Learning some complex arithmetic is the best decision of my entire math career

Messed with this stuff today

solved $x+y=\ln x + \ln y$, $x+y=xy$, and the other stuff is me still figuring out how to exploit the epic properties of $x^2\equiv -1 \pmod y$ and $f(x) - x = \frac 1 {f(x)}$.

I am now addicted to ensuring everything works over the complexes

One step closer to the krabby patty formuler

@Thorgott do you have any reference of the rationalization of top sp for that?

maths.ed.ac.uk/~v1ranick/books/gtop.pdf

I didn't read the contents but the author is Sullivan. 100% trust

oh dear, the book is full of p-adics. Can't even find where rationalization is

@onepotatotwopotato Mmm... yummy $p$-adics.

Seems Balarka gives the definition of Rational sphere (but only for odd $n$?)

im trying to compute this integral using contour integration, but I am confused about why the integral is not well defined for $k'$ in the lower half plane when $x - x' > 0$. I thought maybe because if we take $k' \to -\infty$ which is possible without changing the value of the integral that we get the exponential in the integrand going to infinity? But I am not really sure

@Thorgott Ah "Simple" here is a terminology in AT! I just thought it as an adjective

@XanderHenderson I don't need p-adics thanks to Bakarka. I'll try next time (next life)

(and I don't believe next life)

@Jakobian No, there was much to do that I didn't bring a book.

mathoverflow.net/q/468806/323920 The surface group question is partially answered @Thorgott but not fully.

@onepotatotwopotato yes, because for even $n$, the rationalized sphere is not an Eilenberg-MacLane space

cause $\pi_{4n-1}(S^{2n})$ has a $\mathbb{Z}$ summand (coming from the maps with non-trivial Hopf invariant)

in fact, $\pi_{4n-1}(S^{2n})\otimes\mathbb{Q}=\mathbb{Q}$, coming from that summand

so the Postnikov-tower of $S^{2n}_{(0)}$ has a non-trivial step, given by the fiber sequence $K(\mathbb{Q},4n-1)\rightarrow S^{2n}_{(0)}\rightarrow K(\mathbb{Q},2n)$

for general simplicity reasons, this actually extends one step to the right, so $S^{2n}_{(0)}$ is classified by the $k$-invariant, the element classifying $K(\mathbb{Q},2n)\rightarrow K(\mathbb{Q},4n)$ in $H^{4n}(K(\mathbb{Q},2n);\mathbb{Q})$

which, while I haven't checked, am pretty sure is the cup square of the universal element in $H^{2n}(K(\mathbb{Q},2n);\mathbb{Q})$

Hmm I can't understand since the 4th line but thank you for the explanation.

does anyone know the youtube channel "UndergraduateMathematics"?

it appears to be gone unfortunately

Today I learned that Polish-Americans weren't considered white by Americans

@BalarkaSen I wonder what you think about race now

a lot of americans view the world as black or white

apparently we were "ethnic white"

is the term they coined, of course, stupid term

race is just a tool for division of American people

the us vs them mentality has gotten stronger after the pandemic

weren't there people vandalizing monuments over it

I think on black friday, it was called?

yup

I was watching this video about us being called ethnic white and so on, and okay the girl in the video understood that race just a tool for division and so on. But she said that ethnicity is important, I don't understand that one, why is ethnicity important?

Ethnicity describes a commonality of culture within a group of people. One's ethnicity is typically a fairly important aspect of one's identity.

I think its just that American mindset of your ancestry being important, even though you never actually met any of those people and never practice their culture, is somehow your own ethnicity

For example, while I am non-practicing, it is very important to me that I am Jewish.

@Jakobian I think that it comes up a lot in American culture because there are a lot of people with very different backgrounds who interact with each other on a daily basis. When you live in a place that is more homogeneous, your ethnicity is typically taken for granted.

It is like the joke of one fish asking another "How's the water today?", with the response "What's water?"

Note, also, that "ethnicity" is not the same thing as "ancestry" (though the two are generally closely linked).

Sure but you're still just American of a particular descent

Your culture is that of an American

@Jakobian American culture is not a monolith.

What you have just said is, essentially, the equivalent of me saying "Sure, but your still just a Slav of some particular descent. Your culture is that of a Slav. You are no different from a Russian or a Croat."

To claim that my culture is "that of an American" is strikingly ignorant and rather condescending. :/

No? You are American

And now you are just doubling down. Again, "America' is not a monolith, and there is no single "American" culture.

If someone called my culture that of a Pole I wouldn't be offended

Or that of a Slav

@Jakobian Exactly. You are a Slav, indistinguishable from a Russian.

I never said you are indistinguishable from other Americans

That is certainly how it comes across.

In general, I would suggest that it is a very bad idea to tell someone else what their culture is.

If you're have Polish ancestry and living in USA but you never even tried Polish food, been to Poland etc., then you may say your culture is Polish but its not really true

@Jakobian Says you.

Gatekeeping heritage. Odd.

@Jakobian what do you want me to say lol

i have no opinion on race per se

makes popcorn

It is interesting that you not, above, that "race just a tool for division and so on", yet you are doubling down on classifying some people as "Polish", and others as "not Polish".

Hi @BalarkaSen!

@BalarkaSen You've been pretty active about it before

hi @anak

Am reading today

@XanderHenderson People adopt cultures of countries they've never been to because they lack or doubt in their own

@Jakobian i dont think so, but whatever you say, chief

Some book called Calculus obviously by no one of renown named Spivak

@Jakobian Says you.

I'd agree with you if you said there is Polish-American culture because there is, and they do have their own culinary food and so on

but you seem to be not listening to my points

@BalarkaSen, I was wondering if I could bounce another question off of you? No worries if you haven't got time, but we discussed last time about the use of a Morse-Bott function to write an n-form $\omega$ which vanishes on some submanifold as $\omega = f\mu$ for a non-vanishing n-form $\mu$. To me, this doesn't seem to be possible to do simultaneously for two n-forms $\omega_0$ and $\omega_1$ with the same $f$. Like $\omega_0 = f\mu_0$ and $\omega_1 = f\mu_1$. Do you agree?

@Jakobian YOU are insisting on creating relatively arbitrary divisions between people. MY point is that you should let people identify themselves however they like---it isn't your job to adjudicate who is, and who is not, "Polish".

@anak Yeah, if $\omega_i = f\mu_i$, $i = 0, 1$ and $\mu_i$ are nonvanishing then at least $\omega_i$ have the same zero locus.

Necessarily

okay, I will allow people that maybe have heard they are Polish once by accident and started believing in it, even if its not true, because you said so

And I will emphasize that your division is entirely arbitrary. At what point does someone who has Polish ancestry cease to be Polish? Are the no longer Polish the second they step off the boat into the US? or when they learn English? or is it their children who are no longer Polish? What if they live in a predominantly Polish enclave in the US? Maybe first generation people are Polish, but not the second?

even if none of their ancestry is Polish

they say they are Polish, so they are

@Jakobian Why is it that YOU are the arbiter of who is or is not Polish? Why do YOU get to "allows" certain people to self-identify people as "Polish"?

It isn't about you.

I don't?

@BalarkaSen So like $\omega_0 = (x^2+y^2)dx\wedge dy$ and $\omega_1 = (x^2+10y^2)dx\wedge dy$ have the same zero locus, and $f(x,y) = x^2 + y^2$ is an easy Morse-Bott function for this, but I struggle to write $\omega_1 = f\mu_1$ for some non-vanishing $\mu_1$.

Ja nie ponimaju etu konversatsii

I think identifying who is or isn't Polish is the job of an ID card

And I think that

@anak Yes, what I said is a necessary condition but not sufficient.

@XanderHenderson fundamentally it boils down to me to identify if a person is Polish or not

so in that sense it is about me, if we are talking about me deciding if someone is Polish

If a person says that they are "Polish", what business is it of yours to question it?

@Jakobian Why do YOU get to decide this?

Gotta check the DNA to see if Polskij is written in it

@XanderHenderson why not? Everyone has their right to decide those things

@Jakobian I strongly disagree.

@BalarkaSen Yeah, so this example serves as a counterexample to sufficiency.

Yep, @anak

Thanks for letting me just check I wasn't crazy. :P

Again, ethnicity is an important part of identity. The only person who gets to decide how a person identifies is that person---you get to decide, for yourself, what your identity is.

No one else has a right to dictate or decide that for you.

Learn any new contact-ish stuff lately, @BalarkaSen?

@XanderHenderson I can decide the validity of that claim in my outview

I was looking at optimal transport of contact forms, which was kind of cool.

@anak Oh interesting, tell me more

@Jakobian Sure, you can be a judgmental a-hole if you like. I would recommend that you keep it to yourself.

But who dictated I have the privilege to participate in existence? Clearly we are made, therefore we have no right to existence, yet we exist.

(Just trying to spice up the conversation)

@anak Thinking about high dimensions again after a period of thinking about low dimensions.

@XanderHenderson being Polish is supposed to be a meaningful word

@BalarkaSen I wish I had much more than that, but the analytic underpinnings are kind of a headache. The paper I was reading was just using the Benamou-Brenier formulation as an inspiration for a version with contact 1-forms. Gives a little bit of a dynamics to it, but I find it;s still shrouded in mystery.

Everyone has their high and lows. ;)

@Jakobian I don't disagree. Where we disagree is that you seem to believe that you have a right to decide (a) what that word means, for all people for all time, and (2) who gets to use that label, and who does not.

I only express what it doesn't mean

@anak What does "optimal transport of contact forms" mean, exactly?

I roughly understand what it means for measures

I don't have anything substantial to say we don't have to keep on going

I don't pay much value to my own identity. And if ethnicity is a part of that then I suppose its not part of what I value.

@BalarkaSen Well a skeletal view of it might just be "given two (contact) 1-forms and a cost function that takes as an argument a path of 1-forms, find a path between the 1-forms which minimizes the cost". The problem is coming up with a cost function which is solvable and meaningful.

@anak Cool!

I like the setup

@AMDG what language is that?

@Jakobian The Polish wannabe, Russian

I thought it might have been South Slavic

since they don't use Cyrillic

Russian is probably closer to the original Slavic language

So what high dim stuff are you doing, @BalarkaSen?

Or non-specific for now?

Convex hypersurfaces in high dimensions, mostly

@BalarkaSen Convex a la Giroux?

Yep

Giroux is an ultra nice guy. I also like his papers.

hes very cool

But the nice guy part tops it off.

Giroux elimination was my bread and butter. But that's more low dim. I wonder if there are techniques like it for high dimensional?

Nope. One knows existence of convex hypersurfaces in high dimensions. But it is not easy to manipulate with them

Just dividing hypersurfaces do not have enough information, unlike dividing curves

See Honda-Huang and Eliashberg-Pancholi for more

That's the extent of what is known

Darn, I guess (con)foliation theory is more powerful than higher dimensional stuff?

Certainly low dimensions is way more concrete and easier than high dimensions in that regard

Has higher dimensional convex contact geometry been bumping lately, or is it relatively untouched?

The major foundational works were done by Honda-Huang-Eliashberg-Pancholi

Very recent

I will have to look into those papers. Honda also is a good writer.

I am unfamiliar with Huang and Pancholi, though...

Yang Huang is a student of Ko Honda. Pancholi is Yasha's collaborator, he proved the almost contact h-principle in dimension 5 with Casals and Presas before BEM

I remember when I tried to read some h-principle stuff along side you. I need to get back to that one day to understand it. You were so much more well equipped for it than I was.

I'd be happy to continue reading someday

@BalarkaSen Did you ever finish that paper? It was Eliashberg and Emmy iirc...

and Borman. Nope, but I might have to go back sometime

Bro what have I been missing out on...

Spivak is great so far

So, interesting consideration came to mind... we can consider 0 as the convergent sum of the integers; nay, even the reals and the complexes themselves.

So 1/0 is the reciprocal sum of all the complexes.

at what stage of math should one read, say, Halmos naive set theory, or enderton etc

ive been coasting on whatever set theory a book's 1st chapter shows me, till now

@nickbros123 Never?

jk

@nickbros123 I am not sure that I would necessarily recommend either of those books specifically, but in the American system, I would suggest that a student should have taken calculus, first.

Not because you need to know calculus to read those books, but because completing calculus typically indicates that a student has reached a certain level of mathematical maturity, and it likely to be able to read and understand texts like Halmos.

I have come to realise how weak my set theory knowledge is. It all started with the question: prove that for any set S (infinite) that spans V(over F field), any linearly independent set L obeys ||L|| $\leq$ ||S||. and another question that Ive been trying out is: if S is infinite, the set Gs={set of all finite subsets of S} obeys ||Gs||=||S||

@XanderHenderson I see. What would u recommend?

How can I see $\dfrac{x \sin (\theta)-\cos (\theta)}{\sin (\theta)-y \cos (\theta)}<1$ for $0<x<1$, $0<y<1$ and $0<\theta<\pi/2$

I just want to be non clueless when it comes to questions like the ones above.

@nickbros123 For what?

set theory?

Why do you want to learn set theory? What is the goal?

Usually, in the US, set theory is taught as a gateway to higher mathematics. You take your calculus, and then you take some naive set theory to build your mathematical muscles.

@SoumikMukherjee this but not a joke

@XanderHenderson I dont know, its just uncomfortable?? there are stuff ive seen in probability books that tell that the specification of probability for open sets give the probability for the whole Borel field. I cant even begin to understand that proof. Stuff on different kinds of infinities. another thing is dont know what AC is, let alone its various forms i keep seeing. Its clear theres a lot of stuff I ought to know but I dont

@nickbros123 You aren't going to learn that from a book on set theory. "Open sets" are a tool of topology---you learn about those in a topology class; "Borel sets" are a tool of analysis---you learn about them in an analysis class.

Different infinities are not relevant to most mathematicians, unless you are hoping to study set theory itself. Axiom of Choice is not relevant to most mathematicians because it is just assumed to hold---it is only relevant if you want to study certain niche topics (e.g. foundations).

If you are going into one of those niches, then yes, you need to learn it. But it is not terribly important to most work-a-day mathematicians.

@nickbros123 I suggest to browse it rather than read it in detail, but try link.springer.com/book/10.1007/978-3-0346-0330-0

Up to section 6 of first chapter would be fine

@XanderHenderson its also relevant in general topology

set theory in general is, and even more so if you study set-theoretic topology

which more like a collection of results than a field, but still

in the sense that the results are not really related

they're just set theory questions in topology

there's also descriptive set theory which is also related to general topology, is mainly about Polish spaces and probably more relevant to analysis too

the term "standard probability space" comes to mind

damn, there's just too much to learn :0

You want to learn it?

not everything is relevant to you, you know

it might be better to learn something else depending on your goals

rn i dont have any goals per e

se

nevertheless I find it helpful to know some cardinal arithmetic

general topology is probably not something you want to learn if you want to be an academic

as in, as your main subject or something

some general topology is always helpful

@nickbros123 It is really hard to learn anything in mathematics if you don't have some kind of goal. I learned $p$-adic analysis because I needed examples for a certain result I was working on. I really learned measure theory when it became relevant to my work.

but its probably a good idea to start learning differential equations

Without a specific problem or goal in mind, it is often quite hard to learn a mathematical topic.

differential equations have almost endless potential for research

maybe differential geometry or something too

you are a physics person from what I know, so maybe you want to learn math for some field of physics?

find some interesting (for you) physics, try to learn that?

(I was talking about differential equations and general topology above because I forgot you are a physics student)

I am now a math student though :)

well its still a good idea to learn some math with physics in mind

even for a math student

the reason I dont have goals is cuz i havent even seen that much mathematics

barely scratched th esurface

well you can browse something about mathematical physics, maybe try to learn that once you find the topic you want to learn?

as in, math needed to understand it

other idea, find a professor and ask for what to learn

the professors around you is your closest environment

I don't know what to advise you, I just study what I like

@nickbros123 the world is your oyster at this point. Just pick up a random book from your list of things to read and go with the ebb and flow

@AlessandroCodenotti I am reading a proof about existence of completion of a uniform space (defined with pseudometrics) from Stone-Cech compactification, author said something is "easy to see", and I have it proven, but I'm not sure if my proof isn't too complicated. Would it be something you could take a look on?

@nickbros123 Use | | instead of || || to denote cardinality

@SoumikMukherjee Or \| \|.

or \# A for the cardinality of $A$, or \operatorname{card}(A) (that's my preference, because then there is no ambiguity at all).

\| makes a double line, I think its what Soumik doesn't like

@Jakobian Stone-Cech? Weird I thought one would go through the Samuel compactification

@AlessandroCodenotti yeah

by identifying Cauchy z-ultrafilters that should correspond to the same point

You first extend pseudometrics to Cauchy z-ultrafilters, identify some of them so that the pseudometrics are separating, and then poop out the completion

the "easy to see" part is that a point of the space we are completing is identified to itself only

and I have a proof of that but I think it might be a tad complicated and not sure if that's what author had in mind

Do you want to take a look?

I cannot right now

@Jakobian Yes

This also looks like a needlessly complicated way to show existence of completions to me

@AlessandroCodenotti Hi, following the candidates?

@SoumikMukherjee Religiously

Why do you think its complicated?

Alireza got cursed

Nepo in the candidates gets a 100 elo boost, Firo gets a 100 elo debuff

I'm rooting for Fabi

Uniform spaces here are in terms of pseudometrics, and we have extension property of Stone-Cech compactification, so going off of the Stone-Cech compactification to extend the pseudometrics seems good to me

I feel bad for Abasov, he is holding good against such opponents but his luck is against him

Or are you objecting about the usage of z-ultrafilters?

Yeah he had a couple games that were holdable but he lost

@Jakobian yeah

Why not do the Cauchy business

The book goes for z-filters quite a lot

and since points of $\beta X$ are basically z-ultrafilters, we are home

Also I still maintain that the Samuel compactification is the natural notion for uniform spaces, not the Stone-Cech one

It also has a description in terms of ultrafilters iirc, at least for topological groups

@AlessandroCodenotti it can be done as equivalence classes of Cauchy filters instead

I don't find the approach that different from the one using Cauchy z-ultrafilters

the only difference of z-filters is that they behave in a more rigid way

and primality isn't equivalent to maximality

I think ultimately you end up doing the same thing as talking about classes of Cauchy filters

not sure what the approach is using Samuel compactification since I don't know about it, other than I heard that its about groups once or twice

eh whatever I'll just stick to my own approach

@SoumikMukherjee what is this Naka-Nepo game

I was checking the same, maybe some known draw in the Petroff?

The time usage suggests this is a prep from Naka

It's an insane line to play if it's not prep

Stockfish gives 0.0 throughout, crazy

Usual stockfish stuff

Nepo figuring out all the best moves over the board what a beast

But the position is tough to hold

True but I think Nepo will manage

Alireza plays Rh3:/

Castling is for noobs

hehe

Alireza is out right?

Not yet, the tournament is still young.

who are you guys rooting for?

Fabi

But I wouldn't mind if streamer boy wins

rooting for my buddy eric

frfr

@Thorgott are you into chess too?

i'm rooting for the black pieces because i love a (probable) underdog

@leslietownes are you into chess too?

no not really. i'm probably more "into" chess than i am into most other games or sports, but that average amount of "into" is "not very much"

I prefer card games if anything

or dice

dad used to play chess. Now its bridge

he tried to convince me to play, but I don't like those games

@Jakobian only occasionally

only when his buddy eric is playing

If I assume that $\mu_2>\mu_1$, we have that $-\mu_2<-\mu_1$ and $\dfrac{-1}{\mu_1} < \dfrac{-1}{\mu_2}$ right?

if they have the same sign

and with that i can conclude that $\dfrac{\mu_2\cos{\theta}-\sin{\theta}}{\sin{\theta}-\mu_1\cos{\theta}} < \dfrac{\mu_2\cos{\theta}-\sin{\theta}}{\sin{\theta}-\mu_2\cos{\theta}} < -1$

Yeah, $\mu_1$ and $\mu_2$ are in $(0,1)$

No way this game is gonna end in a draw now, 3 pieces for a queen

what a game

aaaand it's a draw