what is '*'?

hello

hi

this it neat: math.stonybrook.edu/Videos/IMS/Differential_Topology

oh, speaking of, Ted said you should read ch. 8 of his book

oh wow, milnor.

i was about to email ted about some old exams on the topic of differentiable manifolds/stoke's theorem, i havent seen him on here in several days

it's a new course at my university. im desperately looking for old exams on that topic

@BalarkaSen which book of his?

@iwriteonbananas multivariable math

ok

:P

you wouldn't be able to

so ted wants me to buy his book

almost, yes

:P

use your university library

it's over 100 EUR

yeah

speaking of the devil

hi ted

That was extraordinarily poor timing

heh

no, @Balarka, I said you should read it

hahaha

yes, but you also referred that when I said @iwriteonbananas doesn't understand the motivation behind wedge products

(neither do I : warning)

I said it's all determinants

and that "it's in chapter 8 of my book"

Yup

That was to get you to get to work

But it's hopeless

oh.

well.

I just stopped by before I spend the day sorting for movers

yikes

:I

Yup, I'm running out of time

exams in a few months. when that's over, i'll make sure i read mult. calc.

heya @Fargle

Howdy @Ted

ted, is/was there a course at UGA that did differentiable manifolds, divergence theorem, differential forms and stokes' theorem?

@TedShifrin must be a huge pain

or do you have any idea where i might find old exams of such a course?

I'll be glad when the next month has passed

What sort of course, bananas?

oh

here it's called analysis on manifolds, but it's a new course. no old exams here.

@Ted just so you know, I have no intention of not doing multivariable calculus. I just don't have the time this month. for one, I am fascinated by Morse theory.

well, it's in my freshman/sophomore course to a certain extent. But our analysis on manifolds course hasn't run in years because the only students who would take it generally had my course earlier and knew it all.

I'd want to understand those.

You can't do Morse theory without multivariable calculus, @Balarka. You're just getting more and more ridiculous.

@TedShifrin i see

? i think i made it clear that that's one of my motivations for doing multivariable calculus the next month?

bananas, I have problem sets from the graduate differential geometry course in which I do all that stuff at the beginning.

are my messages that ambigous?

yes, @Balarka

@TedShifrin i'd be very happy if you could send me those with or without solutions.

:(

I don't have solutions, bananas.

Did your course cover the inverse/implicit function theorems, bananas?

@BalarkaSen "*" shows cartesian product

what a silly notation

no, ted, we covered that in a previous course

@BalarkaSen For once, I have to agree with ...

oh, ok, bananas

Yes I know I am on my phone so no latex

@BalarkaSen so what should I do for the question

email me, bananas, and I'll send you problems

sure

Some of them won't be appropriate

@Remember $X= \Bbb R$, $Y = pt$, $A = pt$ and $B$ is the nullset. $(X \times Y) - (A \times B) \simeq \Bbb R - pt$ is not connected.

or am I just being silly?

that's fine, ted

Can someone link me with latex....

Proper subsets, @Balarka

aha. well, take $B = pt$ too.

Not proper, either

oh, non-proper

Yes that's also one of the conditions

fair, fair.

I'm convinced this is correct. Shouldn't be hard to prove via defns.

hello @Mike.

Goodnight, @Mike

@BalarkaSen I wasn't able to prove it using defs....

Hello@MikeMiller

Morning.

Putting off going to the airport as long as the hotel will allow (e.g., til 11)

@Ted I'm visiting prof tomorrow. Gonna see if I could use some of his ideas to get my problems right.

He allows visits on Sundays, @Balarka?

for me, sure.

Cool.

@MikeM: Try not to lose your phone or wallet on the return trip.

I'll give it my best shot.

no, I am a bit busy.

That's all I can ask, @MikeM :)

@Mike ps : was this for me or for pjs? just curious, as the former would be nicer of you than usual.

@Balarka: Sometimes not knowing is better.

true

bananas, I sent you some. I can dig up more if you need.

merci

@Ted @Balarka @Mike \o

hi

@Studentmath !!! Glad to know you're still alive!

Hi @Studentmath

Still!

I saw you were here about a week ago, and said hi belatedly.

We miss ya.

@TedShifrin I'm just curious if Mike is even a bit interested in what I am trying to do :P

I saw that, yeah. Miss you guys too! Want to keep my pure-math mind a bit busy, so I try to pop around when I find some free time

So besides military stuff, what're you doing these days, @Studentmath?

I sleep and eat

Ah, exciting stuff :P

I never knew you could work so many hours a day. I never knew there were so many hours in a day..

I meant 35 hours in a day :)

Some like that.. Nah, I do get 6 hours of 'sleep' a day, almost always

Yeah I figured :P

So how's no longer having to suffer ungrateful students?

LOL, ask me again in late autumn ... I'm used to not teaching much in the summers.

I'll try then

I'm procrastinating right now ... should be sorting and labeling stuff for the movers later this week.

Oh right, you're also moving out. Good luck! You have friends in the new place, right?

Yup ... But way too much to worry about for the next 3 weeks or so.

I'm not as young as I once was :D

So, any good math problems you've stumbled across ?

Sadly not, didn't have any time to.. Any to suggest?

I figure I need to get comfortable with tensor products to pursue what I am trying to pursue, @Ted. Any good places where I can read them up from?

I did it from Atiyah-MacDonald, but that's not the right thing I want.

Well, understanding the universal property that bilinear maps factor uniquely through linear map on the tensor product is important.

I can guess, but I am sure you'll enjoy the new house and all... will make it easier to enter the new mindset, too

I think it's worth working some of it out concretely for vector spaces, thinking of multilinear maps.

I am actually working with vector spaces (tensor product of fields)

I came across some very interesting probability problems that stumped me somewhat in the fall, @Studentmath.

Oh, please do share.. I need something for the dead hours of the week

So understand concretely as many ways as possible why $F^n\otimes F^m\cong F^{mn}$.

I guess @Mike could have helped me, as he knows some (a lot) algebraic geometry, but alas - I guess he's not really interested.

I don't think he knows any algebraic geometry. He knows a lot ... but ...

@TedShifrin hm, ok.

I need some kind of tool to visualize tensor products, and explicitly use the visualization in technical computations. Knowing that it's a pushout and how to construct it... isn't really helpful.

@Studentmath: Not that it'll challenge you much, but I can send my final exam. But I'm trying to dig up a few of the problems that were interesting. One thing I discussed with Pedro was a combinatorics way to figure out the expected number of cards you need to turn over (in a random deck) to get your first ace. There are several other ways to do it, too.

I tried to explain to Pedro ages ago that you can think of tensor product in terms of twisting geometrically ... the algebraists talk about extension of scalars. @Balarka

That sounds interesting, will try that out.

@TedShifrin ah?

But it's good to think of the example of the Möbius strip as a nontrivial line bundle over the circle. What happens when you tensor that bundle with itself $k$ times?

And yeah, anything you think can keep me busy at any dead hours, I will appreciate it - I check the mail every couple of days

I don't know how to tensor bundles.

You mean tensor of the fibers (which is a vec. space)?

Yes, that's what it means.

Anyhow, I will go back to my family and get ready for the weeks - enjoy the summer y'all! Bye for now Prof. @Ted

okay.

see ya, @Studentmath. Enjoy!

@Balarka: This is the real version of the tautological line bundle on projective space that is so important all over algebraic geometry.

hmm.

I don't really know how to visualize what happens when one tensors fibers.

You need to think about how you glue together trivial pieces.

You also need to understand how a linear map (say isomorphism) $T\colon V\to W$ induces maps $T^{(k)}\colon \otimes^k V\to\otimes^k W$. What are the matrices? Do it with $\dim V = \dim W = 1$ to start.

hmm, ok. I'll think about all this and get back to you. thanks!

OK.

@TedShifrin can you help me with my question ?

@Rememberme What question?

@TedShifrin hola!

Here are sample questions from this problem.

Hola, @Stan.

Yes, @Deathslice, because they're talking about drawing the cards one at a time, the order matters. Hence permutations, not combinations.

Ok can you offer a rebuttal to a statement that my friend had told me? He said "Since they are drawn without replacement, it makes no difference if you take the three one-at-a-time, or in a single bunch. You're dealing with combinations"

But the wording says "one at a time," which suggests that they care about the order. If you only care about what collection of cards you end up with, your friend is right.

@TedShifrin So in the problem you gave me, I assume $\text{Hess}(f-\lambda g)$ is negative definite. So if I am to utilize the IFT, this must be $\frac{partial \mathbf{F}}{\partial \mathbf{y}}$. But I don't understand why I am paranterizing on $n-1$? This is to give me a function not defined using a constraint right? So is the vector I am rewriting $(\lambda, \mathbf{x})$?

@Stan: Nothing to do with the proof we finally got a few problems ago. You do not assume the whole hessian is negative definite. You only assume it's negative definite when you plug in tangent vectors to the constraint manifold.

We're no longer doing anything with the implicit function theorem to get things as a function of budget. Now budget ($c$) is fixed.

Hmmm.

Ok.

@TedShifrin so if they asked for 2 reds and 1 blue, it would be 10P2 + 8P1 correct?

No, because you have to decide whether you're drawing RRB, RBR, BRR, etc.

alright

@Stan, you done with me for now? I have organizing/sorting to do.

I am sir. Thank you

BBIAB.

I am going to go think some more

Later

Hey @TedShifrin

have you ever heard of this guy ualberta.ca/~vbouchar/VincentBouchard.pdf

very strong

@TedShifrin to really visualize the mobius band as a lines continuously varying with pts on S^1 while twisting around, I have to view the (boundaryless) mobius band as sitting inside a plane bundle on the circle: for a line to twist around once, it needs an ambient space to twist in. But then self-tensoring, we get a line bundle on S^1 sitting inside an R^4 bundle, and I have a hard time visualizing that, or seeing why the lines necessarily remain "twisted" inside even more space to move in.

@Rememberme I am here now, at least for a while (in case you drop by)

Monoid and group rings are cool.

so if they asked for 2 reds and 1 blue, it would be 18!/(10P2 + 8P1)?

yup.

@Dave that's correct when applying any function to anything

applying a function f to an algebraic expression u yields f(u)

okies - thank you :)

(sorry, needed a different letter from x)

lol

how's it going, @Dave?

@SohamChowdhury slow :P i tried one method of taking logs - didn't work so trying method 2

@anon, what's the simplest number theory problem that one can solve with rings?

@SohamChowdhury if possible can i show you my methods if i get stuck

If I'm online, sure.

@SohamChowdhury um, iunno

how hard is the two-squares theorem?

Hello.

oh, nvm, @anon. it's in the second rings chapter of Aluffi.

Is anybody here familiar with the Lambert W function?

what about it?

@anon: I can't visualize tensor products of line bundles except algebraically

@anon I have found something rather interesting.

I have discovered a function that returns the inverse of the function $f(k, x) = k^{x}x$.

Ah, LaTeX does not work here...

@Taylor see the "LaTeX in chat" link on the starboard -->

Yep, I have done it.

@MikeMiller hey mike

@Taylor sure, $y=k^x x~\Rightarrow~ (\ln k)y=(\ln k)xe^{(\ln k)x}~\Rightarrow~(\ln k)x=W((\ln k)y)\Rightarrow x=\frac{1}{\ln k}W((\ln k)y)$.

one can solve a few more general exponential equations using lambert W, that is well-known

hi Stan

@anon Yes, but it does not yield any values, i.e. you cannot work out $W(y)$.

Well, you can.

Anyway, $\lim_{n \to \infty}\left(x_{n}\right)$ is equal to the solution to $f(k, x) = k^{x}x$.

does anyone here have access to the journal of differential geometry

if you're on school wifi you should

$x_{1} = f(k, x), x_{n + 1} = \frac{1}{2}\left(x_{n} + \log_{k}\left(\frac{y}{x_{n}}\right)\right)$.

@MikeMiller Is it a specific and not particularly new paper you are looking for?

@anon

it is!

@Taylor ?

@SohamChowdhury i set up a room so i can show you what i tried

Have you been reading what I have been typing?

@Taylor yes

well, 1987. I don't have access Befause I'm out of town. Let me get the link

It does seem to work.

@MikeMiller yes

@MikeMiller Just the author and title. I might have it in my collection

Would you agree, @anon?

@Tobias: Taubes, gauge theory on asymptotically periodic 4-manifolds

@Taylor dunno, not really interested in numerical stuff

Wow.

@MikeMiller No, sorry. Seems to be mainly books I have by that author

hi, everyone

Hi pal

no worries

maybe @Samuel has access ;)?

I'm at home right now, sorry dude

@zed111: somehow I missed your message. Sorry! my emails in my profile - can you send it there?

if I were at work I might be able to but I won't be there until monday

nah it's no problem. I appreciate it

@SohamChowdhury well, you have to prove that Z[i] is a UFD, and that's about it.

@MikeMiller okay

thanks, I owe you

ah, @MikeM is busy acknowledging debt :P

@anon: Well, of course the interval (homeo/diffeo to $\Bbb R$) sits inside the trivial $\Bbb R^2$ bundle (the normal bundle of the circle as it sits in $\Bbb R^3$, if you like). But I don't know why the abstract tensor product has to be done ambiently. Indeed, it does not. Just think about transition functions.

@BalarkaSen well, you take the fun out of it. :P

@Soham: He does that to get revenge for the number of times Mike and I have done that to him.

I'll let you know how that paper is. maybe you'll like it, @Ted. or maybe you've read it before

@TedShifrin not really (<-- blatant lie)

I know knot whereof thou speakest :P

haha

That was to Mike, actually, but it fits you too.

no, I knew it was to Mike. just couldn't stop loling at it.

I can do an hour of organizing/sorting, and then I really don't want to keep doing it. :(

nah, it doesn't fit me

@TedShifrin organizing anything is a hard work.

it's like doing dishes after eating. ugh.

oh, nope, never read it ... Although we did a year-long seminar on gauge theory in which I gave a number of lectures, it's really quite orthogonal to stuff I've thought about for the most part.

I actually have never minded cleaning up whilst I cook or afterwards.

If you're enough of a slob at it, @Soham, you don't have to take a bath, either.

Who the hell starred that?

you don't have to do that much to not bath.

yeah.

I bath once in a week.

Balarka doesn't shower all winter.

yeah

ahhaa

@SohamChowdhury did I tell you that?

I don't recall

Well, in the most of the western world, many civilized people have a cleanliness — not stinkyness — obsession.

its no problem he wants to be stinky @TedShifrin

No, as long he doesn't visit us, @Karim.

haha

@TedShifrin I don't see how thinking about transition functions would help. They are local. Strips with 0 and 1 twists both look the exact same locally (line segment x arc on S^1), but are different in the global picture where you actually see the twisting take place.

@TedShifrin luckily for you, won't happen in your lifetime. I might not even study math, who knows.