Mathematics - 2015-07-11 (page 1 of 5)

Soham Chowdhury

01:57

@Dave exactly what you think it means: just apply $\log$ to both sides.

for example, if you have $5\times 2^z = 3$, "take logs": $\log(5\times 2^z) = \log 3\implies \log 5+z\log 2 = \log 3\implies z = \frac{\log 3 - \log 5}{\log 2}$.

@Rememberme stretch your hand out as if your fingers formed the corners of a pentagon. :)

Khallil Benyattou

02:14

$\mapsto$

Mary Star

Does someone know if the book algebra of van der waerden is available online?

Dave

@SohamChowdhury thanks for explaining ! :) i will give it a try tomorrow when i awaken from slumber !

Kaj Hansen

02:43

Hello everyone

Karim Mansour

hi kaj

Mike Miller

Morning.

Kaj Hansen

Hey @MikeMiller. You mean evening?

Mike Miller

nah

how's things

Kaj Hansen

03:02

Not too bad @Mike. I'm just relaxing and listening to some music right now while I browse main

Soham Chowdhury

@MaryStar Library Genesis is your friend.

Mike Miller

03:26

sounds like a god night @Kaj

Soham Chowdhury

04:03

@Balarka, I sent a little progress report to SB.

I asked him for permission to stop after finishing the first five chapters of Munkres.

Soham Chowdhury

04:40

four*, actually.

Kaj Hansen

@SohamChowdhury, who is SB?

Soham Chowdhury

A prof who has (unbelievably) agreed to mentor me.

Kaj Hansen

That's awesome! Where do you live?

If you don't mind me asking

Soham Chowdhury

Kolkata.

Nah, not overly into secrecy and all that.

Kaj Hansen

Ah, I live in Georgia myself. Southeastern USA.

Soham Chowdhury

04:42

You attend the uni where Ted used to teach?

Kaj Hansen

Indeed

A good friend of mine from high school visits Kolkata sometimes since his family's from there. He's working for NASA right now in Houston.

Soham Chowdhury

Alex told me you're into metal, so take a look at this. (Turn your speakers down, this is not much more than a proof of technical skill :P)

@KajHansen Cool.

Kaj Hansen

Holy. Crap. That guy has skill.

Soham Chowdhury

Do you play anything?

Kaj Hansen

mhmm, I played French horn for 7 years.

Soham Chowdhury

04:45

Ah. Which reminds me, I have to finish reading A Devil To Play.

Kaj Hansen

I stopped playing once I started college though since it takes up too much time, and it's not much fun if you don't have a group to play with.

I love all sorts of genres, but it's true that metal is one of them :P

Soham Chowdhury

It's about a horn player who picks the horn up after ~15 years and resolves to perform in public after one year.

@KajHansen Oh, I have been listening to this lately.

Kaj Hansen

I could imagine doing that I suppose. The biggest thing is that the muscles in your lips get weak without practice.

Soham Chowdhury

I play guitar, and our patron saint Segovia did a beautiful version of the Chaconne.

@KajHansen yeah, he talks about that a lot. It's a very funny book.

Especially all the bits about water keys :P

Kaj Hansen

So I think I could still play reasonably well after having not played for 4 years, but since my muscles are weak I wouldn't be able to play very long.

Listening to your link now :)

Soham Chowdhury

04:48

First one?

Kaj Hansen

Second. Chaconne

Soham Chowdhury

The first one is the original violin version.

Kaj Hansen

Check out this metal album I recently discovered. I'm starting you out at 6:30 just because the shredding there is incredible: youtube.com/watch?v=upv6_4WT6hY&feature=youtu.be&t=390

Soham Chowdhury

Buffering. Meanwhile, when you're done, give this a listen. It's almost like falling water. <3

Ah, your link is nice.

Kaj Hansen

I love this piano

Added to favorites

Remember me

04:56

@Soham,@Kaj Hello

Soham Chowdhury

@Kaj: And this, also by Ravel. There's a nice tolling Bb meant to evoke church bells. It's something like "the last thoughts of a prisoner before he's hanged", essentially.

@Rememberme Hello, @Rem.

Kaj Hansen

Hey there @Rememberme. I know I've asked you this a ton before, but remind me of your old username?

Soham Chowdhury

Sayan

Kaj Hansen

Yes!

Remember me

haha

Hey @AlexC

user147690

04:56

Hey @Rem @Kaj @Soham

Kaj Hansen

And @Alex here was committing before? Sorry, I've been away for a while :P

Soham Chowdhury

Hey man. Where've you been?

Yes, I found out his old username a while back.

user147690

@KajHansen DN -> Committing -> Incurrence -> Alex

Soham Chowdhury

Did you complete the challenge?

user147690

I certainly did not

user147690

04:57

haha

Kaj Hansen

@AlexClark, you came just in time. If you liked Kalmah, I recently found another metal album I like a lot

user147690

Although technically I am not half way through the prescribed time

Remember me

I need to talk to @Tobias where is he ?

user147690

@KajHansen Oh sweet, hit me with it

Remember me

@AlexC My challenge is similar to yours ... Let me show you

user147690

04:58

@Rememberme Sure

Kaj Hansen

@AlexClark, here it is. I just shared it with Soham. I'm starting you at 6:30 just because the shredding is incredible there: youtube.com/watch?v=upv6_4WT6hY&feature=youtu.be&t=390

Soham Chowdhury

@Rememberme ?

user147690

@SohamChowdhury Pretty much, don't set a challenge that is long, unless it is your top priority

user147690

@SohamChowdhury Uni destroyed it, but in a good way, since I didn't have great direction when I started it

Remember me

Here @AlexC @Soham http://chat.stackexchange.com/transcript/message/22662965#22662965 ..
It is pretty long...

@Soham Started rings?

user147690

05:01

@KajHansen Sounds good man, did Soham ever show you NeO?

Soham Chowdhury

"I have this craze of doing homological algebra which I yet don't know what it deals with"

cough

@Rememberme yes.

user147690

@Rememberme I don't like it sorry haha

Kaj Hansen

Nope, but he did show me this: youtube.com/watch?v=MQtkhWow6G4

Remember me

You dont like what @AlexC ?

Soham Chowdhury

Your "plan".

user147690

05:02

@Rememberme Your plan

Remember me

Oh... any reasons ?

user147690

I put wayyy more planning into it than you have, and mine failed terribly

Soham Chowdhury

It seems you have no idea what you're doing and are fascinated by "big words", as Balarka says.

Remember me

Yes thats one thing.. But lets see

Soham Chowdhury

Just my 2c.

user147690

05:03

If you don't know what homological algebra is, which I don't, you won't be able to plan to do it

Remember me

Thats what I have been feeling....

user147690

Yes, I think he showed me that one as well. If you can, it's a pretty long build up, but I think the build up is vital, give this a listen

https://www.youtube.com/watch?v=-wa4zqBwPVE&t=350

Remember me

@AlexC I require this in a proof...
Any set when given the discrete topology its sets become both open and closed right?

user147690

Discrete metric space?

Kaj Hansen

No problem. Listening now @AlexClark

user147690

05:05

They are progressive, but this is my favourite part of a song ever

Kaj Hansen

Indeed @Rememberme.

Remember me

No the discrete topology... As in the topology of all subsets of the set

Okay then my proof is spot on

Thanks @Kaj

Kaj Hansen

Every singleton set is open, so by taking unions, we see that $\tau_X = \mathcal{P}(X)$

And every complement of a given subset is also in $\mathcal{P}(X)$ of course.

Remember me

Also its complement is open implies set is also closed (singleton)@Kaj isn't it

Kaj Hansen

Indeed. So every subset is open and closed.

user147690

05:08

Dear god @Rememberme Homological algebra starts at page 761 of Lang's algebra...

Remember me

And therefore only singleton sets can be connected and therefore making the set totally disconnected @Kaj

Kaj Hansen

Man, I'm loving the guitar work at 8:39 @AlexClark

That's right @Rememberme

user147690

@KajHansen They've got two main guitarists and a violinist, as well as a basist

Remember me

I know @AlexC ... That is why I want to talk to tobias if he gives me some kind of an insight on what we do in it

Kaj Hansen

Here's an interesting thing to think about @Rememberme : You can add a single point to a set with the discrete topology in such a way that the resulting space is connected.

Remember me

05:10

Thats pretty interesting @Kaj

Kaj Hansen

You can pause and think about it if you want. Otherwise, you can read my post here if you're interested: math.stackexchange.com/questions/1302187/…

Remember me

@AlexC so you like music ... it seems

Have you ever heard of POTF?

user147690

Poets of the Fall?

Remember me

Yup

user147690

@Rememberme I've heard a few songs, I'm into heaps of things, but they aren't really my style

Remember me

05:12

SO whats your style:p @AlexC

user147690

Hmmm I guess I still like posthardcore, hardcore, metalcore, which are really where the roots of my current taste lie, but I am more into melodic death metal and some harder progressive sounds atm

user147690

@Rememberme I think if you are to make a plan, make one where you include each step, since your plan seems more like an end goal only

Remember me

Okay then you might like .......
Eluveitie @AlexC

Kaj Hansen

You like post hardcore @AlexClark ? Do you listen to La Dispute? MewithoutYou?

user147690

@KajHansen Neither

user147690

05:15

@Rememberme folkcore xD

Kaj Hansen

Oh boy. You're in for a treat. Give me a sec

Also, for progressive metal, I can link you to some Protest the Hero if you're unfamiliar

user147690

Also unfamiliar

Remember me

Yes @AlexC So i will be focusing more on topology and algebra in the next few months ..... try to finish DF(Till Algebraic geometry) and finish Munkres

Soham Chowdhury

@KajHansen so much tapping <3

user147690

Oh haha, these are american bands, I didn't know there was any real scene for post hardcore in america

Soham Chowdhury

05:16

Do you know Animals as Leaders, @Kaj?

user147690

Australia is really into post-hardcore, so we have tonnes from cities all around me

Remember me

Nice name @Soham

Kaj Hansen

@AlexClark :

Here's a sample of La Dispute: https://www.youtube.com/watch?v=ppjlFriEOeM
Here's a sample of MewithoutYou (they are in my top 5 favorite bands): https://www.youtube.com/watch?v=9Yi9iCj15i4

user147690

Oh last I heard you didn't like post hardcore haha

user147690

Or maybe it was just chugging

Soham Chowdhury

05:18

And, both of you, give this a look (and a listen!)

Blotted Science - Cretaceous Chasm (HD)

►

Synced to the King Kong canyon scene.

Instrumental tech-death.

Kaj Hansen

Here's a sample of Protest the Hero: youtube.com/watch?v=rvFyw6uOoYc

@AlexClark

user147690

Giving all a listen, thanks guys

Kaj Hansen

I don't mean to bombard you with so many links. I'm just really passionate about my music :P

user147690

Nah man, I sometimes get depressed when I run out of music

Soham Chowdhury

@Kaj, look at the last link.

user147690

05:19

So it really helps

Kaj Hansen

@SohamChowdhury, watching now :D

Soham Chowdhury

"Ta-da, ta-da, ta-da dum dum dummmm . . ."

Kaj Hansen

Wow, I'm actually enjoying this a lot. Clever title too.

Soham Chowdhury

Ron Jarzombek is mad.

The guitar playing is probably meant to evoke insects on your back. :P

Remember me

@AlexC What book did you use for Galois theory?

user147690

05:22

@Rememberme I'm using Morandi

Kaj Hansen

Idk if this is going to have any vocals, but if not I'm totally okay with that. I like instrumentals.

@Rememberme, I was really fond of Artin 2nd edition myself. That's where I learned it from, but it's an introduction to algebra with a chapter on Galois theory, rather than a text on just Galois theory. I don't know what you're looking for in that regard.

user147690

Yeah I used Artin as well, great book

Remember me

I just looked at Artin...@Kaj
Even I thought it will be all about Galois theory but its about linear algebra, Field theory and few bits of Galois

I want a only specific Galois theory book

Though DF has Galois theory but is it enough?

user147690

Well Morandi then

user147690

But you'll want ring and field theory exposure

Remember me

05:26

Yes DF's my savior!! :)

user147690

But you didn't look at Morandi?

Remember me

DF for field and rings @AlexC and Morandi for Galois

user147690

Actually I would probably use Artin for field and rings

user147690

D&F is honestly mostly for exercises in my opinion

Remember me

Hmm... Thats put in dilemma again then

user147690

05:29

Go Artin -> Morandi

Remember me

Okhay

And DF for exercises

user147690

Yep

Remember me

Okay...

user147690

Also while you are doing munkres, it wouldn't hurt to work on Kreyszigs functional analysis

Remember me

@AlexC I always wanted to ask this...
Did you give your talk which you were supposed to give.. As in you were very enthusiastic about it...

Kaj Hansen

05:31

What exactly does functional analysis study?

Remember me

I might do functional analysis from Rudin(basic that is)@AlexC

user147690

@KajHansen Metric spaces -> Normed spaces -> Inner product spaces

Kaj Hansen

Interesting

user147690

Very topological, atleast at first

Remember me

But lets see.. Analysis was never in my list :p@AlexC

But I got to do it....

user147690

05:33

It should be

user147690

It's very beautiful

user147690

Although there isn't much analysis love in here haha

user147690

@Rememberme I have anxiety, and I may have freaked out into being over perfectionist until it was too late

Remember me

If I can finish algebra and Topology in the next 6 months (lets say) Then I might reword back to complex Analysis @AlexC

Because I might happen to require it somewhere

user147690

I shall go now and get to work, Protest the hero seems pretty awesome @Kaj, thanks for that, listening to the full album for Volition to get a bigger taste, and then I listen to your one @Soham

user147690

05:35

@Rememberme Hmmm Complex analysis will definitely help with some of the algebra questions

Kaj Hansen

@AlexClark, I recommend checking out the full album "Fortress" for Protest the Hero. I think you'll like it.

user147690

Atleast it did when I took algebra lmao, he kept giving questions that were very relevent

user147690

@KajHansen Alright, will do, thanks man :). Cya later

Remember me

Hmm.

Kaj Hansen

Great catching up with you @AlexClark. See you around.

Remember me

05:41

Hello@MikeMiller

Kaj Hansen

05:52

It hasn't switched over in chat yet, but I uploaded a new profile picture on here :D

Soham Chowdhury

06:09

The old one (which is what I'm still seeing) looks like a mugshot.

Kaj Hansen

haha, I'm not sure why it takes so long to switch over in chat @SohamChowdhury

What about when you visit my profile?

Remember me

06:35

The pic is different @Kaj

Balarka Sen

@Soham cool

hi @KajHansen

@Remember Morandi is too advanced for you. do galois theory from D-F.

Remember me

06:51

Okay @Balarka

Balarka Sen

do basic galois theory, then we'll see if you want to read further. :)

Remember me

Okay :)

@Balarka How is it in Kolkata ... Floods??

Balarka Sen

don't ask. it's all underwaters.

Remember me

My grandparents house was flooded

Schools are closed?

Balarka Sen

almost.

Remember me

06:54

Hmm.. today here it dropped to 19 degrees

Early morning

Okay bye gtg....

Kaj Hansen

07:18

Hey @BalarkaSen

Balarka Sen

what're you upto, @Kaj?

Kaj Hansen

Just relaxing and listening to music @BalarkaSen. I'm also trying to find whatever .pdfs I can of textbooks I own because I like to also have a digital copy if I'm out somewhere.

Balarka Sen

nice. I don't usually bother for pdfs once I have hard-copies.

Kaj Hansen

07:34

Yeah, I prefer hard copies definitely @BalarkaSen. But I can't lug my whole library everywhere :P

Balarka Sen

True.

iwriteonbananas

heyho

Balarka Sen

hi

@iwriteonbananas I have pinged you stuff on the alg top room

iwriteonbananas

ah, didnt see

you have any idea where i might find algebraic topology oral exam archives/questions?

Balarka Sen

google for alg top qual exams?

iwriteonbananas

07:48

what's a qual exam?

nice, there are some good search results

Balarka Sen

@iwriteonbananas things you have to take after grad school to pursue phd, apparently.

iwriteonbananas

i see

Balarka Sen

here

iwriteonbananas

yeah, just found that. what a gold mine

Balarka Sen

Kirby's question about using vk-t to compute \pi_1 of hawaiian earring is cool

iwriteonbananas

07:52

true

i need to recall vk-t...

Balarka Sen

try to recall the proof as well

iwriteonbananas

@BalarkaSen explain :P

Balarka Sen

what, the proof?

iwriteonbananas

what you were referring to

Balarka Sen

oh, you mean Kirby's question?

iwriteonbananas

07:57

yeah

Balarka Sen

well, an obvious choice for using vk-t on the earring are the circles plus bit of something else. however, all of these open sets are bound to contains small circles in their intersection. in fact, they'll contain a shrunken version of the hawaiian earring inside their intersection

so you have to know \pi_1 of the earring in the first place to apply vk-t

not a promising thought

iwriteonbananas

right

what can we say about the homotopy type of the dunce cap?

Balarka Sen

unrelated : note that wedge of $n$ circles is included in $\mathcal{H}$. in fact, there is an inverse system of such wedge of circles

this induces an inverse system at the level of fundamental groups.

so this gives you a hom $\pi_1(H) \to \varprojlim F_n$, the latter being inv. limit of free groups

turns out this is an injection.

iwriteonbananas

@BalarkaSen "inverse system" ?

Balarka Sen

roughly, a chain of homs $\cdots \to A_{n+1} \to A_n \to \cdots \to A_2 \to A_1$.

iwriteonbananas

08:04

ok

Balarka Sen

inverse limit is the limit that lies at the very end of the chain (don't take this literally)

@iwriteonbananas dunce cap means a hell lot of things

iwriteonbananas

ok :P

Balarka Sen

one being the identification space obtained from the polygon $aaa\cdots aa$

iwriteonbananas

@BalarkaSen i mean the space obtained from $\Delta_2$ by identifying edges in a way that preserves the orientation of the edges

Balarka Sen

other being the identification space obtained from the polygon $aaa^{-1}$

iwriteonbananas

08:06

@BalarkaSen yea that one

Balarka Sen

it's contractible

iwriteonbananas

oh really?

Balarka Sen

you paste a 2-cell to the circle by the word $aaa^{-1}$. this is the same as attaching by $a$.

so your space is a disk/htpy equiv, which is contractible

iwriteonbananas

eh, why is it a disk?

Balarka Sen

attach a disk to a circle by the word $a$. this is just attaching by the id map.

@iwriteonbananas are you having trouble seeing why attaching w/ $aaa^{-1}$ is the same as attaching w/ $a$?

iwriteonbananas

08:09

yeah

Balarka Sen

take a disk, take a circle.

iwriteonbananas

taken

Balarka Sen

attach the bd of the disk to the circle by the attaching map which pastes the one-third to the whole circle counterclockwise, the second one-third to the whole circle counterclockwise again, and the third one-third to the whole circle clockwise.

this is the geometric realization of $aaa^{-1}$

iwriteonbananas

ok

Balarka Sen

now concentrate on the second one-third and the third one-third. it's like looping around all over the circle once, and then looping all over the circle in the opposite direction.

so you can homotope this to a point.

iwriteonbananas

08:13

i see

Balarka Sen

so the attaching map can be homotoped to the map which sends the first one third to all the circle, and second and third one-third to a pt.

this is a disk

iwriteonbananas

okok

how do you answer "find all 2-fold coverings of figure 8?

Balarka Sen

god i don't know.

there're millions of index 2 subgroups of F_2

iwriteonbananas

yeah :(

Balarka Sen

the next question is nice

iwriteonbananas

08:24

right

torus and $S^1\vee S^1\vee S^2$

for homology

Balarka Sen

exactly. and why aren't they htpy equiv?

iwriteonbananas

not sure about the other one

because torus has abelian fundamental group

the other space doesnt

Balarka Sen

yea, well, equivalently, the torus has a simple universal cover. the other space doesn't.

iwriteonbananas

what's a simple universal cover?

Balarka Sen

the univ. cover. for the other space is the cayley graph of F_2 with a sphere attached at each point.

@iwriteonbananas i mean, it's an easy universal cover. R^2.

iwriteonbananas

08:26

oh ok

Balarka Sen

but this is the same as your argument because proving things with \pi_1 is essentially the same as proving things with the universal cover

htpy equiv spaces have htpy equiv universal cover

iwriteonbananas

yeah nvm

:P

do you know an example for (2)?

oh wait

Balarka Sen

yes, there is a standard example

iwriteonbananas

mhh cant think of one

Balarka Sen

$\Bbb P^2 \times S^3$ and $\Bbb P^3 \times S^2$.

universal covers are now the same. as $\pi_n(\widetilde{X}) = \pi_n(X)$ for $n > 1$ where $\widetilde{X}$ is the universal cover of $X$, this is verified for higher htpy groups

for $\pi_1$, note that the universal cover is two-fold. so in both cases, $\pi_1 = \Bbb Z/2$

iwriteonbananas

08:30

oh didnt know that

Balarka Sen

it's an easy fact. read it up from Hatcher sometime

iwriteonbananas

ok

Balarka Sen

it makes $\pi_n$ into a $\Bbb Z[\pi_1]$ module, iirc

because $\pi_1$ acts on universal cover, and $\pi_n$ of space is the same as $\pi_n$ of univ. cover

iwriteonbananas

the first space has trivial 2nd homology, the second space has 2nd homology $\Bbb{Z}$, right?

Balarka Sen

i haven't computed. how do you know?

homology of product is not product of homologies, remember that

iwriteonbananas

08:33

$H_i(S^n\times X) = H_i(X) \oplus H_{i-n}(X)$. you showed me

Balarka Sen

ohh, aha

clever. yes.

iwriteonbananas

"compute $H_*(\Bbb{C}P^n)$"...cellular homology, right?

Balarka Sen

yah.

iwriteonbananas

it has one cell in even dimensions less than $n$ if i recall correctly

Balarka Sen

CP^n has a cell in even dimension and has no cells in odd dimensions

iwriteonbananas

08:36

yeah

Balarka Sen

so H is Z in even dimension and 0 in odd dimension

iwriteonbananas

yep

"Why do you have the Fundamental Theorem of Algebra under algebraic topology in
your syllabus?" ?

Balarka Sen

it's a corollary of theorems about winding numbers and homotopy

iwriteonbananas

yeah

i gotta do some complex analysis exam prep. see ya later

Balarka Sen

bubye

Chris's sis the artist

09:24

@robjohn hey. If you wanna take a look over my solution to that problem let me know. I just prepared the whole file for a magazine, as a proposed problem. I need to recheck it again to make sure all is fine, no small mistakes, and then I send it.

I'll be back in 30-60 min, I leave in a couple of minutes.

user147690

10:22

@KajHansen Mewithoutyou is actually really good, thanks man!

Chris's sis the artist

Back.

@robjohn If you find a solution, don't post it here anymore, but keep it for sending it to the journal I proposed the problem. I'll tell you privately where once it's published.

robjohn

@Chris'ssistheartist I am working on it in between other things. I think I have made significant progress. I have a bit more to go though.

Chris's sis the artist

@robjohn It's perfect, but keep a possible solution for the journal. I'm almost sure they'll accept my problem, it's too amazing.

@robjohn (in case you would like to publish a solution in journal)

@robjohn Or others might take it from here (the solution) and publish it there, so pls don't post a solution. :-)

@robjohn I also developed other similar versions that even Ramanujan himself would love a lot. :D

zed111

11:17

@BalarkaSen hey

Balarka Sen

hello

zed111

You know about Galois theory?

Balarka Sen

I happen to have studied Galois theory, yes

zed111

where do I read it from, for an intro?

Balarka Sen

Dummit-Foote is nice, as an introduction to the theory.

Remember me

11:26

Hello@Balarka

Balarka Sen

hi

Remember me

@Balarka I have a doubt....

You free?

Balarka Sen

almost

zed111

@BalarkaSen Hows Gallian last chapter?

Remember me

Okay...
How am I supposed to show that an infinite set is connected in the cofinite topology ?I am not able to get any ideas

Balarka Sen

11:29

I don't know, haven't studied Gallian, @zed111

Huy

@Rememberme: Use the definitions.

Balarka Sen

@Rememberme Use the definition of connectedness

Remember me

Also can I extend it to cocountable topology?

Balarka Sen

Sure

Remember me

As in does the same thing apply to cocountable topology

Balarka Sen

11:31

if you can't prove that an infinite set is connected in cofinite topology, you aren't paying attention to the defn of connectendess

Remember me

Okay.. Let me try...

Balarka Sen

hi @Huy

how's things?

Huy

hi @BalarkaSen

@BalarkaSen: Reading about Riemannian metrics and Lie groups a bit. I don't like it too much so far to be honest.

Balarka Sen

I have heard that Riemannian manifolds are good stuff. I only know about those from my meager knowledge of geodesic metric spaces, unfortunately.

don't know any Lie theory either

oh, I forgot to ask you -- did you figure out that Hopf fibration problem, @Huy?

Huy

@BalarkaSen: I solved some easy exercises related to it, but I'll have to do the harder ones at some point too. I'll ask you when I'm starting. :P

Balarka Sen

11:40

:P you can ask me if you want to visualize the whole thing, but I probably won't be of much help if you shower me with ten thousand huge formulas for the bundle

Remember me

Done @Balarka

@Balarka Should I write down the proof here?

Balarka Sen

if you don't want to, don't bother

Remember me

Okay I am writing it... :)

Balarka Sen

I don't want to check it right now

Remember me

OK

Balarka Sen

11:46

@Huy The Hopf fibration is of interest because you can derive as a corollary that there are infinitely many "distinct" (rigorous version : distinct upto homotopy) maps $S^3 \to S^2$. This is hard to visualize indeed.

For higher dimensions, maps $S^n \to S^m$ modulo "small perturbations" are more and more wacky. It's still an open problem to determine their behavior (the so-called "homotopy groups of spheres")

Soham Chowdhury

@Rememberme an infinite set can be decomposed as the disjoint union of two sets, at least one of which must be infinite and therefore not open. correct?

Kartik Watwani

people lets do this iit question

Balarka Sen

@SohamChowdhury at least one must be infinite, thus the other is not open.

Kartik Watwani

how many solutions does $\pi^x=-2x^2+6x-9$ have?

Soham Chowdhury

@BalarkaSen wait. I wrote "the other one is not open".

Balarka Sen

11:59

you didn't.

Kartik Watwani

someone there to solve this?

Transcript for

$Mathematics$ Mathematics