Mathematics - 2020-05-11 (page 3 of 3)

Ted Shifrin

22:00

No, you want this universally for all $g,g'$ that are close.

Sha Vuklia

Right so, for each $g$, we find a neighbourhood $U_g\ni g$, such that when $g'\in U_g$, we have $\vert h(g)-h(g')\vert<\epsilon$. We can find a finite cover of the diagonal of these $U_g$'s. And now somehow I have to formulare that there exists a "tube".

Balarka Sen

Group action? You mean a functor from the groupoid of one object to the category of sets?

Sha Vuklia

I guess I would have to formulate/prove the tube lemma for this more general case?

Érico Melo Silva

@BalarkaSen i open chat to see this shit

smh

Ted Shifrin

Serves you right for repeatedly pinging me, @Erico.

Érico Melo Silva

22:09

about GH!

Ted Shifrin

Yes, but I've heard vacant promises before.

Érico Melo Silva

that’s true but i also read more now than i used to

Alessandro Codenotti

@BalarkaSen You laugh but there's people out there who think like that. We showed orbit stabilizer as a consequence of some nonsense about presheaves being a colimit of representable presheaves in the HoTT course

Ted Shifrin

oh geez

Balarka Sen

@Alessandro I am not laughing. I am serious

I don't see an issue with this

Ted Shifrin

22:11

hides in the crypt

Alessandro Codenotti

I tried to find out if there's a notion of projective limit of metric spaces or of coarse spaces earlier today

Balarka Sen

I will be very surprised if there is a coarse inverse limit

Alessandro Codenotti

Looks unlikely even in the easy case of a countable chain

Balarka Sen

@Alessandro Consider $S^1 \to S^1$, $z \mapsto z^p$. Look at the inverse system of circles given by these

Alessandro Codenotti

It's already messed up even for metric spaces so

@BalarkaSen Uhm what does it look like? I have no intuition here

Sha Vuklia

22:16

@Ted could you give one more hint ;( ~

Balarka Sen

It has a natural map from $\Bbb R$ to it, given by the universal cover map to each factor

Ted Shifrin

@Balarka: If you want a good laugh, look at this. Or am I being obtuse?

Balarka Sen

It also has a natural copy of $\mathbf{Z}_p$, the $p$-adics, sitting in it

Alessandro Codenotti

Messy

Balarka Sen

(think of the $p$-th roots of unity inside $S^1$)

The space happens to be $(\Bbb R \times \mathbf{Z}_p)/\Bbb Z$ where $\Bbb Z$ acts diagonally

Alessandro Codenotti

22:17

But I mean, it's not even clear what metric to put on the inverse limit

Balarka Sen

Countable inverse limit of metric spaces naturally has a metric, it sits inside the product space

Ted Shifrin

You know that $\tilde h$ is $0$ on the diagonal of $G\times G$, @Sha. Can you show that there's such a neighborhood $\tilde U$ so that $(g,g')\in\tilde U\iff gg'^{-1}\in U$?

Alessandro Codenotti

Taking products is already messy in the coarse category

Balarka Sen

@TedShifrin Hahah it took me a while to see $N$ is the normal bundle and the manifold both at the same time

I was staring at the first line for a solid minute

Alessandro Codenotti

@BalarkaSen Right, but when working with coarse spaces you really shouldn't change the metric of the factors, which you need to do to metrize the product usually

Balarka Sen

22:21

I don't know coarse I am just giving an example

This space is a natural geometric space called the $p$-adic solenoid on which $\mathbf{Z}_p$ acts

Alessandro Codenotti

I know, but I want to work in a category of metric spaces, not of metrizable spaces

Balarka Sen

It should be the replacement for "Cayley graph of $\mathbf{Z}_p$"

What is a natural geometric space on which $\text{Gal}(\overline{\Bbb Q}/\Bbb Q)$-acts?

Ted Shifrin

@Balarka: Seems the person is reading nLab and totally getting things wrong, too.

Look at the crazy short exact sequence.

Balarka Sen

Hahahaha

Alessandro Codenotti

@BalarkaSen You can still do a "cayley" graph thingy by picking a generating set, can't you?

Balarka Sen

22:22

Swear to god I did not notice

Sha Vuklia

@TedShifrin I don't see it rn, but it might do me good to go to bed. I will think about it again tomorrow.

Balarka Sen

@Alessandro I think that will be completely horrible and irrelevant

Alessandro Codenotti

Very noncanonical I guess

Ted Shifrin

OK, sleep well, @Sha.

Sha Vuklia

Thanks, bya

Alessandro Codenotti

22:23

Also infinite degree is bad for graphs for some reason

Hmmm local compactness gets messed up I guess

Ted Shifrin

@BalarkaSen Is this to me or to Demonic?

Balarka Sen

To you

The short exact sequence is gold

Ted Shifrin

I totally gave up at that point and posted my bitchy comment.

Alessandro Codenotti

You know your notation is bad when even differential geometers complain about it (sorry Ted)

Balarka Sen

@Alessandro Essentially the correct notion of Cayley graphs of pro-groups should be taking Cayley graphs of each guy in the inverse system, and then taking an inverse limit - but doing this in the category of graphs cannot possibly be right, you should do it in the flexible category CHaus

Only then you get interestingly weird spaces on which these pro-groups act

What is the coarse geometry of these spaces? I have no idea

I thought about this once.

Many moons ago

Alessandro Codenotti

22:26

Inverse system of graphs sounds weird

Balarka Sen

That's essentially the $S^1 \to S^1$, $z \mapsto z^p$ example. $S^1$ is, topologically, the Cayley graph of $\Bbb Z_p$ with standard generator :P

Alessandro Codenotti

@BalarkaSen All limits exist in graphs though so why not?

Balarka Sen

This is bad

Ted Shifrin

@Alessandro: In my own defense, if you look at my papers and books, I think you'll find that I think about exposition :P

Balarka Sen

2

Q: Is the inverse limit of simplicial maps between finite directed graphs also a graph?

I think I have an intuitive understanding of why the following statement might be true, but I am not sure how to go about proving it. It's also possible my intuitive understanding is wrong and the statement is not true. Let $G$ be a finite connected directed graph. Let $G_1, G_2, \ldots$ be a se...

It is not a very interesting space

Alessandro Codenotti

22:29

@TedShifrin I looked at your books before, but I don't know if I want to look at your papers, diffgeo is too scary for me!

@BalarkaSen Hm

Ted Shifrin

Most of my papers are more algebro-geometric in flavor, actually.

Alessandro Codenotti

I'm afraid that's even worse if you ask me

Ted Shifrin

My thesis has a few technical pages that would make you cringe, but otherwise it too was mostly readable.

Yeah, but I don't do abstract algebraic geometry crap.

Balarka Sen

Paul Garrett has some notes where he talks about p-adic solenoids you can look into those if you want

Ted Shifrin

But I do use exact sequences of bundles and sheaves and adjunction formulas.

Alessandro Codenotti

22:30

Actually algebraic geometry and differential geometry are the lowest grades I ever got counting both my bachelor and masters

Balarka Sen

I think pro-spaces are interesting and some people already know the story (Sullivan, if I had to guess).

I dunno the coarse side of things

Alessandro Codenotti

@BalarkaSen I already have way too many things to read

I found a very nice book about coarse geometry of Polish groups the other day which is very tempting

Balarka Sen

Same

We combine the Galois phenomena of the previous Chapter with the phenomenon of geometric periodicity that occurs in the theory of manifolds. We find that the Abelianized Galois group actscompatibly on the completions of Grassmannians of k-dimensional piecewise linear subspaces of R∞

From Sullivan, "Localization, Periodicty, Galois Symmetry"

That seems like a great geometric space on which Gal(Q) acts to me

Thorgott

22:58

@Balarka There was a slight error in my uniqueness assertion earlier, but I realized that, once we have existence, uniqueness follows from regular Picard-Lindelöf after fixing the first variable. By translating the differential equations back and forth through the chart, the functions $\Phi^i\colon U_i\times(-\varepsilon,\varepsilon)\rightarrow M$ are each uniquely characterized by $\Phi^i(x,0)=x$ and $d\Phi(x,-)\vert_t=X(\Phi(x,t))$ for all $x,t$. These agree on their respective overlaps also by fixing $x$ and then regular Picard-Lindelöf, so they glue together to $\Phi$.

Stupid Kid

23:09

Well another wierd question. I mean stupid question.

It is telling me use Euler pentagonal number theorem and expand it to 3 factor and ask me verify if it is pentagonal number or not lol.

And it doesn't even tell me what actually is Euler pentagonal number theorem

Stupid Kid

23:26

And for God's sake after some testing I found the formula for pentagonal number itself is wrong lol $\frac {n(3n-1)}{2}$.

for n =0,-1,-2,...

it doesn't give you 1,5,12,.... instead it gives 0,2,7,... lmao

I think it should be positive natural number instead

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