I guess you could use the fact that the polynomials of degree $\le n$ form a vector space of dimension $n+1$. But you'd need to know about the vanderMonde determinant.

@TedShifrin oh. my. god.

That is, by the way, an exercise in the linear algebra book somewhere.

I let the book in an hiatus

I'm trying to focus more on SAT

I'm kinda desperate because I have no hopes to get on an American university

Specially when talking about pure math (not pure meth)

Why is that?

The problem is expense.

That and insecurity.

You're not from a Muslim nation.

Yeah.

How does that relate, tho?

I don't know that college admissions would be affected for someone like you.

I mean, it's not like if not being Muslim $\implies$ good results, hahahah

Depends on the level of universities you apply to.

I don't know about your SATs and grades and recommendations, etc.

I have the dream of studying @ Berkeley

@Lucas you're brazilian right?

Ted means that its very difficult for foreign muslim nationals to get into the US right now

I mean, it doesn't have to be a graduation, tho

thanks to our cheeto in chief

@EricSilva sim

Better to do that in grad school. It is ridiculously huge and anonymous for undergrads.

uchicago has a weirdly big community of brasileiros in pure math my man

I thought you were trying for IMPA, Lucas.

U Chicago is super crazy hard and abstract. I hesitate to recommend it to anyone.

IMPA is like... a research university

I mean, I have to be at least graduated

Oh, I guess that's right. My fault.

that's a p fair criticism @Ted

I knew you wouldn't fight me, Eric. Someone else would.

lol

I didn't even look at Berkeley's Financial Aid, lol

@Lucas it probably suuuuuucks

I told you that I had two people from Athens who'd taken our Spivak course go to UC. One (a woman) switched out of math within a quarter. The other (a guy) did finish a math major but wasn't very happy; switched to a Ph.D. in philosophy. (Of course, his dad did the exact same thing.)

ah well this is not my conversation... i'm staying in my continent

That's why I'm desperate. I mean, I'm in my second year (@Eric knows best), so next year I'm aplying

Plenty of excellent math in your continent, Balarka.

Second year of what, Lucas?

@TedShifrin High school. The American and Brazillian scholar systems are a bit different. If you guys have 4 years, we have just 3.

that is true.

the secondary education is 3 years @Ted

sniped

Oh, I see, Lucas.

Actually, in my day, it was just 3 years of high school. 9th grade was still "junior high school."

I have this year and the next one to study.

whoaoh that's wild @Ted

SATs are a big part of it, Lucas, but there's lots more to it. Grades. Level of courses. Letters of recommendation. Essays.

My grades are pretty... cool...

I remember Andre asked me why there were so many brazilian undergrads in math here and he just paused and asked "Are they all crazy rich or something?"

On a scale of 0 to 10, I never had a grade below 8.

I honestly don't know how most universities/colleges evaluate foreign students for college. There aren't AP courses/exams abroad, for example.

Lucas, Berkeley is super competitive. It's one of the top 5 colleges in the country. So you are competing against people who have all 10s. Give yourself broader options.

@Eric in terms of reais, how much do you pay?

That is a valid question, Eric.

i get paid to go to school my man

@TedShifrin this is super disappointing, lol.

@Ted, what would you recommend me?

im a citizen of both the US and Brazil @Lucas so my fin aid situation is a bit different

wait what

You need to give yourself a variety of options, Lucas.

although I have a good friend here who gets a fin aid package and he's a brazilian national, but im unsure of how much exactly he gets

Yes, I'm aware of that @Ted

i could ask him

Remember, also, that in the US undergraduate education is far more general than in foreign countries. You cannot do all math, or even half math.

just a sec, I'm gonna check something relevant

i think u could probably do half math but it's probably not the best idea

So is this the whole math course?

I did a ton of math as an undergrad, but overall I doubt it was still half

I'm a bit confused..

i guess it depends how many courses are typical at the institution in question

@Ted so far ive actually done more than half math

It's a minor

I think at MIT something like 8 maths were required and I had double that, including a lot of grad courses.

but I also audit a bunch of history/classics so if you include stuff that i did and isnt on the transcript it's not half anymore

I was also a French major, except for a thesis, Eric. Plus 3 physics, 2 chemistry, 1 biology, several philosophy courses and English lit courses.

As a physics major, I took 14 physics courses out of like 36 total courses or something like that

i think i might end up with an incidental medieval studies major @Ted, but i won't actively pursue it

oh, Eric, and I took 6 semesters of German, 2 of Russian (along with 8 French literature courses).

So I don't think math was anywhere close to half, even though I did way more math than needed.

The only exception I know of in the US are engineering majors. Even at some generally liberal arts schools like Duke, engineering majors take something like 28 of 34 courses in engineering-related courses, so thats engineering plus natural and computer sciences.

oh man that's so much language

Last night dream involved some intense stuff related to path integrals where part of the integrand is the Thomae function. I need to check whether this function is injective and thus whether it has an inverse function

Wait what

engineering is always overloaded, Kevin. At UGA they're ridiculously overloaded with hours.

I must say I understand nothing behind what you guys are saying

Do you mean the math @lucas or the course schedules?

Join in the club

lol we have a single solitary engineering major here and it's brand new and has no one in it

I was just arguing that, unlike Europe where people do almost all math, in the US a math major typically takes relatively less math than everything else.

@KevinDriscoll course schedules

@Lucas Sorry, what was confusing?

@Lucas so at an american university course schedules are typically pretty open, you typically would be required to take stuff that isn't math (general education)

@Kevin: That is a lot of physics. I guess I took the same number (approx) of math, but I did a ton more things to "compensate."

i think ive taken 18 math courses so far (including this term) but in total ive taken like 28 courses "officially"

@TedShifrin You can tell in my diff top course. Ask us to do a slightly abstract problem like the one Im still pondering that I asked you and the math grad students blow me out of the water. Ask us to do a computation in local coordinates and I race past most of them suddenly.

Yeah, Eric, you're way ahead of me for college.

but even here at chicago im a freakazoid and no one here has taken a comparable amount (except one kid who left to go to a start up)

As you know, @Kevin, I'm a big believer in computing (just ask Eric). My students learned to do both.

@Kevin: Your problem is not having done any proof-based mathematics in forever. Theoretical linear algebra, multivariable analysis ... all that stuff is a serious prereq.

linear algebra is like the gluten of the loaf of mathematics

Eric: You're procrastinating as badly as Balarka now ;P

oooh, and as badly as Semiclassic.

nah it's impossible to procrastinate as badly as me

i am procrastination personified

I think Semi exceeds you markedly.

i have my paper half written

LOL, oh. And the pset? :D

so im getting stuff done WHILE goofing off

lol rip the pset is as yet unstarted

OK, Eric :P

Which class is the pset for?

alg top

u'll have no prob

so it's not as big a deal, im confident i could work it out p fast

Well, Hatcher problems can be quite challenging.

Maybe I'm just not a good topologist. :P

You're right

On both?

:P

UGA grad students struggled with a lot of Hatcher's problems.

I think I had done 90% of the Hatcher's exercises when I learnt algebraic topology

Enjoyed a lot

You're way more topology talented than you give yourself credit for.

benson made me do a crap load of hatcher probs last winter when he taught me stuff

OK.

Anyhow.

I guess I might have been too blunt to Lucas ... but ...

@TedShifrin Nah, I learn through the gritty way

But topology is certainly my comfy zone

I still remember when you were nothing but algebra.

gross

And knew no multi calculus or analysis.

Which person are you speaking of?

I am not sure I know them

You.

You claim not to know you, yes.

:P

Unless it's a matter of wrong apostrophe's.

Balarka 1.0 is, uh, a normie asshole version of myself

I think I knew 0.8.

roflmao

it sounds like protoBalarka would fit in at Chicago

Yup. He and Demonark could be twins.

roasting UChicagoans @Daminark

I think I'm currently Balarka 4.0, the polished new model who is a massive edge and memelord but somehow manages to not be an ass

You call?

No, still capable of asshood.

LOL

Oh, potato arrives

potato cometh

potato comet

I think Eric found out I was more or less of a kindred spirit ... geometry and food.

@TedShifrin :( Well, capable, but I try my best to not be one

I would not have gone so far as 4.0 ... when did we zip through 2.0 and 3.0?

ya great intersectionality of our interests

I think Balarka 4.0 is also a massive hipster but not the self-deprecating 3.0

Balarka and I are into shitposting and topology so there's that

very different sorts of topology

@Daminark is like my long lost cousin

Yes, I already said you were twins.

Equally horrid punsters, too.

well he May or may not be my mathematical bro though

True, I'm more number theoretic topology

wink wink nudge nudge

Heh

2 May or not 2 May

7 Days in May

One May is enough.

idk dude he's hilarious

he always wears the same outfit

me2

and he roasts people all the time

I love that kind of stuff

Fucking hell that's true

@EricSilva loool

Well, I leave you to this ... See you all anon.

(No comma before the anon.)

Oh I love it when Peter May says "whee"

bye @Ted have fun make good choices

See you @Ted

Going to an MIT alumni cocktail hour :P

See you @Ted!

@Daminark have you ever heard schlag refer to peter may as he who shall not be named

Oh nice!

To hobnob with rich engineers.

lol

I thought that was the Nenomena

But yeah Balarka is defecting to geometry and I'm more into number theory so maybe we're more cousins than twins

@EricSilva nope, I've only ever heard Schlag make one dig

number theory is like geometry for suckers

I plan to die in my topology rabbithole

I don't want a happy ending

When he did the AT with forms business he was just like "Yeah call Peter May and ask him to draw you some commutative diagrams"

neutral game throughout

@DAminark I once heard him refer to Soug as the guy with the "noncompact ego"

Like I'm actually not sure I've heard Schlag shit talk other than that. Like he never dissed anyone else

i heard him diss rand paul

Lolol, Schlag never mentioned Soug except with the whole conductors stuff I mentioned some time back

Aww why did he never make all those jokes with us?

Soug was the main comedian between the three my year

well the last happened when i was asking him pde questions in his office when i was going down a quasiconformal mappings rabbithole and he told me the story of how rand paul was at a ranch that he happened to be at and apparently he's a huge asshole

Ok, I have found the perfect avatar. No memes this time (sorry)

why is everything changing

I think I'll keep this one

@EricSilva the times are a-telling and the changing isn't free

is the revolution upon us tho

probably not. maybe the destruction is though

No. When its here, you wont have to ask

But I think maybe Marx was wrong about the revolution anyway

So it might not be coming

Lol how were you not intimidated by the guy? Like I was scared shit of saying anything in his presence for fear of getting struck by a lightning bolt

meh the revolution thingy is a cliche now. multiple ""revolution"" happened during the past century and they have failed miserably to bring any form of socialism or anything closer

perhaps you mean a sociological revolution or an economical one than a politically motivated one though

@Daminark Idk i just don't get intimidated by these folks, they're normal ppl

@Balarka that capitalist social order too stronk

tru

what about Anarcho-syndicalist catalonia tho

Oh I heard the story

i mean they all got killed because repubs had the big guns

you talking 20s?

36-39

wow sad memory on my part

Fair

@Daminark i think that ive actually been really successful just asking profs for shit, it helps that a lot of the analysts here in particular really seem to wanna educate us young folks

It's strange, like around Laci, Marianna, Peter May, I'm not too afraid. Also Soug

i mean peter may is like, anti-intimidating at least

and soug is an ultra normie

But around Schlag and DCal I'm just like "What if I'm being stupid? Yeah let's not take the chance"

he just wants watch soccer and crack open a cold one with the bois

(I've had less contact with DCal since I've been in two classes but then swapped to Laci, but still I get a slightly similar vibe)

And that's fair re Soug and May

Even now Soug remembers me as being homeless

@Balarka anyway it's a good example that the people who spew neolib propaganda usually just dont know happened

hada rough exam today

@Daminark ok u know benson is actually intimidating but that's like cause he's actually just a crazy person

I'm actually less intimidated by Benson

ok but have u tried to have a math conversation with him in his office lol

Like he feels like he'll just be all "Lol no" and correct you. Schlag feels like he'll just internally facepalm. Though Benson keeps bouncing right up in your face when he talks and that's a bit scary

Not scary as much as it's just like, don't crash into me plex

@Faust oof, sry m8

@Faust :(

yeah i think im taking too many classes

how many are you takin @Faust

5 math classes

@Daminark he fucking just emailed me a bunch of times while talking to me, like too my face

@EricSilva well, that's true. i think i can come up with only one or two examples where neoliberalism definitely have had positive impacts on occasions (eg Deng's China).

@Faust WHOA that's a lot man

compared to Mao's at least

Okay that's actually fantastic

yeah i think ima have to drop one i have been doing really good on all my assinments so i thought it was gunna be ok

you guys really like politics, don't you?

no

@Balarka it's almost like economics is like rlly complicated or smth

that can't be right tho

which?

that economics is complicated

hah

@Faust yeah I mean don't burden yourself

well i can believe it.

I dunno much about economics despite having a father who studied it

to be clear, i was being like ultra facetious

heh

i don't have a lot of intelligent comment to make on economical technicalities. i can merely observe the trivialities

Lol I should start emailing people while talking to them more now that I think about it

you have to force them to email you their responses to what you say

it's a power move

@BalarkaSen if the economy be bad, it don't be good -a great t h o n k e r

@EricSilva I was reading Marx's mathematical manuscripts today lol

like he missed the boat but like other than that it's pretty good

Okay I see, I'll definitely make use of that if I get tenure :P

@Daminark it forces them to realize that what they have to say is inherently less valuable and should be relegated to a later time for reading

Oh now I get what you're saying

Originally I thought it was just like "You can make them do it so do it in order to assert dominance"

Just for general principle, but oof

and so on and so on

even my sniffles are more important than what u have 2 say

sniffles

zizek intensifies

I lol'd IRL

I think my next life project is to try and imitate Benson

walk into class blasting STRFKR

Benson or Soug

me and benson have weirdly similar musical tastes

I liked whatever he was playing. I should get him to listen to Undertale

Neutral Milk Hotel or nothing

because it's the only album I have ever heard

and it's the best fucking album of the world

- hippie

@Daminark, he was playing STRFKR lol

GY!BE released a new album

i wonder how it is

i actually was pretty into neutral milk hotel for a while

am i a hippie

neh

only if you think Aeroplane over the sea is the best album of the world and that's the only thing you have ever listened to

WhAtAbEaUtIfUlFaCeIhAvEfOuNdInThIsPlAcE

the alternating caps is meant to emulate the lyrical stylings of jeff mangum

yeah i got that and kek'd a little

what is a kek?

It's a mutated lel

i mean historically it's the opposite of what @Daminark said

Historically mishmorically

@Eric u should go full hippie and listen to obscure shit with me

like scott walker bish bosch

i always need space to listen to music i can actually play

and captain beefheart trout mask replica

pulse demon is @BalarkaSen's fave album

i have heard of that album

@Balarka or u can go down the rabbithole of my taste and listen to tropicalia

it's fckn g00000d

lol

maybe

in all seriousness -it's good-

@anakhronizein oh my god i'm 2 seconds in and this is already my favorite album of all time

i can feel the aesthetic

I can prove for a fact that that is sarcasm.

no im 100% honest

i love this

but i have to stop listening to this because my ears are bleeding

Can I ask a small group theory question?: the Cantor-Schroeder-Bernstein theorem doesn't hold in the category of groups, and I think I found a counterexample: $G = \bigoplus_{i\geq 1} \mathbb{Z}_{2^i}$ and $H = \bigoplus_{i\geq 2}\mathbb{Z}_{2^i}$, and I feel both are non-isomorphic, maybe because of some divisibility argument, but I'm not sure.

The usual one I see has to do with free groups.

Like $F_2$ and $F_3$

@anakhronizein o'course, you could do almost the same but not worry about divisibility modulo $n$. Hadn't thought about it, thanks!

:)

@anakhronizein So I could do something like $G = \times_{i=1}^\infty F_2$ and $H = F_3\times G$.

Well I was thinking of something a lot smaller looking, no infinite product

I was wondering if I could find an injection from $F_3$ to, say $F_2 \times F_2$.

Ooooh.

@Miguelgondu He's really saying $F_3$ embeds in $F_2$.

You can indeed find one from F_3 to F_2, if you wanted.

what would be the injection from $F_3$ to $F_2$.

lemme think about it.

Indeed, think about it. :)

I always think in terms of covering spaces on that specific problem

But there's a neat algebraic formula

ur a covering space

ur a jet bundle

tbh that's cool

no it sucks

Oh, @BalarkaSen I had a question about jet bundles

actually i dont believe that

nothing pops to mind, really. May I ask for a hint?

dangit i should have said "ur an espace etale"

u missed ur mark my man

something of the form $x\mapsto a$ and $y\mapsto b$ and $z\mapsto$ some product of $a$ and $b$ wouldn't work.

@anakhronizein Indeed?

I can't even find the paper now, hold on

@BalarkaSen people.math.gatech.edu/~etnyre/preprints/papers/phys.pdf

Outlines somewhere about the 1-jet space having a natural contact structure.

I was wondering if it generalizes.

to?

something something tangent to holonomic sections

I know vaguely about this from a workshop talk

wait, wait, wait, $x\mapsto a^2$ and $y\mapsto b^2$ and $z\mapsto ab$?

(because of this question: math.stackexchange.com/questions/531190/…)

@Miguelgondu that's the one I was thinking about. You'd have to show it is an injection, though.

@anakhronizein I'm on it!, thanks!

re jet space, I was wondering about n-jet spaces.

I always wanted to learn more about the jets

I vaguely know the holonomy approximation theorem and how it proves Smale's paradox

holonomy is a word i should know

it means different things in different contexts apparently

sounds right

i dunno if the holonomy approximation theorem has anything to do with the actual holonomy

should that be holonomic approximation theorem?

probably

i dont remember names

googling the latter brings up h-principle stuff

mhm

aka hell-no-i-won't-understand-a-word-principle

I think I wrote up a few things in chat a few weeks ago

let me see if I can find it

Starting here. I have ranted through some messages

kk

i've seen the classic video re: sphere eversion

the holonomic approximation theorem is the reason you need corrugations in that video eg

huh

before everting it

Hey guys - I've been looking around the stuff that I'll be studying for a lecture about analysis on manifolds, just to get a basic idea about the things. Besides my whole confusion around using tensors to define stuff on manifolds, there's one thing I don't really even see the motivation for - there's a whole chapter on integration on riemann and pseudo-riemann manifolds, and a specific mention of there being a notion of volume on those types of manifolds - why is that?

It seems to me like there's a good definition of integration on general manifolds, and I would expect that gives us "volume" too, so I'm sure I'm missing something. Why does integration on general manifolds not necessarily give us a correct notion on volume of a manifold?

Other details of last night dream:

Let $\theta$ be the thomae function and $\mathscr{H}_x$ be hamiltonian densities, and $f$ be (forgot)

Then the first equation is as follows:

$$\int_{(0,1]} \mathscr{H}_x=\frac{f}{\theta^{-1}}$$

Using this the second equation is obtained as follows:

$$\int_{\mathscr{\Gamma}} \mathscr{H}_x = \frac{f(x)}{\theta \int_{(0,1]} \mathscr{H}_x}$$

@John you can integrate a top dimensional form on a smooth manifold, and those things are called volume forms, but they're not unique or canonical, there really isn't a natural choice unless you have a richer structure. On a (pseudo) Riemannian manifold you have a natural pick, so that's why you kind of set it apart.

The problem itself is to calculate some kind of interactions which need to take account of all Hamiltonian densities for all possible pathways the interaction can occur, thus a path integral then become ranged over some uncountable set

In what sense are they not canonical / unique?

Some reality check: The Thomae fuction is not injective, thus it only has an inverse relation, not an inverse function

(where can I pick some basis/something differently, so that I get a different result ?)

a simple example might clarify

like you can just scale the volume

and get something different

but why would you pick one over the other on a general smooth manifold? you basically don't have a reason