.12461696]

>>12458554
Is counting symbolic or geometry? A bunch of +1s arranged on a page is symbols, but also a geometric graph. Is a bunch of dots in your vision in an arbitrary ambient space and dimension, is that geometry or symbols?

[Anonymous No.12461743]

>>12457035
The nerve of a group represents singular cohomology in the sense that [math][X,NG] \cong H^1(X;G)[/math]. Is there a similar theorem for when [math]G[/math] is a groupoid (e.g. the fundamental groupoid of some other space)?

[Anonymous No.12461745]

>>12461526
Just like me!

[Anonymous No.12461943]

I asked in /sqt/ and no one bit. Please help

Details in combinatorics trip me up.
Let’s say there are 32 balls in a bucket. Each ball is one of 4 colours (blue, green, red, yellow) so that there are 8 balls of each colour. Within each colour the balls are labelled from numbers 1 through 8.

I get tripped up when I try to sum each 'type' and 'subtype' of combination.

TOTAL combinations: 32C4=35,960
FOUR of a number (e.g. 5555): 4C4*8=8
THREE of a number (e.g. 5556): 4C3*8*28=896
•with a shared colour: 4C3*8*7*3=672
•with different colour: 4C3*8*7=224

TWO numbers repeating (e.g. 5566): 8C2*4C2*4C2=1008
•with different colours: 8C2*4C2=168
•one matching colour: 8C2*4C2*2*2=672
•two matching colours: 8C2*4C2=168

ONE repeat of a number (e.g. 5567): 4C2*8*28*24/2=16,128
•with different colours: 4C2*8*7*6=2016
•with non repeating numbers of matching colour: 4C2*8*7C2*2=2016
•with a repeating number matching colour with a non-repeating number: 4C2*8*7*2*6*2=8064
•two matching colours: 4C2*8*14*6/2=2016
•three matching colours:4C2*8*7*6=2016

The one below is definitely wrong cause I can't get the subtypes to sum to the types
NO repeats of a number (e.g. 5678): 32*28*24*20/4!=17,960
•with different colours: 8*7*6*5=1680
•with one matching colour: 8C2*4*16*10/2=10,080
•with two matching colours: 8C2*4*6C2*3=5040
•with three matching colours: 8C3*4*15=3360
•with four matching colours: 8C4*4=280

I don't know whether I'm missing subtypes and/or simply just have bad math (I'm leaning towards just the latter). Please help calculate each subtype!

[Anonymous No.12461952]

>>12461518
>the hardest geometry problem
>not even his professor at mit can solve it
>it's a basic calc 1 probelm

why is math in movies so hilariously bad?

[Anonymous No.12462096]

>>12461602
Same, but at least we can share the links to people who could be simultaneously interested and capable.

[Anonymous No.12462113]

>>12462096
Liar, you're a PhD, I know!

[Anonymous No.12462143]

>>12462113
If I ever manage to graduate. I am a total waste of oxygen with no skills whatsoever haha...

[Anonymous No.12462187]

>>12462143
What is your area?

[Anonymous No.12462205]

I baked cookies anons. Is there a sphere packing/tiling theorem for uneven spheres?

[Anonymous No.12462215]

>>12462187
Algebraic adaptations of techniques from stable homotopy theory.

[Anonymous No.12462221]

nachzehrer#3715

I want to speak to someone with a genius level IQ on discord.

[Anonymous No.12462225]

>>12462215
Is it hard? What are the pre requisites for studying homotopy theory?

[Anonymous No.12462230]

>>12462187
Algebraic topology and finite groups, the connections of those via homotopy theory.

[Anonymous No.12462239]

>>12462205
You're supposed to bake cookies fren not fry them

[Anonymous No.12462242]

>>12462230
Oh, then my post (>>12462225) is for you.

[Anonymous No.12462258]

>>12457556
>all of mathematics can be described using english language
nah, it can't.

[Anonymous No.12462268]

>>12462225
Depends on the flavor of homotopy theory. Algebraic topology will teach you the applications of classic homotopy theory and is an essential starting point for understanding the more exotic varieties.

[Anonymous No.12462279]

>>12462242
Unstable homotopy theory is pretty hard, yeah. Even if you make the reasonable choice and restrict yourself to CW-spaces, you will still find a lot of trouble on your path to any results (just think of the homotopy groups of spheres which one would think should be solved by now but nope!), and the finite complexes are even more ad hoc. I don't know. Maybe it's not that bad but my supervisor pities me, maybe it is really hard. For sure, it's way beyond me. For prerequisites, know the basics of group theory and point set topology, develop some knowledge about homology and cohomology, too, as they will allow you to at least disprove certain claims. Also, I am talking from the unstable perspective. This poster makes a good point >>12462268

[Anonymous No.12462288]

>>12462268
>>12462279
Random question:
When you two talk about other types of homotopy theory, do you mean stuff like rational, local and stable homotopy theory or autismo higher categories and combinatorics around simplexes?

[Anonymous No.12462340]

>>12462288
When I speak of homotopy theory, I mean [math]\pi_*(- ; G) \colon \textbf{Top}_* \to \textbf{Top}_*[/math], where the group [math]G[/math] is some [math]\mathbb{Z} /n\mathbb{Z}[/math] with n=0 included for the sake of ordinary stuff. Everything but those are "other" types for me. I could of course think of rational stuff as "non-other", but that would be highly irrational because the homotopy groups I would encounter are mostly finite and then they would be killed by the UCT [math] 0 \to \pi_n(X) \otimes_\mathbb{Z} \mathbb{Q} \to \pi_n(X; \mathbb{Q}) \to \text{Tor}_1^\mathbb{Z}( \pi_{n-1}(X), \mathbb{Q}) \to 0[/math].

[Anonymous No.12462417]

>>12462340
Hum, cool, very cool.

[Anonymous No.12462488]

>>12457035
>Stats exam is tomorrow

[Anonymous No.12462499]

>>12462340
Can you explain why you care about this shit?

[Anonymous No.12462817]

>>12462258
what

[Anonymous No.12462819]

>>12462499
Classifying spaces of groups are pretty important for various reasons. If you take the bundle [math]G \to EG \to BG[/math] for a group [math]G[/math], then first ask yourself the yes/no question "is this group discrete?" and the answer will determine your path.

If the answer is no, then the contractibility of [math]EG[/math] tells you that [math]\pi_n(G) \cong \pi_{n+1}(BG)[/math], so knowing the homotopical properties of the group will give those of its classifying space (and vice versa). An example of this, and a very dumb one, would be the following. Let's pretend we don't know the homotopy groups of [math]\mathbb{C}P^\infty[/math], but we know those of [math]S^1[/math], and we know that [math]\mathbb{C} P^\infty \simeq BS^1[/math]. This tells us the following: [math]\pi_2( \mathbb{C}P^\infty) \cong \mathbb{Z}[/math] and the other degrees give trivial structure. One could of course do this for a group more complicated than the circle and get some more interesting information, but that was an easily verifiable claim about a space that is itself not a group. More generally, we may have a space [math]X[/math] such that [math]\Omega(X) \simeq G[/math], and then we obtain the information carried by [math]\Omega (X) \to \mathcal{P}(X) \to X[/math] from [math]G \to EG \to BG[/math]. However, I don't do this stuff so let's look at the other side!

[Anonymous No.12462822]

>>12462819
If the group is discrete, then there's not too much to be extracted from [math]G \to EG \to BG[/math]. It pretty much only tells us that [math]BG[/math] is an Eilenberg-MacLane space [math]K(G, 1)[/math]. However, should we be interested in group (co)homology, this would now become quite important, as one has [math]H_*(G; M) \cong H_*(BG; M), H^*(G; M) \cong H^*(BG; M)[/math], where the stuff on the left hand side is the group (co)homology of [math]G[/math] with coefficients in a suitable module. This isn't an easy thing to compute in general, though, so a little trickery can be used. Assume [math]G[/math] is finite. Then its cohomology splits into a direct sum of pieces, one for each prime divisor of the order of the group. Choosing the coefficients to be in the field [math]\mathbb{F}_p[/math] for each such prime, we kill everything except the [math]p[/math]-component, and this gives us [math]H^*(G; \mathbb{F}_p)[/math].

This is when we reintroduce homotopy theory. If you have ever heard of the Bousfield-Kan completion, this is it. Since [math]G[/math] was assumed finite, it is "[math]p[/math]-good" for every prime, that is, [math]H^*(BG) \cong H^*((BG)^\land_p) [/math], where [math]X^land_p[/math] denotes the [math]p[/math]-completion of [math]X[/math] and all cohomology is with [math]\mathbb{F}_p[/math]-coefficients. We now know that the cohomology of the completed classifying space is isomorphic to the group cohomology of our group (with the same coefficients!). Furthermore, this completion comes with a nice property: [math]X^land_p \simeq Y^\land_p[/math] iff [math]H^*(X) \cong H^*(Y)[/math], so we may simplify our considerations by applying this. Suppose we have a finite group [math]H[/math] such that [math]H^*(H) \cong H^*(G)[/math] and a much more approachable classifying space. Then [math](BH)^\land_p \simeq (BG)^\land_p[/math]!

[Anonymous No.12462826]

>>12462822
If we can now decompose [math]BH[/math] in a way that is preserved by the completion, then we have a similar decomposition for [math](BG)^\land_p[/math], and this may be used to compute [math]H(G; \mathbb{F}_p)[/math], which may prove useful when dealing with various bundles like the Borel fibration.

A bit rambly, but that's life.

(And that [math]X^land_p[/math] should be [math]X^\land_p[/math], oopsie!)

[Anonymous No.12462867]

>>12462819
>>12462822
>>12462826
based tranny. post your favorite christmas song.

[Anonymous No.12462870]

How would you know if you have some aptitude for math

[Anonymous No.12462876]

>>12462870
If your IQ is at least 130.

[Anonymous No.12462887]

>>12462867
OK, whatever your reasons for asking might be. Either of these, I suppose. The first one is actually by Sibelius and Waltteri is cute. https://www.youtube.com/watch?v=Esub5aDLgQk & https://www.youtube.com/watch?v=ZW8um5Uv6Pw

[Anonymous No.12462889]

>>12462870
If you like philosophy

[Anonymous No.12462906]

>>12462876
Something cheaper to find out please

[Anonymous No.12462908]

>>12462887
Which is your favorite manga and anime?

[Anonymous No.12462924]

>>12462908
I must admit I haven't read too much manga, because there's something in that makes me anxious, but maybe Senpai ga Uzai Kōhai no Hanashi. Best anime is definitely Utena. How about yours?

[Anonymous No.12462930]

Post things about gas dynamics and spiral fractals.

[Anonymous No.12462959]

>>12462924

I would have to say that my favorite anime is Sakurada Reset and if I would have to choose a manga it would be Oyasumi Punpun. Why do you think that reading manga makes you anxious? After getting used to reading math books, mangas are read quite fast.

[Anonymous No.12463027]

>>12462959
>Sakurada Reset
That's been on my plan to watch list for a while. Thanks for the reminder.
>Why do you think that reading manga makes you anxious?
Probably takes me back to being 13-14 and having nasty classmates. Sleepless nights made me tired, being tired made me tense and timid, being timid made me get bullied more, getting bullied made me lose sleep. Then add a volume of Detective Conan which I thought would be a good read during the dark night hours, and then all the sudden gore probably formed a connection between feeling bad and manga. Just a conjecture, though.
>After getting used to reading math books, mangas are read quite fast.
Indeed, and the same for pretty much every other type of literature over there as well. The sheer compactness of mathematical text makes Lovecraft light reading!
Any progress and the mathsy side of life?

[Anonymous No.12463036]

>>12463027
My math life has been frozen for a few years, after graduating I had a lot of problems in the rest of life and lost all my will in doing anything. Since then I haven't found a single thing that can motivate me, had some random urges like trying to prove the collatz conjecture or grabbing random math books to find cool new topics, but nothing has worked so far... One doesn't value motivation until its all gone.

[Anonymous No.12463083]

>>12462887
just wanted to check if you were the finnish postgrad.

[Anonymous No.12463087]

>>12462887
the one by sibelius was pretty good btw.

[Anonymous No.12463157]

>>12463036
Even in all that negativity, you still keep trying. It is important not to give up even if everything seems pointless. Instead of some unrealistic goal like Collatz, you should probably set your mind to something less ambitious that will allow you to regain your strength once possessed and reward you with the sensation of learning. If this seems like an interesting direction, I suggest you read Rotman's book on AT, then supplement that with Whitehead and Neisendorfer's books on homotopy theory. Learn for the sake of learning. Whatever fruits your beautiful garden will give you, those are just a bonus. The important thing is to build the garden.

>>12463083
>>12463087
You caught me then. Take one more. https://www.youtube.com/watch?v=iZ4EGvU3abg

[Anonymous No.12463167]

>>12457035
>Wear a mask
Fuck that.

[Anonymous No.12463183]

math undergrad with zero knowledge in physics. can /sci/ recommend a thorough introductory text to quantum computing that suits my background?

all the sources I could find were either pop science tier (avoiding any mathematical formalism), or assume way too much familiarity with quantum mechanics

[Anonymous No.12463208]

>>12463157
Are you actually trans?

[Anonymous No.12463410]

>>12463208
No.

[Anonymous No.12463414]

>>12463208
Not science or maths.

[Anonymous No.12463419]

>>12462876
My IQ is 145 and I'm fairly certain I would struggle studying math.
t. comp sci chad

[Anonymous No.12463424]

>>12463208
How new?

[Anonymous No.12463451]

why are there so many faggot algebraic topologists ITT? any chad algebraic geometers?

[Anonymous No.12463550]

>>12457055
You are on the wrong site my fren

[Anonymous No.12463717]

>starts proof-based book
>reads the entire chapter
>starts doing the exercises
>get stuck
>looks up the solution
>you need to use an inequality learned during calculus 2 or some bullshit like that that wasn't even mentioned in the chapter
Anyone else tired of this? Holy fuck, they should give us all the tools to solve the problems in the theory, but no, they just throw exercises where you can only solve by remembering stuff you did ages ago, this is honestly making me disappointed in maths. It's ok to make hard exercises, but it's frustrating when they make cheap artificially difficult problems like that.

[Anonymous No.12463731]

>>12463717
What was the exercise?

[Anonymous No.12463768]

>>12463731
You're gonna call me a brainlet because it was a really easy problem, but whatever. It was to prove pic related. The only reason I got stuck was because I didn't remember the Telescoping Sum which I learned more than a year ago and the book didn't even mention it at that point.

[Anonymous No.12463777]

>>12463768
not the same dude, but whats the book? i have an entire course based on proving stuff just like this, and i suck at it. would like a good source (even tho your book isnt the best as it seems)

[Anonymous No.12463788]

>>12463777
It's written in portuguese and it's not that good. I'd rather use Tao, but it's not what the professor uses, so...

[Anonymous No.12463790]

>>12463768
Do you know about the method of mathematical induction? You can prove it using induction.

[Anonymous No.12463806]

>>12463790
Yes, I thought of that, but I'm pretty sure that problem is for [math]\reals[/math]

[Anonymous No.12463821]

>>12463790
Yes, I thought of that, but I'm pretty sure that problem is for [math]\mathbb{R}[/math]

[Anonymous No.12463846]

>>12463790
Yes, I used that, I wasn't sure of it because the solution used the Telescoping Sum.

[Anonymous No.12463853]

>>12463036
>random urges like trying to prove the collatz conjecture
I just reduced the Collatz Conjecture to the problem of adding two explicit large natural numbers. Can you add natural numbers? If so, would you like giving it a try?

[Anonymous No.12463926]

>>12462817
he's right, for example mochizuki's proof of ABC conjecture can only be expressed in japanese language

[Anonymous No.12464111]

>>12457035
Does an enginner need to know anything beyond Stoke's theorem and fourier transforms?

[Anonymous No.12464130]

differential forms are some real shit

[Anonymous No.12464145]

>>12463788
Fuck off, Elon is dandy.

[Anonymous No.12464181]

>>12463208
Yeah thats the idea, learning for the sake of learning but I should first motivate myself to do things to avoid starvation. Thanks for the kind words Anon.

>>12463788
>>12464145
Can confirm that Lages Lima is the gud shit.

[Anonymous No.12464205]

>>12463208
Yeah, I'm a transcedental number theorist, luv that e+pi shit..

[Anonymous No.12464218]

>>12463788
Why use Análise Real when there's Curso de Análise?

[Anonymous No.12464249]

>>12464111
an engineer doesn't need to know anything, lol

[Anonymous No.12464297]

>>12464145
>>12464181
It's a meme.
>>12464218
It's what the professor uses and the author himself said that his other book should serve more as a reference book.

[Anonymous No.12464364]

Should I continue and finish off Tao's Analysis I book or switch to a harder textbook with some applications? I'm currently in Chapter 7, and kinda worry that I'm gonna miss something important from calculus since I skipped it for analysis.

[Anonymous No.12464387]

>>12464364
>calculus
Shouldn't have skipped it. In calculus you learn how to walk, in analysis you learn how to run.

[Anonymous No.12464395]

How much liquid can the average human womb hold? How long of an orgasm would be required to fill said womb?

[Anonymous No.12464438]

I want a picture of Donaldson on the next OP.
That is all.

[Anonymous No.12464465]

>>12464364
Why would you ever skip calculus?

[Anonymous No.12464490]

>>12464387
>>12464465
Well calculus seemed self-contained in analysis, and honestly, I am tired of tedious computations, I prefer proofs.

[Anonymous No.12464494]

>>12464490
Learning to get intuition and insight from computations is a valuable skill.

[Anonymous No.12464504]

>>12464364
you'll be like that cocky guitarist who can play fast solos but is absolutely clueless when he's supposed to just riff in the background

[Anonymous No.12464511]

>>12464490
>, I am tired of tedious computations, I prefer proofs.
Be careful what you wish for. You might just get it.

[Anonymous No.12464568]

>>12464387
>>12464494
fake and gay, there's nothing to miss from a plug'n'chug calc course for enginiggers if you have already taken analysis in [math]\mathbb R[/math]. Why do you think European majors skip calc altogether?

[Anonymous No.12464574]

>>12464568
>Why do you think European majors skip calc altogether?
what?

[Anonymous No.12464599]

>>12464568
No we don't. We have it in high school.

[Anonymous No.12464600]

>>12464568
They don't. If you think you're so smart, then just start calculus with Apostol or Courant, but skipping it for analysis is retarded.

[Anonymous No.12464615]

>>12464568
Yikes. Next thing we know you'll want to skip Riemannian Geometry to jump straight into Metric Spaces of Non-Positive Curvature.

[Anonymous No.12464619]

>>12464615
yikes

[Anonymous No.12464622]

>>12464599
I guess? The calculus we did wasn't proper calculus I don't think in A levels. Never had to pick up a textbook on calculus to get the highest grade available. University on the other hand I picked up Taos real analysis straight away and learnt what doing maths really meant.

[Anonymous No.12464625]

>>12464568
>Why do you think European majors skip calc altogether?
because of differences in naming, there's no calculus here, everything from "what's a limit, what's a continuous function" up to "stokes theorem" is called "real analysis"

[Anonymous No.12464635]

>>12464625
Wow, Europe is trash.

[Anonymous No.12464646]

Is it really bad to skip calculus? I'm doing pretty well on Tao and some additional exercises from Rudin without any calculus background. I'm a little bit rusty on trigonometry though kek.

[Anonymous No.12464664]

>>12464660

>>12464660

>>12464660

[Anonymous No.12464891]

i'm a physics major and i'm switching to math next year but its only in april next year and i got literally nothing to do until then, should i read an analysis book or something?

[Anonymous No.12465689]

tfw no hungarian alien mathwizard friend

[Anonymous No.12466927]

>>12463788
>>12464145
>>12464181
/mg/ full of sopas de macaco, uma delícia.

[Anonymous No.12467428]

>>12464615
>Metric Spaces of Non-Positive Curvature.
nice

[Anonymous No.12468539]

>>12457035
Too cute. Also the qts.