Mathematics - 2020-04-28 (page 2 of 2)

Astyx

22:01

What's PDE ?

Ted Shifrin

I was thinking of numerical analysis, more computational. Although you may not use that much advanced algebra in your work in PDE, it is generally expected that everyone has a good background in basic pure mathematics.

@Astyx: partial differential equations.

Astyx

Ok that's what I was thinking but wasn't sure

Ted Shifrin

@JingeonAn: There are lots of algebraists who don't see why they should have to be bothered learning analysis, functional analysis, geometry, topology, etc. Everyone should have a solid foundation at the level of beginning graduate-level mathematics.

You don't have to be an expert in the inverse Galois problem or algebraic number theory. :P

Alessandro Codenotti

@Thorgott I mean, is there even any difference between $\infty$-groupoids and topological spaces?

Thorgott

you tell me lol

Ted Shifrin

22:05

@Balarka: Did you see Mike's question about numbers of l.i. vector fields on $S^{4k+1}$? As far as I know, this stuff is all serious homotopy theory, and there's no reason he should expect me to know anything :P

Jingeon An

Yeah. But Im afraid of algebra lol

Balarka Sen

Yeah it's super scary

Alessandro Codenotti

@Thorgott The answer is actually "no" with the right adjectives on both sides

Thorgott

I see

Jingeon An

I think there are different brains for algebra and analysis.

Thorgott

22:17

I just realized that a subring without unity can still be a ring with unity in its own right and it makes me feel weird

Jingeon An

One friend cannot really do analysis but he is monster in algebra

r9m

^ I like where this is going .. (secretly gets popcorn)

Ted Shifrin

I have never been algebraically strong, but my first textbook was an algebra text, and I realized I liked it quite a bit.

r9m

@TedShifrin Could you suggest some books on lie groups? Something which starts from very basics would be very helpful for me ..

Ted Shifrin

With what focus? (I no longer have most of my library, so such questions are difficult for me now.)

Jingeon An

22:25

When I was doing my Bachelor, algebra was my first math class (my major was Astrophysics) and I was knocked out. Later I tried intro analysis and loved it.

Ted Shifrin

A very elementary start is a paperback by Curtis called Matrix Groups.

r9m

@TedShifrin more geared towards pde stuff would be helpful .. but for now I want to pick up the very basics

Ted Shifrin

@JingeonAn: Sometimes teachers/textbooks make a huge difference.

I don't have a good answer, @r9m.

Jingeon An

Absolutely agree. I wasn't a good student in Japan, but here in Europe it is much easier to follow the class.

Thorgott

Someone who doesn't love Galois theory clearly can't be a good person

Ted Shifrin

22:27

If you already know a good deal about manifolds, then you should learn the basics on the manifolds side and things like the Maurer-Cartan equations.

In the US a typical undergraduate math major can't make it through Baby Rudin and hates analysis :P

Balarka Sen

@Thorgott Please follow the Be Nice policy of the chat

Jingeon An

I thought in US students cover much more topics.

Ted Shifrin

In the US a tiny percentage (maybe 10-15%) go on to graduate work in mathematics. So math majors are much less demanding and much more varied than in Europe.

Thorgott

@BalarkaSen Why do you take this so seriously ? it's the internet, no one cares except for you

Balarka Sen

@Thorgott Gonna report you too

Ted Shifrin

22:29

@Thorgott: He was being sarcastic to your sarcasm. Both of which are better examples of sarcasm than anything our president claims is sarcasm.

Or maybe I should just ignore everyone.

Jingeon An

Only 10-15 percent? Even in top 20s?

Balarka Sen

He was being sarcastic as well

Ted Shifrin

Not sure what you mean. But even at Berkeley, MIT, Harvard, way less than 50% go on to graduate school in math.

Thorgott

we're multiple layers deep into sarcasm

Jingeon An

That's crazy.

Ted Shifrin

22:31

Just make sure you double-check that it's actually sarcasm and not irony.

Balarka Sen

@TedShifrin Oh you didn't meet SteezO- did you

r9m

@TedShifrin just looked it up .. ok .. I don't know enough about manifolds to understand that rightaway

Ted Shifrin

Well, there's the algebra side (representation theory, structure of Lie algebras, Engel's Theorem, etc.), but you definitely should learn differentiable manifolds, differential forms, etc..

Whom, a @Balarka?

Balarka Sen

Wait I might have a screenshot

You'll love it

SteezO- was taking a take home exam and decided to cheat

Jingeon An

@TedShifrin Yea. I already learn differential manifolds up to De rham cohomology. And it was super interesting.

Thorgott

22:36

I discovered it in the review queue and flagged it, so the site posted one of these automated replies about the guidelines, to which SteezO- replied "Why do you take this so seriously ? it's the internet, no one cares except for you"

Balarka Sen

in another question some guy in the comments went like "this is clearly an exam problem, going to report you" and he responded with "Why do you take this so seriously ? it's the internet, no one cares except for you" (Edit: sniped)

Jingeon An

But not sure I can use it somewhere after graduation, so I choosed to study in analysis.

Ted Shifrin

That was all to @r9m, @Jingeon :P

Oh, thanks for the context, @Thorgott.

You can do geometric analysis, @Jingeon. Plenty of interesting stuff in the overlap.

Thorgott

they also asked a second question, where another user was trying to get them to work it out themselves and after they were being condescending, the other user reported them, to which SteezO- replied "only geeks like you care, gonna report you too haha"

r9m

@TedShifrin ok :) Thanks

Jingeon An

22:37

I never heard of it!

Ted Shifrin

One of my FB friends (and students from long ago) posted a solution in which his calculus student had clearly cheated. @Thorgott @Balarka I told him he could find Leibniz's rule all over the internet (and MSE). He's not as well-versed in how students can cheat as I am!

@Jingeon: PDE relating to geometry of Riemannian or hermitian manifolds. Lots of deep, interesting stuff (for which, for example, Yau became extremely famous in the 70s and 80s).

Balarka Sen

Lol yeah this must be a big issue now that everyone has to take exams from home

Ted Shifrin

Yup, and faculty are clueless.

Jingeon An

Thanks, I'll look up for it. This looks interesting!

Ted Shifrin

Of course, the people on MSE that love doing people's homeworks and exams are still plentiful.

Balarka Sen

22:40

Yeah I hate how 90% of MSE is homework help

Not even actually hard homework, it's just copy-pasting the question and someone writes down an answer

and the student surely copy-pastes the answer back in the homework sheet

It's a cycle of ctrl C ctrl V

Ted Shifrin

Not on Macs :P

It's command C, command V.

Balarka Sen

Welp exposed myself as a Windows user

r9m

@BalarkaSen same on Linux as well .. :P

Balarka Sen

Welp definitely exposed myself now

@Alessandro @EdwardEvans Ulver, "Shadows of The Sun" is an unbelievable album! I can't believe it. I am only 7 minutes in

Alessandro Codenotti

Oh wow it's been a while since I've heard Ulver being mentioned

I'll check it out tomorrow

Balarka Sen

22:45

Yeah post Bergtatt they stopped being a black metal band so I never really listened to anything else by them

Mistake

Thorgott

yeah, it can be quite bad

Jingeon An

@TedShifrin Have any recommendations on the book of geometric analysis?

Just can find a book from Yau

Ted Shifrin

Not off the top of my head. Most of them are newer and so I haven't kept up.

r9m

Peter Li's book seems good :)

Joe Shmo

@TedShifrin, @BalarkaSen, nope, it was given as an informal definition for "reasonably behaved spaces ($= S^{n-1}$, $D^n$)"

Balarka Sen

22:52

That makes sense, @Joe.

r9m

@JingeonAn which one?

Jingeon An

@r9m Thanks!

Joe Shmo

ok, so whats the real definition? and that still does't answer what happens at $* = 0$

Balarka Sen

Anyway, $H_0(X, A)$ is isomorphic to $\tilde{H}_0(X/A)$ for reasonably behaved spaces, to answer your question. The tilde indicates reduced homology

Jingeon An

@r9m Lectures on Differential Geometry.

Joe Shmo

22:54

gotcha

because you remove a point

Balarka Sen

The real definition can be found in the beginning of Hatcher Chapter 2.1

Joe Shmo

thanks

r9m

@JingeonAn That's heavy on Yamabe problem stuff and bunch of more advanced topics. Not a beginner's book. Cheeger and Ebin's book has enough building material before one can start with Schoen-Yau though ..

Balarka Sen

@r9m You've read Cheeger-Ebin?

r9m

@BalarkaSen parts of it .. not the whole thing

Balarka Sen

22:59

Cool. I'm reading it right now

Jingeon An

Thanks, what is the title?

r9m

Nice .. =)

Balarka Sen

Comparison Theorems in Riemannian Geometry, @JingeonAn

Jingeon An

Comparison theorems in Riemannian geometry?

haha thanks

Balarka Sen

You need to have a solid background in foundations of Riemannian geometry to read this though

I find it extremely terse and analytic

r9m

23:00

yes :) .. get the older edition though. The newer reprint is riddled with mistakes :P

Jingeon An

oh :(

Balarka Sen

I haven't encountered major mistakes so far, just one or two typos

Jingeon An

I just read the contents of Peter Li's and Cheeger-Ebin's, and Peter Li's looks more for the begginer, is that right?

Balarka Sen

But I am also not attentive about analytic details

r9m

@BalarkaSen mostly typos .. nothing that's life threatening :P

Balarka Sen

23:03

Nice, that's good

r9m

@JingeonAn yes .. but Li's book has more modern proofs of stuff .. while covering more basic things

Balarka Sen

I don't actually think the analysis matters too much when doing comparison theorems

These are soft estimates for what it's worth

But color me surprised, I guess

Jingeon An

By the way, just a simple question, can any differentiable manifolds be expressed as a subspace of Euclidean space?

Balarka Sen

Yes

Jingeon An

What is the idea of the proof?

Balarka Sen

23:05

For the compact case, you cover it by charts, take graphs of the chart diffeomorphisms after extending it by a bump function, and then take product of these

Jingeon An

Thanks!

Balarka Sen

That gives a map to some $\Bbb R^n \times \cdots \times \Bbb R^n$, where the number of copies of $\Bbb R^n$ is the number of charts in your atlas. And then you add one extra factor of $\Bbb R$ or something to make it an embedding

For noncompact it's a little more subtle and you need existence of proper functions.

Jingeon An

What do you mean by proper function?

Balarka Sen

A function such that preimage of compact sets is compact, or colloquially "sends infinity to infinity"

You can look up a proof in Hirsch's differential topology text

Jingeon An

Thanks!

Balarka Sen

23:14

The cool fact is that any $n$-dimensional smooth manifold embeds in $\Bbb R^{2n}$.

That's the Whitney embedding theorem (usually stated with $2n+1$ but this strong version is also true)

Thorgott

I remember having read briefly about this, but I forgot: do topological manifolds also all embed into euclidean space?

Balarka Sen

Yup.

Thorgott

nice, is this more/less difficult than Whitney?

Alessandro Codenotti

Depending on what a topological manifold is to you

Balarka Sen

Embedding manifolds in Euclidean space isn't Whitney; it's the dimension pushing that's Whitney

It's the same proof as differentiable case, instead you have chart homeomorphism, you take a product, etc

Thorgott

23:20

say, second-countable, hausdorff, locally euclidean

Alessandro Codenotti

Then yes. If you drop second countable the long line is a counterexample

Balarka Sen

But there's a much stronger theorem which says every compact metric space of covering dimension $n$ embeds in $\Bbb R^{2n+1}$. Ask Alessandro for proof

Thorgott

the chart idea relies on compactness so that you only have finitely many charts, no?

Balarka Sen

You can deal with that by proper function stuff, yeah

Alessandro Codenotti

@BalarkaSen really? I didn't know

Balarka Sen

23:22

Oh @Alessandro

It's some folklore fact

Alessandro Codenotti

Cool, It's probably in Engelking's dimension theory book, I'll check it

I never learned dimension theory properly

It's too hard lol

Balarka Sen

Yeah

Thorgott

nice, I might look into that if we don't cover it in my topology course this semester

Balarka Sen

I'll read it with you as well

My topology course is now on separability axioms lol

Alessandro Codenotti

Let $f$ be an open surjection between paracompact spaces $X$ and $Y$ with finite fibers of the same size. Then $X$ and $Y$ have the same covering dimension

Balarka Sen

23:24

Dry garbage

Alessandro Codenotti

I guess? But wtf

Balarka Sen

A point set topology course should be just an all-examples course

Follow Steen and Seebach

Alessandro Codenotti

lol

Balarka Sen

Best way to learn these nuances is to know lots of examples

Thorgott

we defined topological spaces today

Balarka Sen

23:25

@AlessandroCodenotti Dang.

Thorgott

the prof told us that if he ever gives a definition without immediately mentioning an example, we should call his lecture garbage

Balarka Sen

Lol oh @Thorgott

Thorgott

separability axioms are dreadful

who remembers all these numbers

Balarka Sen

T3.5

Thorgott

I know Hausdorff and normal

I think that's T1 and T4?

Balarka Sen

23:29

T2 and T4

Thorgott

damn

I don't have it in me

Balarka Sen

There's Kolmogorov (T0), Frechet (T1), Hausdorff (T2), regular (T3) and normal (T4). Then there's completely normal (T5) and perfectly normal (T6)

And in between you have shit like T3.5 which is Tychonoff

T2.5 is completely Hausdorff I think

Nothing about these are perfectly normal by the way

EnjoysMath

0

Q: The kernel of $h$ knowing the kernel of $f = g\circ h$, all surjective ring morphisms? (relation to Twin Prime Conjecture)

I don't like how the twin prime conjecture is notoriously an analytic problem. Neither should you, and here's why. See: prime Omega extended to mult. rationals for background. Let $\Omega : \Bbb{Q}^{\times} \to \Bbb{Z}$ be the abelian group homomorphism that restricts to the usual prime Omega ...

This is not crackpottery, I swear

2

Balarka Sen

clickbait

Thorgott

what's a typical example of a theorem holding for T6, but not for T5 spaces

Balarka Sen

23:37

Shrug

T5 basically means hereditarily normal, I am not really familiar with T6

Damn, @EnjoysMath. You were rediscovering foundations of Arakelov geometry here

EnjoysMath

@BalarkaSen I'll take that as a compliment. I'm really only wanting to apply it to TPC

lol

@BalarkaSen added short proof at bottom of latest post

(link above)

proof of commutative triangle

Yes \o/ got an upvote and a fav. That's all one can hope for :D

Balarka Sen

I believe in you man. Keep it up, you'll solve TPC

EnjoysMath

@BalarkaSen I'm not seeing how that would rediscover the foundations of Arakelov geometry

Balarka Sen

Honest

@EnjoysMath The answer seems to say so lol

EnjoysMath

Arakelov theory

In mathematics, Arakelov theory (or Arakelov geometry) is an approach to Diophantine geometry, named for Suren Arakelov. It is used to study Diophantine equations in higher dimensions. == Background == Arakelov geometry studies a scheme X over the ring of integers Z, by putting Hermitian metrics on holomorphic vector bundles over X(C), the complex points of X. This extra Hermitian structure is applied as a substitute for the failure of the scheme Spec(Z) to be a complete variety. == Results == Arakelov (1974, 1975) defined an intersection theory on the arithmetic surfaces attached to s...

@BalarkaSen oh, lol

I largely ignored the answer, saving it for later when it can be understood

@LeakyNun :math.stackexchange.com/questions/3649100/…

Eagerly awaits comments on post

If $f = g\circ h$ and we know (have a good expression for $\ker f$)

Then $h(x) = 0 \implies f(x) = 0$

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$Mathematics$ Mathematics