@Ultradark i am struggling with simple things. What about you?

I am struggling with moduli spaces @anakhro

they are strangely beautiful beasts from a bizarre and enigmatic land

@Ultradark what is interesting about moduli spaces?

Can you give me the synopsis? I have never seen them before

@anakhro yeah I will give you a short synopsis and two resources

@Ultradark what's your quest today in this land?

Can I actually differentiate a power series even-though it is not convergent?

@Alucard my quest is such: To A) qualitatively and analytically categorize all cross sections of planes cutting through a 3-dimensional geometric structure with compact support and finitely many geodesics

actually

that will take probably like 5 years

so my goal for today is to just peruse the literature

@quallenjäger you can differentiate any "formal power series" even if you do not know it is convergent. The real problem is what can you then conclude after differentiating a power series when it might not be convergent.

So algebraically, it's a perfectly reasonable thing to do. But analytically, what would you ever claim?

@anakhro How can I make sense out of the derivative of a formal power series?

again as differential quotienT?

converge is usually nice, but remember the harmonic series diverges and can be used to prove the infinity of primes

My issue is, how can we define the derivative of a formal power series?

but that's a whole nother story

The differential quotient as usual in analysis, is not well-defined in this case

As we have $\infty-\infty$

Let $A(x) = \sum_{n\geqq 0} a_nx^n$ be a formal power series.

Then just apply the power rule to each term.

$$A'(x) = \sum_{n\geqq 0} na_n x^{n-1}$$

Ok then the derivative is purely an algebraic expression right? It does not really have a analytical meaning

As you have said before.

Only if it converges, we can show that this expression coincide with the differential quotient as we know in the analysis?

Hey @Semiclassical

Does it make sense?@anakhro

Hey @EricSilva

@quallenjäger this derivative of a formal power series is purely an algebraic expression with no analytical meaning. Yes.

But I can relate them if it is convergent right?

Well if A(x) is a convergent power series with radius of convergence $R$, then I think yes, $A'(x)$ as defined above coincides with the usual derivative of $A(x)$, and is a convergent power series with radius of convergence $R$, too.

it's the notion of a formal derivative vs. that of an analytical derivative

gowers.wordpress.com/2014/02/22/differentiating-power-series

Take a look at this for more.

c.f. en.wikipedia.org/wiki/…

hey semiclassical

hi

Great! Thank you @anakhro @Semiclassical @Ultradark

:)

And another question out of my interest, why it is interesting to define a derivative if the series doesn't converge?

Why is it useful?

the main reason I've seen is because of generating functions

@quallenjäger For the same reason we want to differentiate polynomials over general fields

Because we get some nice properties

(though not necessarily the same ones of course).

Ah I see, that's interesting

Thanks

at the level of generating functions, you start with a sequence {a0,a1,a2,...} and the derivative gives you the sequence {a1,2a2,3a3,4a4,...}

which can be pretty handy depending on what you're counting

Yeah generating functions is where I first saw it.

of course, it is very handy when a generating function actually does have some radius of convergence

since then you can extract coefficients just by contour integration

@Semiclassical I see, that makes sense.

who knows a bit about projections

specifically, using a light source to project a geometric realization in 3-space onto the euclidean plane

what kind of projection is that called

@Ultradark can you clarify what you mean by "light source"

Do you mean physically using a projector (such as an overhead projector)?

yeah

I have a geometric structure

anyway it's not super important, i can probably figure it out myself

something like that

It's only stereographic if the object is a sphere

what if it's an arbitrary shape, then is the projection arbitrary

I don't think it has any special name other than "projection"

thank you secret

it is a very practical problem for stuff like 3D rendering of course

@TobiasKildetoft what i said before did not make sense lol

if you can't incorporate light sources, what you get will be pretty limited

I was trying though

so I imagine that all of this is pretty well-known in the realm of computer graphics.

but, by that same token it's not going to be familiar to most math people

semiclassical that's probably true

see for instance this: web.cse.ohio-state.edu/~wang.3602/courses/cse5542-2013-spring/…

semiclassical i designed a 3d structure and received a 3d model so that's why i was asking about the projections and light sources

it's an architectural design

for a future world cup stadium

$\hat{\imath},\hat{\jmath},\hat{k}$

hmm

and at halftime everybody in the stadium has to do math

@MatheinBoulomenos hey

Heya chat

@Fargle hi

Hi @Fargle

What's everyone up to? Kinda dead in here

F

Hi @Fargle @Alessandro

The semester just began for me so that's keeping me pretty busy at the moment

Hi @Ted

Heya @Ted

For once, Alessandro doesn't have time to run me over ...

Hey Ted!

Yeah, I guess that makes sense, school is a thing

@Daminark u nerd

D:

Hi Demonark, Eric

olá

I'm in the unusually privileged position of only having one class, and almost no outside responsibilities, so I get to "do math" (read: ping-pong between the chapter 1s of like 7 books and then forget because video games) to my heart's content for a few months.

I think @Fargle is disoriented and lackadaisical with all this "freedom."

Solid read, doc. Truly solid.

That reads sarcastically but is not meant as such

@TedShifrin Yeah I might have to drop a class maybe

It didn't read aloud to me at all.

I'm taking five at the moment, four of which have weekly exercises sheets

Hey hey hey

Heya @Balarka

yo @Balarka

I'm not used to the weekly sheets, as long as we know our stuff for the exam at the end of the term we're good in Italy

welcome to the pset life bro

it's hell and we're all dead inside hurray

pset life is ok

I far prefer weekly exercises/homeworks (even ones that get graded). You learn more that way than by cramming for an exam at the end, @Alessandro.

I didn't really cram at the end, but I just like being able to not worry about a course for a week if I need to

Ah, but now you're a grad student ... time to raise your game :)

The exercises here are graded, you need to get at least half of the points in them to be admitted to the exam. They don't actually weight on the final grade though

I'm fine with that. I just believe in feedback so you learn.

I'll also need a few weeks to get used to the level of everything here, is definitely higher than in Trento (which is great overall)

I like psets because I learn more if someone sets fire beneath my ass

Even if it's an easy assignment I'll just want to get done with it and read other stuff. Studying/learning gets very accelerated

@TedShifrin you talked about motivating the vector axioms by thinking scalar multiplication as stretching the plane... but then how does that give you those many axioms? I mean, you can explain each axiom given each axiom, but how do you explain why the list of axioms is what it is?

Admittedly it's not weekly on my end. Maximum number of assignments on three of my full-credit math courses is 8.

Because they all work in $\Bbb R^n$. I mean, you look at the typical field axioms (for $\Bbb R$) and you ask how they adapt.

And a semester is 6 months

That's a hell of a semester, @Balarka.

that's a fucking crazy long term

Everyone at UGA complained that the 15-week semester was weeks longer than any other (civilized) school.

Wait, not 6

growls

turns out it's 8

lol ours are like 8 weeks of proper work

Good luck fitting in two, @Fargle.

@TedShifrin I see

I actually prefer semesters to quarters, Eric.

week 10 is like a study week and during week 1 we just get syllabi

i think quarters are like slightly two short

4.5

@TedShifrin Well, that is one week longer than ours

too

Well, that is ludicrous. I start material the very first day of class.

@TedShifrin Fine, we'll have sesquimesters. 18 months.

We have 15 weeks here too. We had 14 in Italy

i would like it better at a solid like 12 or something

3 months sounds good

Yeah, people at UGA like to whine, @Alessandro.

I think we end up with about 11 to 12 weeks here.

MIT's is something like 13.5.

I forget.

I haven't had too many classes where syllabi take more than 10 minutes or so, though I will say the homework in week 1 tends to be kinda whatever unless it's assigned/due later in the week

I want a lifetime's worth of semester

Same. My first homeworks have generally been like "here show that set theory still works the same as it did when you took intro proofs"

I feel you on that @Balarka

nah dude i want a fuckin break

@Fargle lol ripu

too much school makes me a dull boi

too much break makes me suicidal

notes a conflict

@AlessandroCodenotti teach us CFT :P

Rep theory pset had to have a bunch of modifications. First because he asked for an example of a rep and forgot to exclude the trivial rep, second because it included some problems that referenced characters which we ended up not reaching in time

@LeakyNun I'm not actually taking the CFT course after all

oh ok

Ugh, @Alessandro. Did you tell me everyone else has had a traditional exposure to singular/simplicial/cellular homology?

Wait I missed it

waits for the room to self-remove and evaporate

Hi @Ted

@Daminark lol ripu

(removed)

hi @Karl