Okay, fine. But then 1/2 and 2 are both elements.

1/2 does not have the special label of being "the inverse of 2". It's a group element in it's own right.

oh okay

So on the face of it, your set would have to be: 1,2,3,4,..., 1/2, 1/3,1/4,...

In that case, your set has an identity 1, and for each element $x$ there's another element $y$ such that $xy=1$

However, there's one other property a group must satisfy: It has to be closed under the group operation

So for instance, 1/2 * 1/2 should be another group element

Which, it is: 1/2 * 1/2 = 1/4. So that case is fine.

What about 2 * 1/3 ?

2/3 is not in the list of elements

Right. So this set is not closed under the group operation, and {1,2,3,4,...,1/2,1/3,1/4,...} is not a group under this operation

What is a group, though, is if you take your set to be all positive rational numbers.

In that case, you've still got 1 as the identity element; you have p/q * q/p = 1 for any rational number p/q (and so you have inverses). Finally, any product of positive rational numbers is itself a positive rational number, so this set is closed under the group operation.

You also have a(bc)=(ab)c for any positive rational numbers a,b,c. So the group operation is associative, and that's another thing you need for it to be a group. (In this case it's trivial, but in others this property is not obvious.)

These properties together make the set of positive rational numbers a group under multiplication.

Thanks I understand this

Something amusing: I found the following footnote in a book just now

"A common trend is to call $\text{Diff}^+(S^1)/\text{PSL}(2; \mathbb{R})$ the physicists universal Teichmüller space and $\text{QS}(S^1)/\text{PSL}(2; \mathbb{R})$ the Bers universal Teichmüller space."

Now, that's a fairly obscure statement and isn't funny in and of itself

What is funny is that, if you search the phrase "physicists universal Teichmüller space"

It shows up exactly twice in Google, and both are from instances of that same book

lol

i mean, maybe it was common in a certain context that wasn't documented online

@Semiclassical did I hear Teichmüller

but absent knowledge of that, you've got a "common trend" with no instances other than itself :P

@LeakyNun ya

is it inter-universal though

oh hell no

It's not immediately clear to me what Diff+(S^1)/PSL2(R) parametrizes. PSL2(R) acts on S^1 by acting on H^2 by isometries and then extending to the boundary del H^2 = S^1. Equivalence classes of circle diffeomorphisms upto those which have a conformal extension to the Poincare disk?

@BalarkaSen hey!

h

the exam was easy

nice

I think what they've got in mind with PSL(2,R) is the Mobius group on S^1, for concreteness.

That's the same as the embedding I wrote

@BalarkaSen I tried to get them to add "nonempty" to a question but they didn't :c

I figured, just wanted to check

because for some godforsaken reason the course requires topological spaces to be nonempty

I be like "can you be consistent with that in the exam then"

silly

I don't really have a good answer for what it's supposed to be, tbh

semiclassical I think that non-group is a groupoid because a groupoid doesn't require closure

I feel you want to glue two copies of D^2 (with the standard holomorphic structure) along the boundary by a diffeomorphism of S^1 - the resulting thing is not going to be a Riemann surface, it's some weird S^2, two different complex structures on the upper and lower hemisphere, whatever that means - and then the diffeomorphism is in PSL_2(R) would imply that it's "standard" because that diffeomorphism extends to a conformal diffeomorphism of D^2

This can be a pile of bollocks

Oh maybe not. Let $S^2_s$ be the standard Riemann sphere, and $U_s$ and $L_s$ be the upper and lower hemispheres with the induced complex structures. Let $S^2$ be the 2-sphere with an arbitrary complex structure with a marked equator $S^1 \subset S^2$, and $U$ and $L$ it's hemispheres. $f : U_s \to U$ and $g : L_s \to L$ be two diffeomorphisms; $gf^{-1}$ is a diffeomorphism of $S^1$.

If this in $PSL_2(\Bbb R)$ it extends to a biholomorphism $gf^{-1} : L_s \to U_s$. If this was $z \mapsto 1/z$ that gives the standard Riemann sphere; I think any fractional linear transformation gives that, just reparametrize appropriately or something.

Bottom line $\text{Diff}^+(S^1)/\text{PSL}_2(\Bbb R)$ should be the parameter space of complex structures on $S^2$ with a marked equator $S^1 \subset S^2$.

@Ultradark I dunno about groupoids. So I’ve got nothing to say on that account

@BalarkaSen iiinteresting

@Semiclassical Here's a better way to parse the parameter space I was trying to say, as pointed out by Mike elsewhere: It's the configuration space of all embeddings $S^1 \to S^2$, upto identifying two if they are related by a Mobius transformation of $S^2$

i.e, $\text{Emb}(S^1, S^2)/\text{PSL}_2(\Bbb R)$

Hmm

Do ring considered to have an multiplicative identity?

can i get some help?

@taritgoswami Rings don't have to have multiplicative identities. The only example that comes to mind is the $C^*$-algebra $C_0(X)$, which is all the continuous complex-valued function on $X$ vanishing at infinity, where $X$ is a locally compact Hausdorff space. The ring $C_0(X)$ has a multiplicative identity if and only if $X$ is compact.

@user193319 Probably that theory was developed before 1960.

Kind of a complicated example (comes from topology/analysis), but it's the only one i can think of at the moment

It may have been developed before 1960...I don't really pay attention to the history of mathematics.

Wikipedia saying, before the 1960s, rings were considered not to have mult. identity, and now we consider rings to have mult. identity

Really, I consider this to be an example from operator algebras, which is the coolest branch of mathematics.

Hands down the best.

@user193319 were you replying to me?

@Mathphile No, I was responding to tarit goswami. But I'll also urge you to look into operator algebras...hands down most amazing branch of mathematics.

okay

In order to do operator algebras, you basically have to know everything in mathematics.

well im far from that level

and there will always be something to learn

That's what makes it so cool: category theory shows up in it, algebraic topology, obviously functional analysis, measure theory, etc.

@user193319 can you help me in my question?

What's cool is that Oxford University just recently hired an Operator Algebraist, so the field is gaining respectability among mathematicians.

@Mathphile What's the motivation for thinking about this? I don't have any feelings for this. My philosophy is, if it isn't obviously connected to physics in someway, then I don't really care about it (I'm very pragmatic in my learning).

Luckily for me, operator algebras is where cutting edge fundamental, theoretical physics is happening.

Guys, how cna I bound $|\frac{(-1)^{n-1}x^2}{(1+x^2)^n}| < M_{n}$

such that $\sum M_{n}$ converges

in order to prove the Uniform convergence

Weierstrass M-test?

That's what you're trying to use?

@BAYMAX Does $x$ have any particular domain?

ohya

its whole real line