@anakhronizein: I guess the most immediate way is to consider $\frac12(\alpha+\beta)\otimes(\alpha+\beta)$ mod the ideal.

It's bijective because it has an inverse, if you want to finish that argument rigorously

I usually mod out by $\langle \alpha\otimes\alpha\rangle$ for all $1$-forms $\alpha$.

Well it works out to be the same

$k\mapsto k+m$ has the inverse $k\mapsto k-m$

Anyhow, what's the big deal?

helloo

hellooo, Antonios

hell, Zlatan

how's it going

hell, balarka

Hel

Daminark has reached the peak

he @Daminark

Hi @Antonios

.

hi @AlessandroCodenotti

Zlatan isn't a Greek name ...

It sounds more slavic than greek

It's his dank persona

It's a football/soccer player

hey Ted do you have an topology exercises or stuff on manifolds?

who was one of my childhood heroes haha

I think it's Serbian/Bosnian, yeah, Alessandro.

he is bosnian @TedShifrin

@Faust: For point-set topology, I mostly used Munkres with an occasional extra problem. For manifolds, there's Guillemin & Pollack (for which I wrote some extra exercises, but there are plenty in there).

k ill order the book

Which, G&P?

@Faust I recommend tu's book for manifolds from a more algebraic slant

(if that's your thing, but do GP first)

@TedShifrin what should I be deducing from that. I get $[(1/2)(\alpha+\beta)\otimes(\alpha+\beta)] = [(1/2)(\alpha\otimes\beta + \beta\otimes\alpha)]$ from plain distribution.

It's fine to do G&P to see all the concrete stuff in $\Bbb R^n$ before doing abstract manifolds.

Im looking for like s econd course in topology level book @TedShifrin

Guillemin & Pollack is my prime recommendation, @Faust. It will also make you solidify basics in analysis (inverse function theorem) and linear algebra.

It's my absolute favorite undergraduate course.

G&P has fun exercises too

(And hardest.)

And I can send you more exercises, too.

that would be great i just found out topology and calculus on manifolds is gunna be offered next semester

I guess I was trying to show that in the quotient $\alpha\wedge\beta = -\beta\wedge\alpha$, @anakhronizein. What exactly are you trying to do?

@TedShifrin We defined the tangent space for an abstract manifold in the last diffgeo lecture... I'm starting to appreciate embedded stuff more :P

You mean in the fall, when you'll be healthy, @Faust?

Yeah, @Alessandro. The technicalities are a bit overwhelming at first.

I was trying to show specifically for 1-forms $\alpha,\beta$ that $\alpha\wedge\beta(v,w) = \alpha(v)\beta(w) - \alpha(w)\beta(v)$.

@TedShifrin yeah but im stupid so it better for me to start early :P

Oh, right, but that's just showing that $\alpha\wedge\beta = \alpha\otimes\beta-\beta\otimes\alpha$.

@Faust, my blue book that we keep referring to here (the YouTube lectures) has all the stuff you need for the multivariable analysis. What does the calculus on manifolds course do?

$\alpha\wedge\beta := [\alpha\otimes\beta]$ is the definition for the exterior product I am using. So you are suggesting showing that $\alpha\otimes\beta - \beta\otimes\alpha$ has the same equivalence class?

So I think this definition needs a factorial in it.

Your definition isn't going to give that formula. It's going to give half it. Which I hate.

Heh.

By half you mean what I was looking for, divided by two?

There are all these $k!$'s that fly around exterior algebra.

Well, we've established that $[\alpha\otimes\beta] = -[\beta\otimes\alpha]$, so $[\alpha\otimes\beta-\beta\otimes\alpha] = 2[\alpha\otimes\beta]$.

@TedShifrin Differentiable manifolds and smooth maps. Topics may include embeddings, submersions, fibre bundles, vector bundles, connections, differential forms, differential geometry, Lie groups, transversality.

Oh, that's not calculus on manifolds. That's a graduate manifolds course.

You definitely should do Guillemin & Pollack first. That's going to be a super hard course.

Is this an incorrect definition I have been using this whole time, then? Or just a convention then, @TedShifrin

@anakhronizein: There's no "correct" or "incorrect," but one ends up with different conventions. I personally prefer $\alpha\wedge\beta = \alpha\otimes\beta-\beta\otimes\alpha$, but your definition won't get that unless there's a factor of $1/2$ somewhere.

Maybe when they go to the quotient?

Hmm @TedShifrin the prof teaching is notorious for giving easy grades so i may still take it but thats discouraging

That's definitely a graduate course.

Standard first course in manifolds. Usually quite challenging.

ya well dif geo felt like a graduate course too

$\alpha\wedge\beta = \frac{(k+\ell)!}{k!\ell!}[\alpha\otimes\beta]$ for $\alpha$ k-form, $\beta$ an l-form, is that it, @TedShifrin

Nooo ... this is way more advanced/abstract.

and i got an A-

having not done math for 4 years

hmm not taking it then

dam i was excited

But Guillemin & Pollack is a great project, and I'll be glad to chat with you about it (or exchange emails about it) if you work gently on that.

Or is the fraction the reciprocal...

@anakhronizein: Those are the factorials that appear if you do complete skew-symmetrization of the tensor $\alpha\otimes\beta$, yeah. I'm not sure that's what your quotient does.

Just figure it out in the case we were discussing.

Figure out where the factor of 2 goes?

Did you follow what I said?

time to go to the gym, have a nice evening/day all

see ya, @Antonios

Tell me what they said about the syllabus sometime.

No, sorry. I am trying to figure out how this 2 can disappear through factorials that come from the quotient. I don't see it.

OMG this convention!!

It is the devil

I'm saying that when you pass to the quotient you need to put in a $2$ or $1/2$ for your desired formula to be right.

Say what, Mike?

Someone should have written a clean note explaining how the different conventions translate

Whether A ^ B requires a 1/2 or not, in particular in starting the curvature change formula

Ohhh ...

(in that case a ^ a)

Kobayashi-Nomizu and Guillemin&Pollack all end up with the volume of the unit cube being $1/k!$, a fact which I shall never accept.

I just don't pay attention until I'm forced to. I hate conventions.

@MikeMiller in my case it does require a 1/2?

Scalars, orientations

Having taught this stuff zillions of times, I'm obliged to pay attention and get it right. :P

@anakhronizein It all comes back to which definition you use

Which definitions matter?

But I can't translate and certainly not any better than Ted cab

I would have said just the wedge product but I bet I'm wrong

So is there a difference between the different quotients you can take for the exterior bundle? Or is the difference only at the definition of the exterior product?

If you just define $\alpha\wedge\beta = [\alpha\otimes\beta]$, just descending to the quotient of the tensor algebra with no factors, then you will have $\alpha\wedge\beta(v,w) = \frac12\big(\alpha(v)\beta(w)-\alpha(w)\beta(v)\big)$.

Ok, so there are also conventions in the (anti)symmetrization products

Warner writes a whole little essay in a footnote about how you get isomorphic results and the deRham isomorphism does not depend on which convention you use.

I think the two conventions should be closely tied

Personally, I don't want that $1/2$. I want the unit square to have area $1$, not $1/2$.

Okay, so what I was finding was impossible to find because I need that factor of 1/2.

I showed you why that $1/2$ is there unless you change the projection to the quotient by a factor.

Yeah, you did.

Is this Warner's Lie groups book?

Or which book is this essay in?

There's no such thing. Differentiable Manifolds & Lie Groups.

Yes, that one, thanks.

That said, I hate some things about his book.

I did not keep it.

@Alessandro Let $\alpha=\frac{1+\sqrt{-31}}{2}$. Then $\mathfrak{p}_1^3=(2,\alpha)^3=(8,4\alpha,2\alpha^2,\alpha^3)$. The minimal polynomial of $\alpha$ is $x^2-x+8$, so $\alpha^2-\alpha+8=0$, so $\alpha^2=\alpha-8$, so $\alpha^3=\alpha^2-8\alpha=-7\alpha-8$, so $(8,4\alpha,2\alpha^2,\alpha^3)=(8,4\alpha,2\alpha-16,-7\alpha-8)=(\alpha)$ (skipping doing the whole double inclusions), note that $\alpha$ is contained in the LHS and $\alpha$ has norm $8$ which is the norm of the ideal

Thus $\mathfrak{p}_1$ has either order 1 or 3

Are there other books that get to this definition via the quotient of the tensor algebra that you prefer, @TedShifrin?

I have actually never seen it anywhere but wikipedia or in class. My professor for tensor analysis did it this way, but I don't remember all the details.

This reminds me a bit about index notation, since you have to do (anti)symmetrization there as well

Most differential geometry texts actually do the sub-module of skew-symmetrized forms.

In which case I do like having the 1/n! since it means that (anti)symmetrization on indices acts as a projection operator

Yeah, sure, but then you have to compensate later with a $(m+n)!$.

Hmm

Every book I ever pick up looks at exterior algebra as antisymmetric maps instead of algebraically through the quotient.

Algebraists prefer the quotient. Geometers prefer the skew-symmetric tensors.

I actually am looking at some of my lecture notes (I scanned some of them). I thought I'd done it both ways in different times.

@AlessandroCodenotti suppose that $\mathfrak{p}_1=(a+b\alpha)$, then $(a+b\alpha)^3=(\alpha)$, so $(a+b\alpha)^3=\pm \alpha$ (since $\pm 1$ are the only units). wlog $(a+b\alpha)^3=\alpha$. I assume that you can get a contradiction by expanding and comparing coefficients, but I'm too lazy right now

These things get even more confusing with complex manifolds :P

I like the quotient a lot more than the maps.

As I said, for differential geometry, the submodule is far better. You don't want to be choosing representatives to do geometry.

When you say submodule, what do you mean exactly?

I mean a sub-module (over the smooth functions) or subalgebra of the tensor algebra.

Hmm, so I'm trying to understand something. In class we talked about the characters of $\mathbb{Z}/5\mathbb{Z}^{\times}$, and as it turns out, if you take a fourth of the Gauss sum of the sum of characters, you reobtain $\zeta_5$. Of course this is something that's easy to just see write away, but the rationale behind why this ought be true isn't completely clear

cut-the-knot.org/fta/boas.shtml

this should be the standard proof given in lectures of fundamental theorem of algebra

I remember being confused by this whole viewing elements of the quotient as bilinear maps things until I realised what people are doing is identifying a subspace of the tensor algebra which maps isomorphically to the quotient, and work with that subspace instead..

Yes, sure, @loch.

hi @loch

But the constants plague one no matter what.

@TedShifrin do you have a favourite book for this treatment?

hey @LeakyNun

No, @anakhronizein. Spivak, Boothby ... they're fine. I do recommend you look at Warner, because he specifically addresses your concern.

Although I'm not sure he addresses the quotient specifically.

Boothby being the intro to diff manifolds and riemannian geometry?

Yup.

I have not seen this one before. I will have a gander. Thanks for your help!

All the times I taught graduate manifolds/geometry, I never closely followed any text. I'm too picky about what I want to include.

Sure thing.

Heh, it's either that or sit at the front of the class and read verbatim from Kobayashi & Nomizu.

That's a tough book to read unless you know everything to start.

I've taught a few sections from it over the years, but it's tough going.

And I continue to be upset that you can find almost in no standard graduate text the recipe for the tangent bundle of a homogeneous space $G/H$ in terms of associated bundles and basic representations. I think it's hidden a bit in K-N.

Geometry is one of those subjects I find to be surprisingly disorganized.

Algebra is super organized, of course.

And analysis, it seems there are books that include everything any analyst ever wants.

Algebra can be dry and dull if taught badly.

Geometry isn't disorganized unless someone renders it so.

But with geometry everyone seems to think "no book teaches geometry the way I want to"

I wrote my own algebra book, too. Other than Artin, I'm not fond of undergraduate texts at all, and Artin's book is way too hard for the undergraduates I was teaching.

I wrote my own undergraduate diff geo text, even though there are several standard ones people like. I prefer my own pedagogy. I'm egotistical.

I like free lecture notes online usually.

I find they usually have the best quality.

I don't.

@TedShifrin K-N is so scary :(

But if you choose random free lecture notes, no wonder you think things are disorganized. The average professor writing his own lecture notes doesn't do a very good job.

I agree, @Eric.

nary a picture to be found

Heh.

Most of my "breakthroughs" in terms of understanding stuff finally comes when I find a small lecture online that finally explains things in a clear way.

how terrifying

Well, there's no pictures in Lang, either, but I don't find his texts that bad, in general.

It's actually from books such as Lee (x2), Tu, and countless other books that I actually get the impression geometry is disorganized.

Not only does it appear disorganized, but it also appears that people don't care for detail.

And often forget about pedagogy.

But in the end it's the subject I love the most I guess, so that's why I stick with it.

Lecture notes are free tho

"You get what you pay for," in general, Demonark.

Lee prides himself on being very careful, @anakhronizein. But I don't know his books very well.

I just know him.

I think some of Lang's books are ok

I actually like his analysis book. His manifolds book is a bit absurd for doing it all for infinite dimensions.

the idea that Lee's smooth manifolds is extremely careful is wild to me

and not disorganized like, at all

it's also mostly not a geometry book tho

Right, not what you and I call geometry.

lol my first statement is the opposite of what i meant

That isn't what I remember reading a moment ago.

that's what i get for doing two things at the same time

Only two?

ya

My favourite books are usually Dover books. Fantastically written in most cases, cheap prices. What's not to love?

Kosinski's book is now sold through Dover I think

You and I have very different taste in what makes "fantastic" writing.

Some books are exceptional, but most ...

I think it's quite clear we never agree, @TedShifrin ;)

Yeah, I think I'm done.

Lmao

luv u 2

As Demonark knows, I have strong opinions on what constitutes good teaching/pedagogy.

the only dover books i have that i like are a couple by cartan and pedoe's geometry book

For me the best thing is when there are Springer books but available on campus wifi

Yeah, Pedoe's book is fabulous. I also got Henry Edwards to get his Advanced Calculus republished by Dover. It's excellent.

Heh, Daminark. I know how that feeling. That moment you check springerlink

@Ted is that the one with the boxy circle on the front

In the absence of that I'd usually just settle for whatever free notes there are online or get something from the library if available

Yeah.

that was the text for my first quarter of honors analysis

so i read it

idr it well

I found one substantive error in it (regarding operator norms for the proof of Change of Variables), but it's a good book.

I like Lang's algebra book. It's where I learned a lot of algebra for the first time, although that wasn't easy

Dover has Mendelson's topology, Jacobson's algebra I/II, Kosinski's book, Shilov's real/complex analysis, Willard's topology, Kauffman, Hodel's logic book, etc.

They really have quite a few gems.

Of those, I vaguely know Jacobson, and it's good. I knew it from hardback long before Dover.

I agree, Jacobson is good. Lots of interesting material, and good exercises

Yeah, they acquired a bunch of these after another publisher.

Kosinski was also published elsewhere.

I used Jacobson 1 and think it's ok

All of Dover was originally elsewhere, I believe.

i think that's the point of dover

I think Jacobson 2 is better than Jacobson 1

i dont have many strong book opinions other than some geometry books

Jacobson 1 was the reference for our LA course

Goldblatt's Toposes book, Kreyszig's differential geometry, Curry's logic book, a few books by Smullyan, counterexamples in topology by Steen and Seebach...

kreyzig is hard to love

I'd love a "counterexamples in algebra" book. There is some stuff where the only sources of counterexamples are original papers

or maybe some sketches with all the details as exercises

I don't think all of dover is republications.

I haven't often used textbooks except in cases where the instructor was kind of unclear and/or nobody in class TeXs notes, so I don't have many strong opinions

Is $\lim f(n)$ always supposed to be the limit as $n \to \infty$? Is it standard notation?

Emily Riehl had a category theory book published there, and that was definitely original.

nerver saw that notation

Looks like just abuse of notation/shorthand, @MultivaluedPersonality

I saw in a lot of places where the index is missing.

but how do you know if it's a limit or a colimit without the index?

Context. :P

Flip a coin

Or context that works too if you're a nerd or smth

Hi yall

Makes sense.

It's like when people write $\prod_i $ etc.

@MatheinBoulomenos mathein :D

@Daminark dami :D

@KasmirKhaan hey

@MatheinBoulomenos btw did you get the book ?

I never asked you that

I got rid of my Counterexamples in Analysis and Topology books, Mathein, but you should write an Algebra one.

@TedShifrin Ted :D

@Kasmir: Have you seen the doctor?

@KasmirKhaan yes, I got it, thanks a lot!

@MatheinBoulomenos you welcome :D

@TedShifrin yeah, maybe I'll do that when I'm a grad student or something

Not yet Ted !

there are some weird examples in algebra

But I got a time :D

Damn, Kasmir. You're trying to kill yourself.

my sister is a doctor btw

haha forgot to tell you that

Oh ... how far away?

she told me its pollen allergy that caused the breathing problems

and the caugh because of cold

but I dont need antibiotics just something to help me breath

i dotn know its name

I take antihistamines year-round because of allergies.

@MatheinBoulomenos mathein ! pls tell me if you are able to do those problems this week or not ! just need to know :D happy with u either way :D

well its only summer thing for me Ted

allergic to many things that grow on summer in sweden

other countries am ok

Well, I hope your sister is 100% right.

@KasmirKhaan maybe this weekend

Well it is 9 years of medicine

so if she is wrong , not only me will be in bad shape :D

Without examining you in person or taking any bloodwork.

Hey @Kasmir, what's up?

ted is still here wow

Also h @Antonios

@Daminark

we should have an internet database of counterexamples in math

@MatheinBoulomenos thanks ! i really need to have them this week because otherwise i wont be able to do the HW, ionly sent u 3 grouo theory problems!

like OEIS but way harder to build

@Daminark a bit sick but all good and you ? :D

@Antonios: I did ask you as you were leaving ... Any word on the course/syllabus?

oh sorry

There is someone who made an internet version of Counterexamples in Topology.

@Antonios-AlexandrosRobotis antionios :D

ehm, none.

hi @KasmirKhaan :)

You might need to go talk to the undergraduate chairman or his secretary in person, Antonios.

Ah, hope you feel better Kasmir

I guess the database of ring theory covers some counterexamples in algebra

thanks ! :D

Are there any course websites from previous years when the class was taught?

I will, it's still a way's away. right now I need to talk to bogomolov and figure out what I need to learn

yeah there are @Daminark

but there are not proofs on that website

If $A$ is a coherent (say commutative) ring, then $A[x]$ might not be coherent

the only example I saw was $\Bbb Q^{\Bbb N}[[x,y]]$ and I haven't seen a proof that it's coherent or that $\Bbb Q^{\Bbb N}[[x,y]][z]$ is not coherent

All that is rendering me incoherent.

@TedShifrin that's good

LOL, good timing, Eric. :D