Each curve corresponds to a term in the von Mangoldt function.

Does anyone know the name of the game described here? (Please, tag me so that I get a notification)

Can someone help verify if my proof is ok?

gyazo.com/be958458ce7aba9ce5983899b0e9abfa

Here's the problem

and my proof below it for part 1 and 2

Mostly i want to see if part 2 is ok.

I think I need to also show -S is bounded below... But apart from that IDK if there's more

When book says Moment about a point. When is the point is it fixed, CM or something else.

Because it don't say anything about that point. Pls clear....

Really simple question

$f^{-1}(0)$ for $f(x,y) = x^2$ should be $\{ (0,b) \vert b \in k \}$

Right?

yes

dunno what $k$ is but as long as there are no nilpotents

$k$ is just a field yeah

$Z_f(\overline{k})$ is just the set of zeroes of $f$ in the field $\overline{k}$ as you would imagine

I just thought of the $x^2$ example and it seems like a trivial enough counterexample because of what I said

why would it be a counter-example?

I imagine he wants to say that it involves a power of $y$ greater than zero?

bc the preimage of 0 is not finite

or I guess I must be missing something really dumb

ah, I misread, you're right

oh ok cool

was suddenly wondering how much of a fraud I was

do you agree he must've meant "have some positive power of y tossed in there" or something to that effect?

that would suffice, though I'm not sure if it's necessary

Hmm what would be the minimum necessary requirements for you to get the preimage of that first coordinate projection function to be a finite set?

you want $p(a,y)$ to be a non-zero polynomial in $y$ for all $a$

which should be the same as asking that the polynomials in $x$ that are the coefficients of $p$ when thought of as polynomial in $y$ aren't coprime

Coprime with what? $p(a,y)$?

no, just coprime

I don’t understand. What ought not be coprime with respect to what?

@Thorgott You mean the polynomial has non-trivial content?

@Astyx fuck working for the UK government

@AlessandroCodenotti Bell Witch + WITTR

@user2103480 Probably but I am only thereon aufmerksam becomen through Penny an der Reeperbahn

Wonderful documentary

en.wikipedia.org/wiki/Giacinto_Morera why the hell is there one whole paragraph praising moreras personality lmao

he's been dead for 100 years nobody that remembers him is still alive

no, I have a collection of polynomials and I ask they are not coprime

i.e. that they generate a proper ideal

you dont have them

they belong to the people

@user2103480 this is hilarious

sorry, WE have a collection of polynomials

happy comrade?

yes comrade

@BalarkaSen I know right. Somebody has a crush on morera and I might look into the wiki editing history to find out who

oh ok now I understand

Somebody's really into Giacinto

who wouldnt have a crush on morera

look at that stache

thank you- I guess now that they are the people's polynomials it makes a lot of sense

@Thorgott magnificent

@EdwardEvans Apparently they knew about RSA encryption years before anyone else

yeah but the English would claim they knew about fire before anyone else if they thought it'd make their dicks bigger

Lolol

get yourself a friend like somigliani

talking his buddy in the afterlife up

a real bro

@Balarka the degree of a canonical divisor on a Riemann surface is the negative of its Euler characteristic. i can show this by some calculation. is there a TOP topological reason to expect this? can this be related to Poincaré-Hopf?

this one?

probably

great i used his name now he finds this when he googles it

lmao

lol

I think it's related to the homology groups rank alternating sum right?

@Thorgott

Yeh so that guy worked on the article from 2010 straight up to 2016 with a few edits per year

is each transcript page of this chat cross referenced to google?

yeh prolly

@BigSocks how do they relate to the degree of a canonical divisor?

@user2103480 lmao

great now I mentioned the 'babou here, gotta delete that message as well

time to enter voluntary self-censorship

when ababou googles your name you've done a lot more wrong than just a lack of self-censorship

oh hmm well idk a lot about that yet

@Thorgott yes and yes

but its my religion and I gotta stick to it now

england is my city

can you be less concise

No

can you be more concise

thanks, i hate u

wow

Wtf

Top 10 anime crossovers

who is that on the right again

David Lynch

ahh yeah

...wtf

?????

😂

@Thorgott The canonical bundle of a Riemann surface is simply its cotangent bundle. The degree is the first Chern class, which is the Euler class for line bundles.

thats not some deepfake shite is it

No clue I randomly found it

I'm not gonna check out peoples earlobes there

@BalarkaSen w h e r e does one randomly find that

Google images

...

ok I dont even wanna know what you googled

"misha gromov david lynch fanfic"

Gromov j chillin

Lmao

@BalarkaSen wtf

I googled it's real

howdy, demonic @Alessandro and a @Balarka and @BigSocks

Any context? Where did they meet?

Hi Ted

fondationcartier.com/en/editions/…

Hi Ted

heyo, Ted- already in chapter 2 of Lorenzini's book

pretty neat, learning stuff well enough

Cool.

Also look up AMS article by Michael Harris, "Sudden Disorientation in a Paris Museum"

Im going to read it now I think

@Astyx @Semiclassical degruyter.com/view/journals/fca/20/1/article-p7.xml

"un dépaysement soudain"

Right, all those are equivalent because of uniqueness of functional calculus (or whatever the terminology is in english)

Mornin' chat

Just thought it might be interesting after yesterday's discussion

Heya @Fargle

How goes it Ted?

It is, I didn't know about most of those

Still bumbling along, and you?

Salut, @Astyx

Salut

Pretty good. Just got a drawing tablet so I can write/draw math without needing paper, so that's been fun.

Yup, and you can save anything worthwhile to a file.

Now, instead of drawing incomprehensible and wrong topology pictures in MSPaint with my mouse, I can draw incomprehensible and wrong topology pictures in Sketchbook with a pen.

MSPaint is pretty bad.

It doesn't exist any more does it?

It does, there was a period where they were just gonna nuke it completely but that got reverted.

Phew

I don't know that they're still updating it, so it's probably some form of abandonware, but it's still available.

@TedShifrin I take it that I should actually get to finally learning characteristic classes

For this you don't need all that fancy stuff.

What's the log of an operator? I'm having trouble interpreting an exercise

I guess it's defined through the series expansion like exp is?

I guess (?) you could define it with $\log(1+x) = x-x^2/2 +...$

@Thor The canonical divisor is the divisor of a global meromorphic $1$-form. This tells you that the line bundle is the dual of the tangent bundle.

If I see this right, in the right hand side, $\Phi$ has values in $L(U,H)$ so the concatenation with its adjoint is an operator with values in that space as well, and so the evaluation of the bochner integral is an element of $L(U,H)$

You can use the pairing with a meromorphic vector field to see that the degrees are negatives.

@Astyx @AlessandroCodenotti that sounds reasonable

that would also explain that 1 which should be an identity I guess?

Maybe it's functional calculus again? Are all the eigenvalues of the operator positive?

and then the $\varepsilon$ makes sense too, probably to make this statement be well-defined

@Astyx I'm not sure. For the stochastic integral to be well-defined, values in $L(U,H)$ are too general anyways. Needs to be Hilbert-Schmidt-valued (it's a slight bit more complicated but that's the gist)

Ugh so I also need to prove that this logarithm is a trace class operator

@Fargle I found out that I can use the Notes app on my iPhone to scribble things. It does save a lot of paper.

Since I haven't invested in an iPencil, I just type notes, @robjohn.

I also like it because sometimes I find myself rubber-ducking to try to solve a problem, so now I can quickly sketch a diagram and send it to said rubber duck.

You have a rubber ducky?

@TedShifrin I use a stylus with a round circle at the end. It does decrease the precision with which you can place a point, but continuous drawing can be pretty precise.

Only in a manner of speaking---I don't literally have one, sadly.

Ah, I have not one of those, either, @robjohn.

@TedShifrin I got one from Meko not too long ago. I don't think it was very expensive.

If I were teaching in the current day, I would definitely have an iPencil. But when I last taught math live on-line, I used a great webcam and got decent at it.

@Thor: Did you catch my final remarks?

Here it is

I use a Wacom

It's cheap and does the job

It's really good with Windows Journal once you fiddle with the settings

@BalarkaSen I can pull out my iPhone much easier than a Wacom tablet. I used to have one of those.

Yeah I imagine iPhone is way better

main thing is that you don't use onenote since it makes ridiculous automatic titles

My iPhone is too small (I opt for the smallest possible), but my mini iPad would be stylus-able.

@Ted: I learnt last week that for percolation with edge probabilities $p$ on a general graph $G$, in general there are two phase transitions: If $p = 0$ everything is closed, then small clusters of bounded size form as $p$ grows, and at some point $p = p_c$ abruptly large unbounded clusters start forming; even further down the line at $p = p_u$ these unbounded clusters merge to a unique unbounded cluster, and at $p = 1$ this unique cluster becomes all of the graph aka everything is open.

Oh, interesting chaotic behavior.

Does the topological complexity of the graph determine anything?

@user2103480 ooh what app is this? looks nicely organized

@BalarkaSen one of the physics profs i worked with did stuff on percolation theory in the 70s-80s. mixture of physics arguments and simulations iirc

That's Microsoft OneNote

It is in principle, and it seems to have a lot of good tools, but these automatic titles just bug me

e.g. link.springer.com/chapter/10.1007/978-3-662-02403-4_5

oh oops thought it was on an ipad or something. I use goodnotes mostly

@TedShifrin I think so. For example, for graphs like Euclidean lattices which has lots and lots of cycles, $p_c = p_u$; whenever there's an unbounded cluster you get a unique one.

automatic tiltes?

look in the middle

Interesting that you don't get more "critical" probabilities, a @Balarka.

oh you did not choose this rubbish looking text

"6. The cohomology ring" -> "6.Traf (OHMLOGO AN"

typically the graphs you'd use in physics would be lattices

@BigSocks no onenote does those automatically

idk, I thought germanic languages were possible very... consonant rich

square lattice, triangular lattice, hexagonal, etc

lmao what

You thought this was german

I'm crying

lol

Your handwriting is pretty execrable.

AGAIN

hahahah

hahaha well not exactly german, but something similar

solid state physicists in general like to ask questions about disorder

Yeah. It actually has to do with the isoperimetric constant of the graph; you call a graph amenable if it has isoperimetric constant 0. All transitive amenable graphs have $p_c = p_u$

@TedShifrin that's my prof's

Even worse!

but sike, my handwriting is actually worse

E: er, a *2 merz-er

is a german proverb

Oy.

@Semiclassical That's cool!

Physics has had so much impact on percolation theory

@Astyx I said that to my mum today and it was a beautiful moment

My students were always impressed with my handwriting on the board, especially given the speed.

yeah. can't say i ever became much of a master on it

@BalarkaSen on characterizing it, at least

@Astyx i didn't put it past them to have regular expressions in german, you know

as far as proofs go, no, but they'd consider that as largely irrelevant

of course, the proverb regaling how you could have some arbitrary amount of 2's in the middle of your sentence, yep

Germans don't say "I love you", they say "E: er, a *2 merz-er" and I think that's beautiful

@Semiclassical yeah they're good at guessing what should be correct

lmoa

yeah

eg conformal invariance of bernoulli percolation at criticality

@BalarkaSen physics had a giant impact on all of probability. It's probably because it's the only real-world subject where it's possible to somewhat-precisely define complicated distributions

conformal invariance stuff is crazy

i've had some contact with that in work i've done

physicists worked with SPDE and SDE before mathematicians made strict sense of those

but conformal field theory stuff? over my head

@user2103480 makes sense

@BigSocks hahahaha wow I didn't even think of that interpretation

like, i know the phrase Virasoro algebra

but don't ask me what i means

Also, Einstein defined BM rigorously. Bachelier did the same 5 years earlier but still, shows again the impact of physicists

BM?

Brownian Motion

brownian motions

ah yes

big money

nvm

@BigSocks salvia

I see you are an erudite gentleman

i wonder a bit about "rigorously"

but he certainly did enough to give it a foundation

@Semiclassical He did not show that it exists but laid down the defining properties I think

right

Wiener's work was inspired by Einstein's iirc

he wanted a proper foundation

"rigorous" always depends on what standard you want

and I doubt Einstein would've cared about math-level rigor, especially at that stage in his career

I guess @Thor disappeared. Time for lunch for this bonzo.

actually a lot of work on trying to make sense of Gaussians on infinite dimensional Hilbert spaces or whatever is about trying to make QFT precise

Einstein didn't even formalize it in Lean... dont hmu for a while...

yeah

@BalarkaSen On the other hand, machine learning

yeah how dare him

@BalarkaSen practically a lot of that comes down to Monte Carlo methods I think

lol Leaky appears out of thin air at the mention of Lean

which brings you head-long into the agony of the sign problem

the basic issue being that most of the integrals you care about are highly oscillatory in many variables

and those are not easy to handle

it's one reason I've long been curious about so-called resurgence theory, as a way to handle that messiness. but that gets over my head fast

MCMC is a smart idea but I dont know how broad it is

@BalarkaSen does that have to do with this stuff? pub.uni-bielefeld.de/record/2936016

@user2103480 dude possibly do I look like I know QFT? I think the point is physicists write a measure on the space of connections on a bundle, which is some nonsense like $e^{-\int S} dA$ they're famous for writing

in the context of paths with the energy being the action functional, the Wiener measure is what makes things precise

hahaha sorry I didn't expect that you actually know many details

part of what makes it confusing is that it's hard to get a handle on what QFT is supposed tobe

like, there's algebraic QFT, functional QFT, and perturbative QFT

with most of what physicists do being the last one

with the main problem being "well wtf is non-perturbative QFT supposed to be"

@user2103480 here's a talk that's worth watching: youtube.com/watch?v=JSI0Im3DauE

this guy's an influential mathematician in this domain

nice

his approach is to discretize Yang-Mills theory; they do something called lattice gauge theory which is like association of matrices to each oriented edge in an oriented configuration on $\Bbb Z^2$ such that if you go around a loop ("Wilson loops") the matrices product to identity

to draw a connection with my earlier point: the reason people like lattice gauge theories is because, in principle, you can apply Monte Carlo methods

then you scale this, $\delta \Bbb Z^2$, $\delta \to 0$ and hope it converges in some appropriate sense to the Yang Mills theory

and in this discrete space you can write down a measure

There was a day where I tried to watch this video (youtube.com/watch?v=1QrlIc4395w&t=2698s) about the connection between SPDE and QFT but since I know neither I didnt get much from it

@Semiclassical ah ok

unfortunately, there's a lot of cases where that leads to the so-called numerical sign problem

in which case Monte Carlo doesn't work

sometimes you can make MC work

but it was uplifting to hear gubinelli say that he had to learn a lot of physics first himself, after already being quite advanced in spde

but there's a lot of problems where we don't know how to make MC work, and therefore don't have a lot of ways to actually do any computations

see for instance the first few sentences of the abstract here: arxiv.org/abs/1603.06458

@BalarkaSen

Looool

the age-old tradition of independent long-form podcasting

don't know why he capitalized "independent"

the Weinstein brothers suck

Eric W Weisstein is the real dude

@BalarkaSen you know that time when bret weinstein tried to explain chess to the international chess federation

although Stephen Wolfram is as bad as Peter Thiel maybe

@user2103480 no lol dont care to know

hes an idiot

@BalarkaSen I don't have a clue what wolfram's political views are and I don't want to know

they just can't be good

lol yeah

these CEOs man

the world is full of CEOs and it sucks

CEOs explaining how they work so hard and deserve every penny to a single mom with 3 jobs

lol

I got pretty pissed at a friend for applying (and getting) a scholarship for this semester even though he really really doesn't need any money (full bank account from a student's perspective, living at his parents', and nowhere to spend the money at since its covid)

The interesting thing is that it is completely out of naivité since he doesn't really have any struggling people in his circle and doesn't realize the money could have actually helped a person

Although chance is, someone equally well-off would have got it

yeah i wouldnt blame the guy. the problem is systemic with how the institute decides who gets scholarships

thank god my institute doesn't do a coin toss on who gets scholarships

@BalarkaSen Fair point. I probably got more annoyed because he's full on "I'm a buddhist oh I act in loving kindness for all beings and do retreats"

lol

honestly india sucks but im happy to have colleges which are funded by the government enough to waive tuition fees and have a working healthcare mechanism

not for long maybe

lol

@BalarkaSen that's a solid feat tbh. My father is from the middle east and I'm just astonished at how "well" india works for the sheer size of the country, the number of ethnicities and the economic troubles

rohingya genocide is probably the biggest crisis that's going to emerge in the next 20 years because bangladesh will be underwater very soon

climate? or is that a metaphor?

nobody will take the refugees

@user2103480 climate yeah

a LOT of it is already underwater tbh

a lot of mangrove forest isnt it

yuuuup

@user2103480 well it still works, somehow. not for long maybe!

maybe these idiots will realize that free healthcare is not such a good thing after the meltdown due to COVID cases

which is not a fault of the healthcare system but because government lied lol

It is not? I mean, define free healthcare

what are you referring to? i mean there's a narrative which is floating around that says because we have free healthcares the hospitals are overflowing

people cant book beds rn because everything is packed

of course this s a nonsense narrative but maybe they'll buy it

I'm very very happy that we have basically free healthcare in germany. Insurance isn't very expensive and if you can show that you cannot pay, government pays. But probably I just misunderstand your point

oh thats excellent

i was just saying that they might abolish free healthcare

soon

Oh what

Harsh

yeah, idiots run the country so

@user2103480 i actually saw it happen with my own eyes. i have lived on the coastlines of bengal and when i was a kid used to go on trips to the beach i'd see there's a thin strip of mud cutting through the middle of the beach which would get wider and wider with every year

I'm not sure whether I know one single country where I think "thats a pretty smart government"

@BalarkaSen oi wow

We certainly don't have "free healthcare" in the US — far from it. But our emergency rooms and ICUs are all overflowing regardless. Corpses being left wherever ...

this is the mangrove soil from the other side which would store near the beaches, which means the mangrove was going underwater rapidly

i have seen places which used to be beaches but are now many kilometers wide mud banks far from the ocean

things decay rapidly, pretty much by law. its only natural that this is happening to society, politics, whatever... as well

Did you watch the newer south park episodes? The climate change episode was great. "Old people made a deal with a demon to get expensive cars and premium boutique ice cream". Then when they had to get out of the deal, they disagree with a deal that forces them to give up soy sauce and red dead redemption 2, and instead made a deal that the demon can come back 50 years later and get 1 million african children

which sums up the situation cynically well