Suppose we partition the set of functions $\Bbb N\to\Bbb R$ into equivalence classes, calling $\Gamma$ the partition. There are many ways to do this (via various asymptotic notations, or even just $\Bbb C\sqcup\{\rm div\}$), but I don't know which way would be most appropriate. Any conditionally convergent series $\sum_{k=1}^\infty a_k$ induces a function $\phi:{\rm Perm}(\Bbb N)\to\Gamma$ given by $\sigma\mapsto \left[\sum_{k=1}^n a_{\sigma(k)}\right]$.

Has anybody studied the range and fibers of $\phi$? (This would include figuring out what $\Gamma$ to use.) @robjohn Since you're in chat I'll ping you, you might find this question interesting.

@arctictern In the real projective plane question, you seem to comment on a map $D^2 \to \Bbb{RP}^2$. But it seems to ask whether RP^2 covers D^2 (which is also of course even more impossible).

well, then they're mixing up which space is being used to represent which

I'm converting it to a longer answer anyway

Fair enough.

hi

I was stuck at this question math.stackexchange.com/questions/1874482/conjugate-function. Any ideas?

anyone have ideas about this? math.stackexchange.com/questions/1875673/…

@ForeverMozart Remarks shouldn't be bold. Maybe the "Remark" would be bold but the content, not really

But I don't care much about typesetting anyway.

I also agree with the close votes, fwiw.

@BalarkaSen Can you help me?

I'm having trouble explaining to this guy why both answers are correct.

Depends. What do you want help with?

Sorry, not really interested in engaging with that.

@BalarkaSen no need to apologize

Hi @Huy

hi Balarka

I am trying to understand something but more or less failing.

what would that be

Properties of immersions of manifolds and their transverse self-intersection.

ah, I've seen you talk about that earlier, I don't think I'll be able to help tbh

it's okay, i should try to figure it out on my own

i am sure it'll be obvious after i understand it finally :P

as is always the case

An example of when you tried to build a lattice with generators <a,1>, only to find it collapses into the integers

A resource containing many detailed solved examples, would also help

@BalarkaSen Supoose the fundamental group has a torsion element. What does that imply for the universal cover?

@DanielFischer: can you help with this? I'm looking at this answer. basically, we consider horizontal paths in the fundamental domain for $\mathbb{H}^2 / SL_2(\mathbb{Z})$ and by moving them further north, realise that they become shorter. these are loops in the quotient and go around a puncture (y -> infty). now Pete Clark states in the comments, that this quotient is a Riemann surface, namely $\mathbb{C}$. where's the puncture in $\mathbb{C}$?

I'm guessing it's $\{\infty\}$, "too"? the point we need to add to get stereographic projection of the Riemann sphere?

oh, or is H^2/SL(2,Z) actually the thrice punctured sphere? that would make more sense, because that's a hyperbolic surface whereas the once punctured sphere isn't hyperbolic

no, something's still wrong

@Huy I think he meant to write $\mathbb{C}''$, which is a common notation for $\mathbb{C}\setminus \{0,1\}$. Since $\mathbb{C}$ is simply connected, it has no nontrivial coverings. Also, $\mathbb{C}$ is not hyperbolic.

yes, $\mathbb{C}$ itself doesn't make a lot of sense

The three vertices of the fundamental region (one of which is $\infty$) are mapped to $0,1,\infty$ respectively. Which goes where depends on your mapping.

thanks, that makes sense

Have you already heard of $\lambda$ (the modular function)?

Although, I think that was for $\Gamma(2)$, not the full $SL_2(\mathbb{Z})$.

no, actually not. but wait, which third vertex do you mean? the one very north is a puncture, the one on the lower right is another one, but the one on the lower left belongs to the fundamental domain, and the point $i$ also belongs to the fundamental domain

People

what is the name of this theorem?

$\dbinom{kp}{p}\equiv k\mbox{ (mod }p^3\mbox{)}$

I may have recalled it incorrectly

Yeah, I think I mixed it up with $\Gamma(2)$. For the full $SL_2(\mathbb{Z})$, how were things again?

@DanielFischer: the left vertical line belongs to the fundamental domain, and the left arc of the half-circle including the point in the middle, too

the rest of the boundary doesn't anymore

@LeakyNun Methinks you mean Wolstenholme's theorem.

@DanielFischer Thank you very much.

Did you pull it out from your memory?

googling gives the name very quickly

@Huy Googling what?

binomial coefficient mod p^3

@Huy doesn't work for me, whatever

@LeakyNun: one of the first results is a MSE post with an equation that I think is equivalent

@Huy oh, thanks

I'm never sure when to post a question to MO instead of MSE.

I think I'd be okay with doing it in the case I'm considering---it's a topic which isn't discussed, as far as I can tell, on either site---but ehh.

@DanielFischer: so I have two punctures, so I need at least one hole or a boundary component to get a hyperbolic surface, right? is the one on the bottom right even a puncture?

I think we actually only have the puncture in the very north, but then I don't really know what surface it "is"

some sort of cone with a puncture at the top? ö_ö

@Huy Not sure. I forgot how things were with the full $SL_2$. Let's see, we glue together the vertical sides, and the two halves of the circular arc. Hmm, I think topologically that does look like a disk/the plane.

ok, when I draw it it again looks like a once punctured sphere which would be $C$ but that doesn't make sense

@DanielFischer: one can think of H^2/SL(2,Z) as the moduli space of the torus (defined as Teich(S)/MCG(S)) and several pages claim that the moduli space of the torus is the open disk or the upper half plane. how does that make any sense?

in Farb&Margalit, they claim that the moduli space is topologically a punctured sphere and has the "structure of an orbifold with signature (0,2,3,infty)" where infty signifies the puncture. "that is we can think of M(T^2) as a punctured sphere with cone points of order 2 and 3"

@Huy See Tong's string theory notes for a proof.

Wait, open disk?

Never did any Teichmüller theory, I don't know what Teich(S) or MCG(S) are. But two tori $\mathbb{C}/\Lambda_1$ and $\mathbb{C}/\Lambda_2$ are biholomorphically equivalent if and only if there is an $\alpha \in \mathbb{C}\setminus \{0\}$ with $\Lambda_2 = \alpha \Lambda_1$. Thus you can always assume that one of the basis elements for the lattice is $1$, and the other - call it $\tau$ - lies in the upper half-plane.

Hence the family of complex structures on a torus is parametrised by $\mathbb{H}/SL_2(\mathbb{Z})$.

That's, as far as I recall the moduli space of the torus.

@DanielFischer: I looked up what "cone point order" and "signature" means in Farb&Margalit, and they only introduce the notion for orbifolds, so again it seems H/SL_2(Z) is only an orbifold, so I don't really see why Pete Clark claims it is a Riemann surface, C

I don't know either. You could ask Pete.

ok, I will

Though topologically it is a manifold. But that doesn't rhyme with the upper half plane being hyperbolic.

Something's going on, but I don't know what.

the satisfying feeling when you type up your comment and it says "1 character left"

@BalarkaSen ...torsion elements in the fundamental group?

@0celo7 I don't see anything in those notes apart from the standard fundamental domain for H^2/SL_2(Z)

@Huy wait, which part are you confused about

Do you know what the standard fundamental domain is?

@0celo7: yes, of course

@0celo7: PeteClark claims H^2/SL_2(Z) is the complex plane

Oh, I misunderstood

Hmm

and I don't see how that makes any sense

The upper half plane is certainly not homeomorphic to that

problem being that if it was homeo to C, then H^2 would be a cover of C

Yeah

however when I try to glue stuff together and bend etc., it does look like a punctured sphere, which is diffeo to C

hey guys, I have a theory I want to confirm: if a % n = b then ca % cn = c*b (% is moduli sign) - Am I right?

my proof: a % n = b <=> a - b = n * k (for some k \in N). then ca-cb = c(a-b) = cnk <=> ca % cn = cb

@slallum the first statement is not true

you should mention that 0<=b<n

then 0<=cb<cn and ca-cb=n*ck

@slallum a%n=b is not a===b (mod n)

oh sorry, didn't know there's a difference. then I change my question to a === b (mod n)

(and thank you for the fast reply :) )

both are right

I don't even know why I made a Vector object in my program. Isn't a vector just a matrix with one dimension only having one row/column

both - you mean for both sides? i.e it's if and only if

@slallum both theorems are right

a%n=b <=> ca%cn=cb

a===b (mod n) <=> ca===cb (mod cn)

oh great :)

thank you very much!

welcome

@TedShifrin: hey, can you maybe help with my earlier confusion?

see my comment here

hi. i want to apply mathematics formula for find location on map , is that i have a location and i want to find the location is on which side of a pointer , that is left side or right side.

@BalarkaSen Do you know about this alternative definition of a smooth structure in terms of functions (mentioned in Milnor & Stasheff, used as main definition in Nomizu - Lie groups & Differential Geometry)?

@Danu Is it the same as the one in Bredon?

If yes, I'm fairly sure he has a proof that its equivalent to the standard one.

@Danu Yeah.

As 0celo said, Bredon talks about this.

I don't really find this particular definition transparent enough. He's thinking of a smooth structure as a sheaf, in a way.

@0celo7 I haven't been around the whole day. What do you want to know about the universal cover?

It can be a lot of things

Having just a couple of torsion elements is not sufficient to say anything decisive. Not off the top off my head at least.

@Huy H^2/SL_2(Z) is topologically C.

H^2 does cover C - note that upper half plane here means the open upper half plane.

That it's homeomorphic to C can be just seen by gluing the fundamental domain appropriately, as I presume you were doing correctly.

It'll be a sphere minus point.

well, C and H^2 are not the same Riemann surface

Topologically, I meant. But yeah.