What are $\pi_4(S^2)$ and $\pi_6(S^2)$? I've forgotten.

$\Bbb Z/2\Bbb Z$, $\Bbb Z/12\Bbb Z$.

Classification of 4-manifolds is difficult even if you restrict yourself to Kähler surfaces :P

Ah ... Not enlightening, then. Presumably a submersion would represent a nontrivial element?

@TedShifrin: For some fixed fundamental groups, there is a complete topological classification of 4-manifolds with that given fundamental group. Cyclic is one case.

Surely it would. It's not obvious to me why, but it seems absurd to think that the projection map $M \to N$ of a fiber bundle of closed manifolds could ever be null-homotopic.

Looks like the OP is satisfied by my answer, even though I'm not. I don't have time to work on this anymore, though.

I'll ponder slightly later.

Let me know if you think of anything. I'll probably think about this again this evening.

@Ted: Qiaochu answered. I was being silly.

what is a "discriminant" of a quadratic expression ?

ive never seen that term before

@Dave $\Delta = b^2-4ac$

Tells you what the roots are like

i've not seen the triangle used before

It's the capital Greek letter delta.

Your text might use $D$ instead.

what does it represent

It represents $b^2-4ac$...

@SohamChowdhury i see ! okay ill try solve it now i think i know what its asking for

Just ask here if you need any more help.

eh, most of the questions in the [algebraic-topology] tag are either too hard or too easy.

Simmons and Hatcher?

Simmons and Hatcher are of different flavors, really.

Simmons is a point-set topology book, while Hatcher is algebraic topology.

Yes. I guess Hatcher is alg-top.

Yeah, but you have to do point-set first right?

Yes.

I knew that.

(I have also done a bit of Munkres, for both point-set and algebraic topology)

Is it imperative to do a bit of analysis before trying topology? I might try after I finish four or five chapters of Aluffi.

@SohamChowdhury can i just check so i understand, for the equation: 3x^2 - 2x + 3 i got a discriminant of -32

Aye.

ok thanks i understand it now !

@Dave Go to wolframalpha.com and type your equation in there. It'll tell you a lot about it (some of which you'll understand in a year or two).

Cheers.

oh thats a neat website!

thanks

@BalarkaSen ei, answer my question please.

I'm sleepy

@SohamChowdhury I didn't know essentially any analysis when I did topology.

Mhm. Jumped into Munkres?

I'll be off to bed now.

Night people

The only thing I thoroughly knew when I did topology was algebra.

Most people benefit quite a bit from starting with real analysis.

I second @Mike.

@hippa

Seems like you're in a class by yourself, @Ramanewb.

@ted Hi, sorry I didn't understand what you mean

It's an English expression, which usually suggests that you are without equal. But I like to use it when some student is alone in the classroom :P

@ted Ok I didn't know that. Do you say that because Hippa didn't answer ?

No, I said that because you're here alone.

@ted Yes, why is nobody here at this hour of the day ?

Well, I don't know. It's getting late where you are and it's night in Australia.

@ted But most people are from the US aren't they

not really ...

in the US this is a big holiday weekend starting now ...

@ted In France too

@ted I have no class on Monday, it's the Pentecôte

what holiday in France?

^ @ted

'Pentecôte'

oh ... that's not a national holiday; it's religious ...

@ted Yes. But nobody wirks on that day

je ne prête aucune attention à la religion :P

@ted Does that mean you're agnostic ?

yes

@ted Then keep searching ^^

Nope.

@ted Do you strongly believe God doesn't exist or do you just don't know ?

For me, none exists.

@ted What makes you think that

I'm not interested in a discussion :)

@ted ok nvrmnd

FREEEEEEEEEEEEEEEEEEDOM! for about a month

congrats, @Clarinet

THANK YOU @TedShifrin! So excited to finally be out!

I know. We've heard nothing but this from you for about 4 months :P

Yep that is true :P

I have also stopped going to the actuarial forum that I used to frequent. Apparently they don't like anyone who leaves the field.

How bitchy of them.

and yes, I do have the sanity to not complain about the field on that website

@TedShifrin Hey Ted!

heya @Stan

A lot of them seem to have this inferiority complex about them. The topic of who can make 6 figures comes up quite frequently. I remember reading about one person who said that anyone who didn't make six figures was "pond scum."

It's as bad as a business fraternity, @clarinet.

Firstly, I think I have fixed my notation and repaired Part A. Secondly, question: so I thought I did state $F$. I said F was $\nabla L$ but clearly that is wrong.

Yes, we agreed that you needed to map n+2 to n+1.

Hint: You are forgetting the constraint itself.

I didn't forget. I just was careless, which is equally bad lol. I thought it was implicit (hahaha).

growls

@Ramanewbie ?

@TedShifrin o/

lol that's a new one.

@hippa IRC real quick, python serious quick question

I am glad to be among real mathematicians for a change. You should see my economics class....

what's that supposed to mean, @stan?

oh

I warned you.

@StanShunpike I FEEL THE SAME! I just left actuarial science!!!!!!!!!!

:D

complex mathematicians are cooler still

Yes, @anon, we are :)

The professor instead of writing $\Omega$ writes W

That one drives me nuts.

W is not omega!

@StanShunpike However, I will say... graduate-level econometrics is fascinating :)

LOL ...

Yes, it is. Good math + econ is a lot of fun..

@StanShunpike Are you sure it wasn't lowercase omega, $\omega$?

YES

That's why it was nuts!

Dang

He was writing lower cases denote endowment. and to denote aggregate, he used W instead

and I was like...that's not capital Omega!

Just be grateful that you've even seen Greek letters recently @StanShunpike :P

lol true.

My life has all been VLOOKUP, IF(BLAH, BLAH), +, -, , and *expected deaths for the last year :P

I will never, never take a job again which uses only Excel and Access to process millions of rows of data.

Wow, totally moribund.

@TedShifrin Heya =)

I feel silly

I just realized I don't know if $\Bbb Q[[x]]$ can embed into $\Bbb R$.

hi guys

@anon seems unlikely, but i can't show otherwise either.

@anon I don't think so, because note that an element $f(x) \in \Bbb Q[[x]]$ is invertible iff the constant term is nonzero, so you can have different series, $f(x),g(x)$, that have the same constant term, then $f(x)-g(x)$ is not invertible, so it would have to be mapped to zero.

@PaulPlummer nonunits are mapped to zero? well, $\Bbb Z\hookrightarrow\Bbb R$ and has many nonunits mapped to nonzero elements.

Ah yeah

I suppose the problem is equivalent to $\Bbb Q((x))\hookrightarrow\Bbb R$ too

Isn't there something wrong with transcendence degree over $\mathbb Q$, @anon?

Oh, never mind.

$\Bbb Q((x))$ has at most uncountable transdegree, $\Bbb R$ has uncountable transdegree

Mmmm :p

Yeah, I was thinking iso, not injection.

@TedShifrin Question: Is the bordered hessian the Jacobian of the gradient of the Lagrangian in this problem?

Actually that doesn't make sense. Nvm

@Stan: it's supposed to be part of the derivative of $\mathbf F$, right?

is the following true ?

$\tau$ = $\tau^{-1}$ iff n = 2?

for $\tau \in S_n$

Is the statement, "Every permutation in $S_n$ is its own inverse, if and only if $n = 2$"?

My point is that I think there are some quantifiers (or even words!) missing there