Don't quote me on that though.

There's a MO post by Thurston somewhere where he discusses this problem

It's like Thurston's top post

Acuriousmind often said I am not talking to him, but thinking at him. More generally, (until recently) most people often saw me as talking my stream of consciousness instead of actually talking to people

@BalarkaSen thanks!

If anyone is interested, the IMA at the UMN is hosting a conference today (which I'm sitting on) and there's a livestream here. topic is "Liquid Crystals, Metamaterials, Transformation Optics, Photonic Crystals, and Solar Cells", and the schedule can be found here: ima.umn.edu/2017-2018.3/W2.27-3.2.18#schedule

@r9m No problem! I actually picked up a lot of intuition from that thread

Also you get to know the story when Terry Tao rolls on the floor to understand gauge transformations

What can possibly top that?

@Semiclassical If you rush the stage I will venmo you 50 bucks.

@0celo7 yeah no.

@0celo7 lmao

Minnesotans are too nice

I'd do that for 50 bucks

riiight

Don't they descend from German settlers?

more Scandanavian

though there are a lot of German settlers as well

viking blood boiling yo

Exactly. Just think of the stage as a village you need to plunder

yeah, uh

i'd rather not be an asshole in front of profs from around the country, so

put on a mask

kindly f*** off?

A VIKING HELMET

@0celo7 aren't you of German descent

that explains a lot

@BalarkaSen Not Nice.

lmao

@Semiclassical are you going to be talking?

I was just talking about how you're so good at math

Hilbert gang Hilbert gang Hilbert gang

Godel gang

spent 10 racks on a new impossibility theorem

@skullpatrol nah

:(

@BalarkaSen I see you're a connoisseur of the finer arts

I'm "sitting in" not participating in the conference

I see.

@BalarkaSen Ok, outline in the positive case: one has to show that the map $\mathrm{scal}:\mathscr M\cap H^s\to H^{s-2}$ is locally surjective

to do this one linearizes, then this becomes an injective elliptic operator

by the Fredholm alternative of elliptic operators on VBs, one has to show that the adjoint is injective

this follows from the maximum principle and some other stuff, including some mysterious commutator identities

done!

The funny thing is that while Minnesotans were traditionally Scandanavian/German, my mom didn't grow up here and my dad's family background is eastern Europe

@0celo7 mmm, commutators

oh and somewhere one fucks $s$ off to $\infty$ so that elliptic regularity applies

mind your language

this is a christian channel

that part isn't so clear yet, but should be easy if you sacrifice a goat to gilbarg and trudinger

@slereah are you part of the bro army or what

who watches pewdiepie in 2018

mostly for the end of that

@BalarkaSen I have never watched pewdiepie

@BalarkaSen So in a sense the Kazdan-Warner theorems are far easier than the Yamabe problem.

who even uses "christian channel" outside of pewdiepie fans

the hbar did for a while

but the atheist communist mod known as "ACM" told us to stop

@BalarkaSen it's an old joke

@0celo7 I learnt a little about Fredholm operators from Salamon's gauge theory text

Seemed useful

Also you can substitute "family channel"

@BalarkaSen Hodge decomposition is literally just the statement that $\Delta$ on forms is Fredholm

watch out for drive by flaggers

@0celo7 Ahh

they are usually from the "family channel"

So what I'll be doing is proving a type of Hodge theorem for the scalar curvature.

Well isn't Morse theory about the statement that $d^2 \mathcal{E}$ is Fredholm? :)

It's gonna be interesting.

@BalarkaSen Hmm.

Well, yeah, and you get a nice expression for the index.

Yeah

I wonder if Atiyah-Singer implies Morse theory.

Something to think about.

<-- careful not to fall down a rabbithole

I'm going down the Atiyah-Singer rabbit hole this summer

So who is the Lubos Motl of MSE

I don't think anyone compares with Lubos

mathematics is a shit branch so we get all the shitty crackpots

Simon Brendle if he were here

what about the real guy

They have similar reputations, except that Brendle is actually damn good at what he does

Wildberger

@BalarkaSen you stole that from ted

@0celo7 what if Simon Brendle had a children named Victor

but root 2 is shorter?

But real numbers are a scam

irrationals you mean

I'm not sure what Wildberger considers wrong

he uses $\Bbb Q$

I don't know if he only wants integers or if rationals are okay

a path is a map from $[0, 1] \cap \Bbb Q$

according to his lectures

I'm sure you could make decent math with that, but it would probably be stupidly awkward

Like that book about doing newtonian mechanics with geometric axioms

a circle is $\Bbb Q/\Bbb Z$

@BalarkaSen what is its area tho

Q/Z is horror, we don't even have a way to visualise that

@Secret it's the group quotient, just the torsion points of S^1

the union of all roots of unity

of all degree

the topological quotient is worse

R/Q?

@Secret have you ever talked to him?

R/Q is an indiscrete space

@skullpatrol who?

Wildberger

@skullpatrol Well he once did some consultation back in 1st year. and I end up asking some tut question and then interested in learning about his rational trig stuff. It's kinda convenient, though, because taking cos and sin of not so nice angles can be annoying. He also mentioned about how in case one felt uncomfortable, there is always a conversion two and from his system of qudrants back to lengths by using sin^2 and cos^2 identities

@BalarkaSen idget it

@0celo7 victor brendle

We have not discussed anything about reals (or the lack of it) though, and at that time I don't even knew about infinties like I do now

vector bundle

ah, lol

this meme is an instant classic

this one's even better

@skullpatrol The recent discussion in the logic room with other users, however suggest we can still do pretty fine with finitist maths. We breifly discussed about ultrafinitism there, such as having a N large enough since we don't expect our universe need to compute countable information. If I met him again, I think I will ask him about more details on how to think about finitism

Meanwhile, I hope someone can find a natural way uncountable sets and well ordering arises, so I can get to keep the weird $\omega_1$

I see.

> Secret it's the group quotient, just the torsion points of S^1

Looks messy just imagining how all the roots of unities piled up in the circle

@BalarkaSen something to think about: a compact manifold, $n\ge 4$, such that $0< \frac{1}{4}K(\pi_1)< K(\pi_2)$ for every $p\in M$ and 2-planes $\pi_i\subset T_pM$, and is simply connected, is diffeomorphic to the standard sphere

Diffeomorphic!

for $n=2$ it's trivial and for $n=3$ nuke it with Poincare

Oh is that like the quarter pinched sphere theorem

it is, but Brendle's version

diffeomorphic is what's new and perhaps surprising

Am I right in thinking that $\mathcal{B} = \{[x,x+1) \mid x \in \Bbb{R} \}$ is not locally finite in $\Bbb{R}_\ell$, the reals with the lower limit topology?

Wait did you mean there exists two planes pi1 pi2 at each point p such that 0 < 1/4 K(pi1) < K(pi2)?

@Secret It's a dense subset of the circle

Like the rationals in the reals

hmm... the first 8 iterations of the sequence that leads to $\Bbb{Q}/\Bbb{Z}$...

@BalarkaSen for every pair of 2-planes in each tangent space

Hmm

or normalize the maximum curvature to be 1, then all sectional cuvatures lie in $(1/4,1)$

Ah

Ok that's a better parsing

Is simply connected a hypothesis or consequence

Surely the former?

@BalarkaSen right I think I can sorta... imagine that

hypothesis

OK

Strange

Is there an $n$-dimensional generalization of left and right derivatives

@BalarkaSen um, otherwise RP^2 would be a counterexample

I guess it would be the derivatives in a direction

RP^4, but yeah I getchu

But $\Bbb{R}/\Bbb{Q}$... I wish there is a sequence that converge to it, it will make it easier to understand

@BalarkaSen well yeh I did say n\ge 4

but it's true in all dimensions

kkkk

This is a very interesting theorem

if you just care about the top cat, then there's a morse theory proof

I'd like to learn that one

do carmo

Actually I’m not sure how general his proof is. It’s also in Cheever and Ebin

Autocorrect really don’t like those names

I learnt the Ricci curvature from Milnor today

the Meyer's theorem

is nice

I don't really get it though

I am trying to find a nondiscrete space that has a countably locally finite basis but does not have a countable basis. I could use a hint. I was thinking of $\Bbb{R}^\omega$ with the uniform topology, but I'm having trouble even finding a countable locally finite basis/.

@BalarkaSen it’s averaged sectional curvature

And Ricci curvature comparison means you have local comparisons to space forms

I'm doing the Taylor expansion computation rn

of the metric in geodesic coordinates

yolo

So geodesic coordinates work by taking a orthonormal basis $\{X_1, \cdots, X_n\}$ at $T_p M$ and extending them to a ball around $p$ by parallel transport (so these are coordinates coming from the exponential map). Then $\nabla_{X_i} X_j = 0$ at $p$.

By orthonormality $g(X_i, X_j) = \delta_{ij}$ at $p$

And by parallelity, $X_i g(X_j, X_j) = 0$ at $p$, yes?

So I need to figure out what happens at the second order

$X_i X_j g(X_k, X_l)$

$= X_i g(\nabla_{X_j} X_k, X_l) + X_ig(X_k, \nabla_{X_j} X_l)$

$=g(\nabla_{X_i} \nabla_{X_j} X_k, X_l) + 2g(\nabla_{X_j} X_k, \nabla_{X_i} X_l) + g(X_k, \nabla_{X_i} \nabla_{X_j} X_l)$

oof

@BalarkaSen I do it to 4th order in my thesis

It’s not pretty

Your approach isn’t the best. You need to use Jacobi fields

I can see why Riemann curvature tensor should pop up somewhere from there but I'll look at it later

Need to run now

@0celo7 ah ok

i shall try that

cya for now

@BalarkaSen I believe Zee’s GR book has a computation from first principles. But Jacobi fields give a ststematic way of computing the expansion

@BalarkaSen if you want to see it, check section 3.3 of my thesis. It’s not obvious what to do

Is it possible to find two constants $c_1,c_2 \geq 0$ such that given three vectors $x,y,z$ we have $$c_1 \left( ||x||^2 + ||y||^2 + ||z||^2 \right) \leq ||x+y+z||^2 \leq c_2 \left( ||x||^2 + ||y||^2 + ||z||^2 \right)$$

?

I've found $c_2$ being equal to 5, using the cauchy schwartz inequality

maybe still using the same inequality I can find $c_1$ but I can't see how

maybe the other way around it's better, given three vectors $x,y,z$ can I find $c_1,c_2$ such that...

well $c_1 = 1$ by triangle inequality no?

@user8469759 if the first inequality were true for some $c_1 > 0$, then $x+y+z = 0 \implies x = y = z = 0$ (assuming, $\lVert \rVert$ is a norm), which is not possible.

yeah sorry I meant $c_2 = 1$ by triangle inequality, and $c_1 \leq 0$

Is Cauchy Schwartz inequality understandable without knowing much calculus?

How can i use AM-GM inequality to show that there does not exist any sequence of positive reals $\{a_n\}$ such that both $\sum a_n$ and $\sum \frac1{a_n}$ coverge?

$\frac1{a_n}$?

@Abcd I think so. I was presented its proof in Linear Algebra course. Also, Rudin presents it in Ch 1, before presenting any 'calculus'.

@AlessandroCodenotti oh, sorry, you are right.

If $\sum a_n$ converges $a_n \to 0$, but then....?

I think it's immediate from the observation that $\sum a_n$ converges $\implies a_n\to 0$, without invoking AM-GM or snipers

flashes a sticker with "SNIPED" written on it in red paint attached to the shirt

@BalarkaSen yes, I can see that $a_n<1$ for all n greater than some N and hence $\frac1{a_n}>1$, so does not converge, but I am supposed to use AM-GM somewhere

@BalarkaSen lol

How did the exam go

Any sources (=sites/links) from where I can study C-S inequality?

Very well, the professor asked me stuff about Noetherian normalization, the weak nullstellensatz, primary decomposition and some observations on exact sequences, all topics I was prepared in

Oh nice!

@Abcd you want only statement and proof or much more?

@Silent statement, proof and example problems.

I have actually forgotten primary decomposition

hey guys :)

hey hey hey @lush

@Abcd I know a book called 'cauchy schwarz master class' , but that might be overkill

It's either available for free online, or I pirated it. I don't remember

@BalarkaSen Do you remember what irreducible and primary ideals are?

@AkivaWeinberger what? the book?

What's the difference between irreducible and prime again?

Cauchy Schwartz Master Class, yeah

In either case, pdfs are obtainable

@BalarkaSen Irreducible means that if $I=I_1\cap I_2$ then either $I_1=I$ or $I_2=I$

Hi @Ted

ma.huji.ac.il/~ehudf/courses/Ineq09/…

This seems to be the pdf

Hi Alessandro, Balarka, DogAteMy

@Alessandro Aha right

Prime and primary are different, Balarka. :)

That wasn't my question though

@BalarkaSen So prime ideals are irreducibles

Yep

Well, you did type "prime" rather than "primary." Just butting in.

The other implication is false, even in Noetherian rings, but irreducible ideals are primary in Noetherian ring

No no I was actually asking what was the difference between irreducible and prime

I always forget that

@Abcd: What do you mean by studying C-S? Proof? Lots of exercises to use it?

Moreover every ideal of a Noetherian ring can be written as $I=I_1\cap\cdots\cap I_n$ with $I_i$ irreducible (hence primary)

Oh, OK, @Balarka. I'll shaddup.

@TedShifrin C-S = Cauchy Schwartz.

Prime generalizes the $p|ab \implies p|a \text{ or } p|b$. That's how I remember it.

@Alessandro I mean eg (4) is irreducible I think.

yo

@Abcd: I know, silly. I've written three books with C-S. in them.

Yo, Eric.

$(4)$ in $\Bbb Z$??

anything cool happening

@BalarkaSen It is, in a PID primary ideals are of the form $(p^k)$ for $p$ prime and primary ideals are also irreducible

@Ted Yep

@Alessandro Right

Ah, right, $2^2$ is different from $2\cdot 3$.

Ok, so are primary ideals of the form $\wp^k$ where $\wp$ is prime?

Where that means it contains elements of the form $a_1 \cdots a_k$ where $a_i \in \wp$

@TedShifrin Yeah that's how I intuit it too.

good ol' Euclid's lemma

Ah, btw, the Nullstellensatz+the decomposition of varieties in irreducible varieties gives a way to write a radical ideal as a finite intersection of prime ideals, the whole primary decomposition business is a generalization of this fact

Aha

@AlessandroCodenotti So assuming this fact (the proof is in Reid's and is not that bad) we can speak of minimal primary decompositions, that is we write $I=I_1\cap\cdots I_n$ with $\sqrt{I_i}$ pairwise distinct and no $I_i$ is redundant (if you remove it from the intersection you get something which is strictly bigger than $I$)

Ah alright

Ah, another fact I forgot: if $I$ is primary then $P=\sqrt{I}$ is prime and $I$ is also called $P$-primary, if $I$ and $J$ are $P$-primary so is $I\cap J$ (Reid literally writes "proof: do it yourself" for this last fact lol)

Which is why you can take the $\sqrt{I_i}$ to be pairwise distinct

I guess "do it yourself" sounds better than "left to the reader" :P

All of this is like a theory of doing arithmetic with ideals

I should have learnt this point of view better

@BalarkaSen the Sobolev space of metrics...I’m gonna define it as being the obvious thing

@TedShifrin I suppose

Exercise for the reader to figure out what I mean

@BalarkaSen Reid's ends the chapter with two "uniqueness" results (basically the minimal decomposition of an ideal is not unique, but some pieces of it are completely determined by $I$) and some geometric interpretations (which I'm afraid I don't understand) :/

Gotcha

I should pick it up at some point

Thanks for the intro!