By the way @Mike your suggestion regarding that octonion linear algebra stuff (with the $\ominus$) was correct, it seems. Thanks.

@TobiasKildetoft how does that work?

i mean, how can usay that this "method" will get me what i want

@Danu Glad to help. Sucks they didn't explain their notation, though.

I hate that.

@NV-US It will constantly make sure that the first $m$ columns of the matrix are upper triangular, so when you are done, the entire matrix will be

That paper makes me pretty angry, still.

How's the writing of your paper going?

ok. i can find an eigenvector, but then i need to pick a basis for V with the eigenvector as the starting vector in the basis. what if dimV is large? i wont be able to just "pick". am i misreading?

@NV-US Just extend using the standard basis except for one of them

ok got it, thank you. i will start now.

@TobiasKildetoft

@NV-US You can also speed this up by finding multiple eigenvectors at a time when they exist

@Danu I have an algebra appendix mostly done except for two technical lemmas. The first I figure I'll ask an algebraic topologist who will probably know. The second, though, I had an incorrect proof and thought it would be easy to fix and I'm on week 2 of that easy fix.

oh, upto the nullity of that eigenspace

@MikeMiller EZ lyfe ;)

I'm less frustrated when I dont know what to do than when I think I do but can't see how to make my intuition work

I understand that sentiment

bad joke

I feel that this is the reason I get insanely angry at plane geometry lemmas which I somehow NEVER manage to prove.

I feel you

(I'm trying to help someone working through a book on it, but I am failing to be of any help)

Seriously, plane geometry is dark fucking magic

I will be soooooo glad when these proofs are done and written

@Danu Just normal Euclidean geometry?

It's been months since I did any gauge theory

@Jasper In the plane.

@MikeMiller The first comment in the mail I got from Humphreys yesterday was that the first proposition in my paper, which I have a very detailed proof of, was probably just a special case of a fairly easy and well-known phenomenon

Hmm, I don't know what kind of results you are looking at, but I once read "Geometry Revisited" by Coxeter and he proves many high school geometry results there.

@Jasper that's exactly the book my friend is working on ;)

The usual Pappus, Desargues, Menelaus, Ceva etc is there.

I'm honestly having quite a bit of trouble with some of the exercises when I'm asked to help

Yes, I find geometry very hard too.

(I keep telling myself this wouldn't be the case if I really sat down and properly read it myself etc etc, but still feels bad haha)

It becomes like Olympiad type problems, you know.

@Jasper The weird thing is that I consider myself decent at "more difficult geometry" (grad school stuff)...

Yeah, I see what you mean

@Danu Yes, where very often the geometry is just analysis or algebra!

Or topology :D

Well, there is another small book I like. Agricola and Friedrich's Elementary Geometry with slightly different topics.

I had an email exchange with Agricola a while back ^^ She does work on $G_2$

Yeah, these authors also worked on another book together, Global Analysis.

@Danu The Dynkin type?

Those topics sounds pretty nice; maybe nicer than Coxeter's book which is more classical

@TobiasKildetoft The Lie group

(not so much the algebra)

@Danu But isn't there more than one Lie group with that type?

I mean the automorphism group of the octonions (the compact, simple Lie group)

ok

I regrettably know very little Lie theory

First topic I'll have to really work on when I start my PhD.

Classic geometry can be quite challenging. I also recommend Pedoe's book (easier than Coxeter, but still beautiful) "Geometry: A Comprehensive Course." He has amazing stuff in there.

Thanks, Ted

@TobiasKildetoft Ah, that's unfortunate. I feel my appendix is similar.

My friend is studying to be a high school teacher so she'll have plenty of time in her life for elementary geometry books; the more recommendations the better ^^

Surely everyone who actually does meaty algebraic topology knows this, I mean.

@MikeMiller Yeah, it is a bit annoying, but not a huge deal.

@MikeMiller I feel like there's a lot of value to spelling things out in many cases

(particularly for readers like yourself, who might not already know ;D)

@Danu For sure, this is good at least for me

@Danu: By all means, tell her to look at Marcel Berger's 2-volume book written for (French) high school teachers. Geometry I, II.

G'night, @MikeM.

@TedShifrin Oof, French... Not sure I'll be able to convince her about that ;)

It's in English.

I just wish I could look at a paper from 1973 or some crap and cite it all away if I don't think it's original.

Beautiful book

Oh, it's been translated? Cool

I just meant European high school teachers as opposed to US.

ahhh

Now I see

It's got all sorts of fabulous stuff in it.

I misread your first sentence

But beyond usual high school level.

Sure, but so is Coxeter's book it seems

Yes.

I, for one, am sure I did not know 95% of the material in his first chapter, haha

I have some of that stuff (affine, projective geometry stuff) in the last chapter of my algebra book.

Cool

If you know any other stuff on high school kind of mathematics, let me know!

Serre Arithmetic

(I cannot get over that book)

> high school mathematics

Actually what are some books whose titles are blatant lies?

All of your parodies of me.

Hi all, I've got a puzzle that I'm not sure about how to solve. There are three cities and I am to connect them by railroad. I want to use as little rail as possible, so I'm going to make three straight lines from the cities and have them intersect at some point in the middle. Is that middle point the center of mass of the triangle?

Ah. Great classic question — one of my favorites. Answer to your question: Only if the triangle is equilateral. :)

mmm

You can actually make a physical model to understand this and find the point.

This is one of those plane geometry questions, haha

I've given numerous lectures giving different proofs of this particular one.

Oh, another great book for your friend, @Danu. Pedoe also has a book called Circles.

Noted!

@TedShifrin my prof gave me the 4-city case as an exercise in symmetry breaking :)

The 4-city case is quite different from the 3, yes. Welcome to varifolds and to soap bubbles. (Ask @EricSilva.)

don't know about varifolds, but I know what you mean w/r/t soap bubbles

huzzah for minimal surfaces

Varifolds are one way into geometric measure theory, but think manifolds with singularities and multiplicities.

Oooh, we're doing the soap bubble thing with cities for our "children university" workshop at my uni :3

huh, interesting

what I really like about the soap bubble version is that you can use it to demonstrate hysteresis

Does everyboy know what this is about? :(

I don't like hysteria, Semiclassic.

rolls Ted's 17 eyes

I'm forgetting what the name of this problem is, though.

oh, Steiner tree problem

@Danu I don't

@MikeMiller Good :)

I don't.

I meant, like, I don't know what four cities means.

I'm not looking for an explanation, just responding to Danu.

Put four cities at the corners of a square. (doesn't have to be a square, but it's the most obvious case to start with)

LALALALA NOT LISTENING I need to proofread my shit

@Semiclassical I really meant I'm not going to read a response, I was trying to save people from typing

oh

for anyone interested, my mention of hysteresis is motivated by this old blog post I liked: thatsmaths.com/2015/01/29/the-steiner-minimal-tree

specifically the last few sentences of the section on "Solutions for 4 Points in a Rectangle"

> Jakob Steiner was one of the greatest geometers of all time

I am now suspicious of the contents of this article :P

lol

wonders when Danu finished his proofreading

BTW, 17 is an awful lot of eyes.

^just about 30 pages more! And then I need to write a 1-2 page introduction (including Acknowledgements mentioning you and Mike ;D)

You trying to get Mike and me fired?

I can get myself fired easily enough

just need to never prove this lemma

@TedShifrin :'(

(jokes aside, I appreciate it)

I suppose I appreciate something ...

hi DogAteMy

Hi

In an English class finally?

Yeah

Heart of Darkness

Never read Joseph Conrad.

(Or maybe I screwed that up entirely.)

@Danu, did you see what happened yesterday?

That's the book that inspired Apocalypse Now, isn't it?

Start here: chat.stackexchange.com/transcript/message/40049871#40049871

There were flags on chat.stackexchange.com/transcript/message/40049832#40049832 and chat.stackexchange.com/transcript/36?m=40049927#40049927

@bwDraco No.

Why should I look at this?

It's ongoing flag abuse.

Petition to rename the chatroom to "Mathematics & Drama" :^)

I didn't see any flag abuse during the time that I've spent here today, and I don't think it's a good idea for me to try to intervene after the fact given that I wasn't there.

It's an ongoing issue...

Oh oh, now I'm in trouble again ...

I think that, for now, it's best to just drop it (bringing it up will only incite further drama).

rolls a few more eyes for good measure

Also, can we pin this?

@Ted is it sigma-finite?

Demonark, what precisely? The set of all eyes to roll?

Yeah

Do I even want to know where most of those eyes located...?

Just think of me as a super-cyclops.

Hey everyone!

Rolling Eyes: A Geometric Approach

Yo @Perturbative

Heya! @TedShifrin @Daminark @SteamyRoot :)

@SteamyRoot don't look behind you

Hi @Perturbative

Ohi

@Daminark what's behind me? D:

How goes it with all of you?

Pretty damn good, I guess, looking forward to tomorrow

@Perturbative: So you got it sorted out that for an abstract manifold it doesn't make sense to talk about smoothness of the charts? That's one of the main advantages of the simplification in Milnor and G&P.

@SteamyRoot you said you don't wanna know where the eyes are located

Well, I assume they're on Ted's body

But Ted looks pretty normal in all pictures/videos I've seen...

@Perturbative it's going pretty well, derping around. You?

@Steamy what do you have going on tomorrow?

hitting on a student Date

Oh right

@TedShifrin Ahh I see you saw my question on main, yep I didn't know that one can only talk about diffeomorphisms after we know that the object we're defining is a smooth manifold

If the manifold sits in a space where you know how to do calculus, then life is easier.

@TedShifrin Aren't we all super-cyclopses

What with the twice as many eyes and all

I was actually meaning to ask you yesterday, if I'm losing out on much by working through G&P and having all my manifolds sitting in Euclidean space, am I?

Because I was thinking of working through Lee's book on the side

No, @Perturbative. It's not that big a deal to adapt ...

There's powerful stuff in G&P that you won't get in other books.

Plus some of the exercises are just downright cool in G&P

Turns out there's a theorem that every manifold can be thought of like so

@Daminark Things are going pretty well with me, I'm strangely getting some work done in the holiday for once

@Daminark Whitney Embedding Theorem?

Yup

@SteamyRoot Pls no garlic tomorrow before the big event

Hahaha

I won't :P

I hope you'll be 'smooth' tomorrow

Badum tss

That was quality

Perturbative I think you're officially my disciple now

Did I snipe your pun there Dami?

Lmao :P

I didn't even think of it

Also apparently the only homomorphism on complex L^1 with convolution is the Fourier transform

The more Demonark forgets about puns, the happier I am.

The more time goes on the more I realize that I was super wrong

I used to think Fourier was just a gimmick

Gimmick? Without Fourier you wouldn't have the internet.

Hey! @AlessandroCodenotti

Yo @AlessandroCodenotti

And I mean, not that it wasn't useful, but that it was some slick trick that you'd pull when you needed to do some differential equations

In physics we did a tiny bit during the waves element and it was basically presented like so

Signal processing is full of fascinating mathematics.

Nifty

Hi

Hi @Alessandro

Hi @TedShifrin

hi zed

Demonark: Can you answer this question? (Don't read my solution.)

Is there a reason why the degree of a map $f : M \to N$ is only defined when $M$ and $N$ are oriented $n$-dimensional manifolds without boundary, and where $M$ is compact and $N$ is connected?

What are you really asking? Are you asking why all the hypotheses are necessary?

Which ones can you answer?

Yep that was what I was asking

Without oriented, you can only do mod 2.

Well orientability has to be one of the conditions met because the sign of $df_x$ at a point $x \in M$ is defined to be $+1$ or $-1$ depending on whether it preserves or reverses orientation

Same dimension obviously necessary.

Compactness needed to get a finite number of points to count.

If you look at well-definedness of the degree independent of homotopy (and of regular value), you'll see where you need no boundary and $N$ connected.

I'll have to take a look at the last one (about no boundary and $N$ connected) in more detail

Thanks @TedShifrin :)

Sure, @Perturbative. Let me know when you want me to send you a midterm exam. :P

I'll be ready soon :P

I'm going off to bed now, night everyone!

Night :)

Also goodluck for tomorrow! @SteamyRoot

hey!

Heya Lucas!

Okay @Ted modulo proving that RP2 - RP1 is connected, I've got it

Hey @Lucas!

OK ... That should be immediate, but maybe you haven't learned cell structure of $\Bbb RP^n$.

I also mentioned it to you because of vector bundles.

how are you @Ted?

doing fine, Lucas ... you?

Yeah I haven't seen that yet. But this is a bit elementary. Basically, let's say RP1 = f^{-1}(0). Since RP2-RP1 is connected and f can't be 0 on it, then 0 has to be either a global max or min

Well, actually, the right question should only be on a neighborhood of $\Bbb RP^1$ ... the function won't be defined on the whole thing.

You're not using regular value.

Well the derivative would be 0 at that global max or min

Well at the pre-image, which is RP1, so 0 isn't a regular value

I'm telling you the question should be about a function on a neighborhood, not all of $\Bbb RP^2$.

Oh those two comments were connected

Well, okay thinking about it using vector bundles

Okay so the gradient of a function is orthogonal to a level set, so if it's not 0 I think that'd mean the normal bundle is a product one, right?

Product, yes.

MikeM has to give it away.

Harder for $\Bbb RP^n\subset\Bbb RP^{n+1}$ :P

I didn't understand how that gave it away. It's the same question more generally with the same proof...

Sorry though, deleted

Sup chat

I don't think Demonark yet understands what the topological make-up of $\Bbb RP^2$ is.

Hi Eric.

Yo @Eric and @Mike

Demonark: At some point you should understand the tautological (universal) line bundle on $\Bbb RP^n$.

Isn't he 20 or so? I thought everyone came to understand RP^2 as part of puberty.

I'm not at all funny today.

LOL, MikeM. I actually drew a picture in my algebra book when I was introducing projective (plane) geometry.

Lmao

Good choice. Was it a triangle?

I missed that part of puberty, I was focusing too much on Besicovitch

Well, I drew the algebraic geometry triangle picture, but also the hemisphere + Möbius picture.

After me own heart.

Well, you know I have no heart.

@Mike btw I'm a couple hours into ff15 and I gotta say I like the boiz a lot more than I thought I would

But yeah I've seen the hemisphere but we just identified antipodal points on the boundary. I'm don't think I see yet how this links into Mobius stuff

They're a really fun group

@Dami Ifnore my comment then

@Daminark what are you trying to prove

@Eric If we talk about math Ted will yell at us, so we'd better keep it to ff15

Ahahaha

@EricSilva: I told Demonark to look at this question, but not at my answer.

That RP^1 is not the regular value of a smooth function defined on a neighborhood of it in RP^2 @Eric

I also told Semiclassic to ask you about varifolds. :)

Demonark: REGULAR VALUE.

Ah mmk

Whoops

You can certainly make it be a zero-set.

I could talk about varifolds for days tbh

I wrote like 300 pages of notes last month on them

To make a long story short, we were mentioning 2-D versions of the soap bubble problem.

Ah cool cool

Read any more of Chern?

Nah I've had to hang out with my significant others parents since they're in town

Haven't had much time to do anything

Ah, the old parental unit excuse :P

Hoping to at least be able to get through sheaf cohomology before school starts up

It's Monday morning here in Antarctica.

Hello everyone.

Heyo

Hi @ALannister.

Ha ha @ Antarctica

Now I have a new name, lol.

Let me expand on that.

NO more names.

I was thinking of using Antarcticaman.

NO.

Then I thought of the Italianised version, Anataracaticamana.

Do one for a few weeks, then the other :P

@Jasper Must be really warm

@Avantgarde It's a bit cold here now actually.

@Daminark You can be Daminaraka.

@Jasper Wear your sweater

Oh Lord

I'm out.

Night.

Welp gotta go entertain folk

Bye chat

See you!

Nifty!

In order to prove $\sigma$-additivity implies finite additivity, essentially, I want to show $\sigma$-additivity for any $n$, with $A_{1},A_{2}, \cdots , A_{n}$ in my $\sigma$-algebra, where $A_{i}\cap A_{j} = \emptyset$ for $i\neq j$ and $A_{i}=\emptyset$ for $i\geq n+1$ and reduce it to finite case?

so, do I need to use induction for the finitely many $A_{i}$ that are nonempty?

Anybody around?

Yeah, sorry I don't know how to answer that

Ted left :(

Took me so long to type that out on my iPad.

That's okay. You won't ever lose that. Paste it next time around

There are less than 10 users in this room now, a rare event.

seems like i missed out on all the good stuff

So it seems

gah whatever i am not gonna catch up with transcript

long chat is fucking long

Will it ever stop?

Anyway, wanna Forster or something?

Let's do it. There was a power cut, so I started reading a bit about sheaves. Good stuff

let's just DO IT

Nike.

I like Shia and his wife Mia.

@Daminark How much have you Forstered so far

I haven't gotten too far beyond what I hit yesterday

Parents often call and I'm often outside

Ok, cool

So until school starts I'm only gonna have those rare pockets of time

So yeah right now I'm on the identity theorem for a Riemann surface

hi @Daminark! Are you familiar with methods, classes, objects (programming)?

right

Yo @Kirill!

I never did object oriented programming