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Mike Miller
Mike Miller
1:09 AM
Hi @AlexW
 
Prototank
Prototank
1:29 AM
hello
 
Alex Wertheim
Alex Wertheim
Hello there, @Mike.
How goes it?
 
Mike Miller
Mike Miller
It's alright. Got a cold, which isn't so hot.
 
Alex Wertheim
Alex Wertheim
Aw, I'm sorry to hear that. How long've you had it?
 
Mike Miller
Mike Miller
About 24 hours now.
 
Alex Wertheim
Alex Wertheim
Damn. Well, here's hoping it's over the 24 hour virus variety, then. Feel better!
 
Mike Miller
Mike Miller
1:38 AM
Haha, thanks.
I still got some work done today, luckily. Got someone to cover my teaching tomorrow - the second day is always the worst for me.
 
Alex Wertheim
Alex Wertheim
That's good, I always find it near impossible to study well when I'm sick. What are you working on?
 
Mike Miller
Mike Miller
I'm in the algebraic geometry course right now, I'm doing exercises for it. I also need to write up the first bit of my talk, but I've given myself until Monday to do that.
 
Alex Wertheim
Alex Wertheim
Neat. You're working out of Hartshorne, I'm guessing?
 
Mike Miller
Mike Miller
Yeah, sort of. The course is going much slower than the book, and I'm not taking the sequel (too busy with other things), so I'm getting less out of it than I hoped.
 
Alex Wertheim
Alex Wertheim
That's a shame, I'm always frustrated when my expectations for a course aren't met. What are you taking in the spring?
 
Mike Miller
Mike Miller
1:44 AM
227, though I'm mostly hoping to learn some new things from it, as I know most of the "stated" topics, I'll be participating in the topology seminar (I think I heard mutterings we'll be talking about the Vafa-Witten equations?), and I'm doing a reading course with Manolescu.
 
Semiclassical
Semiclassical
ooo, witten
 
Mike Miller
Mike Miller
Shoo, physicist!
(PS: I forgot to mention the other day. Formal prerequisites aren't really a thing. If you want to take something, and you think you have the background knowledge, you can take it. There's no demand that you have taken the geometry qual or 225C to do 227 if you, like, know the contents of 225C...)
 
Semiclassical
Semiclassical
lol
 
Mike Miller
Mike Miller
@Semiclassical At some point I'd like to learn some of the "physical" motivation for lots of these topics; why a physicist might believe they're true. But that's a long way away - down in a future where I've time.
 
Alex Wertheim
Alex Wertheim
Ah, excellent. Manolescu does very cool work. Are you thinking about him as an advisor? He does geometric topology, if I recall correctly.
 
Mike Miller
Mike Miller
1:47 AM
Yes and yes. Doing a reading course is essentially step 1 to getting an advisor.
The course is basically, "Here's 4 topics I want all of my students to know. Here is a combined 12 references on these topics. Now go, and learn!"
 
Alex Wertheim
Alex Wertheim
Haha, thanks for the heads up. That is good to know, though in truth, I have an exceptionally weak topology background, so I'd really like to learn these things. I actually love taking courses, I just am concerned about whether the workload will be too much, and retention as well.
 
Semiclassical
Semiclassical
witten stuff is pretty much above my pay-grade, to be honest
 
Alex Wertheim
Alex Wertheim
Plus, there's only so much time you can devote to courses before you have to get to the more important task of research, I suppose.
That's awesome though. Do you know already what you'll be reading out of?
 
Semiclassical
Semiclassical
i don't do particle / high-energy physics, so the context i know is less esoteric
 
Mike Miller
Mike Miller
well, he's basically a mathematician that doesn't waste time proving things, @Semiclassical. that lets him develop more ideas that we can then steal whole-cloth :)
so i strongly appreciate him!
 
Semiclassical
Semiclassical
1:50 AM
lol, yep
 
Mike Miller
Mike Miller
Yeah, @AlexWertheim, I have all but two of them as ebooks on my chromebook right now. I'd have started already but I've still got stuff to do this quarter.
 
Semiclassical
Semiclassical
actually, some of the stuff we used in our stuff (namely picard-fuchs equations and monodromy) is stuff we knew of b/c witten used that stuff. (obv. mathematicians knew about it a lot longer)
 
Mike Miller
Mike Miller
Probably I'll start a day or so after my talk. :P
@Semiclassical it's really great the symbiotic relationship math and physics is developing in recent years.
 
Semiclassical
Semiclassical
well, it's been there in different periods
there's a reason why physicists are the ones who call the divergence theorem "gauss's law", after all
 
Alex Wertheim
Alex Wertheim
Awesome. It's a shame I'll just narrowly miss that talk :(
 
Mike Miller
Mike Miller
1:52 AM
right
@AlexWertheim: I'll send you my notes after I'm done with them, if you'd like. They'll be fairly detailed because I really doubt I can remember all the details on the fly.
 
Semiclassical
Semiclassical
though sometimes you do stumble upon 'new' ideas/applications
 
Alex Wertheim
Alex Wertheim
That'd be great, @Mike. The detail will be extremely helpful/necessary, haha.
 
Semiclassical
Semiclassical
even in the fairly simple realm i deal with
 
Prototank
Prototank
Suppose $f$ is meromorphic on $\mathbb{C}^*$ and suppose $f$ has no zeroes. Then $f$ is a constant. Is this true?
 
Mike Miller
Mike Miller
What is $\Bbb C^*$? The extended plane, or the punctured plane?
 
Prototank
Prototank
1:57 AM
extended
 
Mike Miller
Mike Miller
Then yes. Theorem: every meromorphic function on $\Bbb C^*$ is a rational function.
@AlexWertheim: I've asked you a hundred times, but: visit is 3/11-3/12, right?
 
Alex Wertheim
Alex Wertheim
Haha, no worries @Mike. I owe you big time after asking you about eight thousand dumb questions. That's right.
 
Mike Miller
Mike Miller
I've heard about eight thousand questions, but no dumb ones yet.
 
Semiclassical
Semiclassical
the piece of math which i wish i knew something about is index theory
mostly so i could really understand what all this stuff in topological insulators is about
 
Mike Miller
Mike Miller
try bleecker-bavnbek
 
Semiclassical
Semiclassical
2:07 AM
hmm, that's a promising title
 
Mike Miller
Mike Miller
it's written for someone with only, say, a background in basic facts about banach and hilbert spaces
but who's willing to learn a lot
and is pretty much self-contained
 
Semiclassical
Semiclassical
nod
and i'd suspect that at least some stuff in there would be familiar as physics if not as math.
 
Mike Miller
Mike Miller
it's sort of stratified into a mathematical half and a more physical half
the physical half was rough for me
in the sense that i stopped reading it
 
Semiclassical
Semiclassical
heh
 
The Emperor of Ice Cream
The Emperor of Ice Cream
Is it a common expression to say an exam "raped you"?
?
in english?
 
Semiclassical
Semiclassical
2:14 AM
eh, the phrase i've heard is 'raked me over the coals'
 
Mike Miller
Mike Miller
it's not uncommon, but it's also not really acceptable in polite society
 
The Emperor of Ice Cream
The Emperor of Ice Cream
So you have heard the phrase "damn, that test raked me over the coals"?
what do you mean with polite society?
 
Semiclassical
Semiclassical
i'm not sure how common in day-to-day usage it is, but it's a known idiom
 
Alex Wertheim
Alex Wertheim
Does anyone have a satisfying intuitive interpretation of why $S_{6}$ has a nontrivial outer automorphism group? I've seen the construction, and understand the proof. I also understand the proof of why the automorphism group is strictly inner for $S_{n}, n \neq 2, 6$. That being said, I've never had a good understanding of what makes $n=6$ combinatorially/geometrically special.
 
Semiclassical
Semiclassical
and comparing a painful exam to rape is, well, not something i'd do in front of my parents/authority figures/people who i think would be sensitive to it
(it's not a phrase i'd use in general, but that's me)
 
The Emperor of Ice Cream
The Emperor of Ice Cream
2:17 AM
oh ok
So you would have a bad impression if someone used it with his friends?
 
Semiclassical
Semiclassical
well, i'm not sure how much impression it'd make on me. but i'd expect some would definitely think ill of it
 
The Emperor of Ice Cream
The Emperor of Ice Cream
k
It's a really common expression in my country.
 
Semiclassical
Semiclassical
2:32 AM
@MikeMiller: i should note that, for my purposes, i'm interested in the index theory of difference equations rather than differential equations. (which i'd think would be more elementary)
 
Mike Miller
Mike Miller
2:48 AM
Dunno anything about that.
 
Mary Star
Mary Star
3:23 AM
Hello!!

A mass of 1 kilogram ($1$ kg) that lies at $(0,0,0)$ is hanging from ropes tied at the points $(1,1,1)$ and $(-1,-1,1)$. If the gravitational force has the direction of the vector $\overrightarrow{-k}$ which is the vector that describes the force along each of the ropes?

Could you give me some hints??
 
infinitesimal
infinitesimal
Making a sketch may help.
 
anon
anon
force diagram
 
Mike Miller
Mike Miller
nice old avatar
 
infinitesimal
infinitesimal
Classic
 
Pedro Tamaroff
Pedro Tamaroff
@anon Hello.
 
anon
anon
3:28 AM
hwello
 
The Emperor of Ice Cream
The Emperor of Ice Cream
@anon hello
 
Pedro Tamaroff
Pedro Tamaroff
@anon Do you know a proof of the implication $S^2-U$ connected $\implies U$ simply connected for $U\subseteq S^2$ open?
 
anon
anon
no. I assume you would contraposition it.
@TheEmperorofIceCream hello
 
Mary Star
Mary Star
@infinitesimal @anon Is it as followed??
 
Pedro Tamaroff
Pedro Tamaroff
@anon The Riemann mapping theorem gives that $U\subsetneq S^2$ simply connected implies $S^2-U$ connected.
 
Mary Star
Mary Star
3:32 AM
i.stack.imgur.com/1Sdel.png @infinitesimal @anon
 
Pedro Tamaroff
Pedro Tamaroff
But I couldn't get the other direction.
I wonder if I'm missing something very obvious.
 
Mike Miller
Mike Miller
pedro: explain your argument, exactly?
 
Pedro Tamaroff
Pedro Tamaroff
@MikeMiller Take $U\subsetneq S^2$ (connected) simply connected.
By Riemann, $U$ is homeomorphic to $B(0,1)$.
 
Mike Miller
Mike Miller
yes?
 
Pedro Tamaroff
Pedro Tamaroff
OK.
 
Semiclassical
Semiclassical
3:35 AM
@MaryStar: two hints, and that's all i'll say: 1) the two force vectors should each point from the location of the mass to the point that they're tied. 2) the sum of all three force vectors should be zero since the mass is in equilibrium.
 
infinitesimal
infinitesimal
Where is (-1, -1, 1) @MaryStar?
 
Pedro Tamaroff
Pedro Tamaroff
@MikeMiller Right. You got me.
 
Mike Miller
Mike Miller
hehehe
 
anon
anon
@PedroTamaroff wait if you put an open cap around the north pole and an open cap around the south pole and call that U isn't S^2-U connected but U is not?
 
Pedro Tamaroff
Pedro Tamaroff
@anon Right, when I say "simply connected" I do assume connected.
 
Mike Miller
Mike Miller
3:38 AM
simply connected means $\pi_0 = \pi_1 = 1$
 
Pedro Tamaroff
Pedro Tamaroff
@MikeMiller Yes, yes.
 
anon
anon
one of those is the fundamental group. the other is the set of connected components? don't remember what they are.
anyway, the implication is S^2-U connected => U simply connected, but with the example we have S^2-U connected and U not simply connected, so what's up?
 
Mike Miller
Mike Miller
yes
path components
 
Pedro Tamaroff
Pedro Tamaroff
@anon Your set is not connected.
(Note also that in $S^2$ (or $\Bbb R^n$, $S^n$) connected and path connected for open sets is equivalent)
 
Mike Miller
Mike Miller
or on a manifold, yes
 
Mary Star
Mary Star
3:41 AM
Let $F_1$ and $F_2$ be the two force vectors.

At x-axis we have $F_{1x}+F_{2x}=0$.

At y-axis we have $F_{1y}+F_{2y}=0$.

At z-axis we have $F_{1z}+F_{2x}-mg=0$.

Is this correct?? @Semiclassical @infinitesimal
 
anon
anon
@PedroTamaroff that's what I'm saying, U is not connected even though the implication says it should be
 
Pedro Tamaroff
Pedro Tamaroff
@MikeMiller You and your fancy manifolds.
 
Mike Miller
Mike Miller
$\pi_n$, as a set, is the number of based maps up to based homotopy $(S^n, *) \to (X, x_0)$; maps from $(S^0, *)$ are just maps of the leftover point; basepoint preserving homotopies are just paths
 
anon
anon
surely I am not getting something ofc
 
Semiclassical
Semiclassical
right. but the components of $F_1$ and $F_2$ need to be in the correct proportion to point in the right directions.
 
Pedro Tamaroff
Pedro Tamaroff
3:42 AM
@anon Oh. Sorry.
Right, so what I should have written is this, @anon.
Assume $U$ is open connected in $S^2$.
Then $U$ is simply connected if and only if $S^2-U$ is connected.
 
anon
anon
k
 
Pedro Tamaroff
Pedro Tamaroff
Although it should be true that $\widetilde{H_0}(S^2-U)=0\iff H_1(U)=0$.
 
anon
anon
if U ain't simply connected then put a noncontractible loop in it and argue maybe that S^2-U has one component on one side of the loop and one on the other?
 
Mike Miller
Mike Miller
that is indeed true, since those are isomorphic (as long as $S^2 - U$ isn't fucked up)
 
Mary Star
Mary Star
@Semiclassical Do you mean that the signs are wrong??
 
Mike Miller
Mike Miller
3:45 AM
@anon sounds like you want your loop to be embedded
 
Semiclassical
Semiclassical
nope, they're fine
 
Pedro Tamaroff
Pedro Tamaroff
@anon Right, we need to assume our loop is homeomorphic to $S^1$.
 
Semiclassical
Semiclassical
what i mean is that (for example) $F_{1x}$ and $F_{2x}$ aren't independent quantities.
 
Pedro Tamaroff
Pedro Tamaroff
That is, we have to show that if $U$ is not simply connected, there is a nontrivial loop inside it that is $S^1$.
 
Mike Miller
Mike Miller
that's true but i don't know an elementary way to prove it
 
Pedro Tamaroff
Pedro Tamaroff
3:46 AM
@MikeMiller What nonelementary way do you know?
 
Mike Miller
Mike Miller
there's a classification of multiply connected regions. see, e.g., ahlfors.
pick one such canonical region it's isomorphic to and you can just draw a loop.
but if you're doing that, you're already done
hm, it actually assumes of S^2 - U isn't fucked up, too
 
Mary Star
Mary Star
It stands that $F_{1x}=-F_{2x}$, right?? @Semiclassical
 
Mike Miller
Mike Miller
argument doesn't work if that's a cantor set. (though again you could just draw a loop then.)
 
Semiclassical
Semiclassical
oops. yes, that's correct, but that's not what i meant to say above
what i meant to say was that (for example) $F_{1x}$ and $F_{1y}$ aren't independent
 
Mary Star
Mary Star
Which relation do these two have?? @Semiclassical
 
Pedro Tamaroff
Pedro Tamaroff
3:56 AM
@MikeMiller
 
Mike Miller
Mike Miller
yes
 
 
2 hours later…
Maximilian
Maximilian
6:01 AM
Hello
 
Balarka Sen
Balarka Sen
6:36 AM
@PedroTamaroff Transfer to $\Bbb R^2$ by assuming $S^2 - U$ is nonempty. If $U \subset \Bbb R^2$ is not simply connected, then there is a loop all over $U$ that cannot be homotoped. This means that the standard straightline homotopy fails, i.e., one or more of the straightlines passes through something that's not in $\Bbb R$. $U$ is not homeomorphic to a convex subset. I guess this should imply that $U$ has nontrivial genus (?) which would mean $\Bbb R^2$ is disconnected.
Of course, this isn't a rigorous proof and probably can only be rigorously stated if $U$ has finite genus.
If $U$ has finite genus, then it deformation retracts onto some finite boquet and the complement is disconnected.
Actually, even if $U$ has infinite genus, just pick one of the holes and pull it towards the boundary.
You still have a disconnected complement.
 
Maximilian
Maximilian
Hello!
Would you mind helping me with something really simple?
 
Balarka Sen
Balarka Sen
What is the question?
 
Julian Rachman
Julian Rachman
Are you talking to me or Bala
?
 
Maximilian
Maximilian
Anyone
 
Julian Rachman
Julian Rachman
Just ask; don't ask to ask.
 
Maximilian
Maximilian
6:47 AM
The question is just what is the domain and range of f(x)=-4tan(2x+pi), I googled a bit, and Its probably something super simple
I put -infinity, +infinity for range
 
Julian Rachman
Julian Rachman
and the domain?
 
Maximilian
Maximilian
and I did an inequality with 2x+pi in the middle and got x by itself and had -3pi/4 and -pi/4 so I put that for domain?
I don't know if its correct though
 
Balarka Sen
Balarka Sen
what's the domain of tan(x)?
 
Maximilian
Maximilian
{x|x/=(2n+1)pi/2, where n is any integer}
According to my book
 
Julian Rachman
Julian Rachman
then how would to correlate that to "2x+pi"
 
Balarka Sen
Balarka Sen
6:50 AM
Right, so what should be the domain of tan(2x + \pi)?
 
Maximilian
Maximilian
Well I guess its smaller, the asymptotes are closer?
 
Balarka Sen
Balarka Sen
just sub in 2x + \pi instead of x in your domain, @Maximilian
 
Maximilian
Maximilian
Well damn, thats easier..
 
Balarka Sen
Balarka Sen
=)
 
Julian Rachman
Julian Rachman
^^ I didn't want to say it @Balarka
the magic word: subsitute
 
Maximilian
Maximilian
6:55 AM
Lol, noob at math here!:P
Thank you
 
Balarka Sen
Balarka Sen
No problem.
 
Julian Rachman
Julian Rachman
:-)
 
Balarka Sen
Balarka Sen
@Julian So, have you finally proved that $\tau_c$ is a topology?
 
Julian Rachman
Julian Rachman
Well. I actually sort of found a very similar proof that I had built off of. it is in the next section
 
Balarka Sen
Balarka Sen
It's not hard.
See if union/finite intersection of your open sets has countable complement or not. That's just it.
 
Julian Rachman
Julian Rachman
6:59 AM
Yes. it does
I always make everything seem so complcated when really it isnt. that is something I have to fix about myself...
 
Balarka Sen
Balarka Sen
@Julian If you're having trouble with this highly abstract idea of topologies, I would really recommend you Simmons and starting off from metric space. It introduces topology way later.
No, topology is abstract. It's not your fault, you're just a beginner.
 
Julian Rachman
Julian Rachman
Ok. I will take a look at it. Will it get me a good transition to alg top after finishing the entire book?
 
Sayan Chattopadhyay
Sayan Chattopadhyay
Hi@BalarkaSen
 
Kaj Hansen
Kaj Hansen
I'm learning topology starting with metric spaces :P
 
Balarka Sen
Balarka Sen
@KajHansen approves of that kind of learning
 
Sayan Chattopadhyay
Sayan Chattopadhyay
7:04 AM
Hi @JulianRachman
 
Julian Rachman
Julian Rachman
@Kaj I was just about to ask @Balarka if your classes' notes good for my learning
 
Balarka Sen
Balarka Sen
@JulianRachman Maybe, maybe not. If you know topology well enough, you'd be able to learn altop too.
 
Julian Rachman
Julian Rachman
can you link the class so he can check it out?
 
Balarka Sen
Balarka Sen
Pete's notes? Definitely!
 
Julian Rachman
Julian Rachman
@Balarka how well? when? Any details?
@Sayan sorry. Hi! Hows everything?
 
Sayan Chattopadhyay
Sayan Chattopadhyay
7:05 AM
@BalarkaSen u have any idea of the power set question......
Great@JulianRachman
 
Balarka Sen
Balarka Sen
Well, @Kaj will be able to recommend you those, @Julian, since I haven't seen them.
 
Kaj Hansen
Kaj Hansen
Here's my course webpage: math.uga.edu/~pete/MATH4200S15.html
 
Julian Rachman
Julian Rachman
@Balarka Ok. so any suggestions on when you think I will be able to understand alg top as you started to discuss above?
so i should scrap munkres?
 
Balarka Sen
Balarka Sen
@JulianRachman You still need to understand algebra well enough. And that could take years.
Don't scrap it. It's a useful book. Come back to it after Simmons.
 
Julian Rachman
Julian Rachman
thank god I only use electronic copies and I print pages one by one
 
Sayan Chattopadhyay
Sayan Chattopadhyay
7:08 AM
Same here@JulianRachman
 
Balarka Sen
Balarka Sen
Don't hurry, @Julian, and don't set your goals for algebraic topology either. You might just like something else while you're studying.
 
Julian Rachman
Julian Rachman
@Balarka I heard that you need to like be good at abstract algebra first
 
Sayan Chattopadhyay
Sayan Chattopadhyay
The master says everything right....@BalarkaSen
 
Balarka Sen
Balarka Sen
For example, about two years ago, I was sure I would become a number theorist, but that was just because I had neither studied number theory nor topology :P
@JulianRachman Er, kind of.
 
Julian Rachman
Julian Rachman
ya. I know I am hurrying too much. I just want to achieve writing a research paper before I graduate because that will be a plus on my apps for me to get into my "dream" school
 
Sayan Chattopadhyay
Sayan Chattopadhyay
7:10 AM
So u dont like number theory @BalarkaSen
 
Balarka Sen
Balarka Sen
I never said that, @Sayan. I like it, but I don't know enough number theory to claim I'd want to study it.
@KajHansen How're you doing with the problem set?
 
Julian Rachman
Julian Rachman
Wait how old are you @Balarka?
 
Balarka Sen
Balarka Sen
51.
 
Julian Rachman
Julian Rachman
@Kaj your problem sets are hard!
Oh right.
 
Sayan Chattopadhyay
Sayan Chattopadhyay
Oh....@JulianRachman u will be shocked don't ask that question
 
Julian Rachman
Julian Rachman
7:12 AM
@Balarka ^^
@Sayan ? ask his age?
 
Kaj Hansen
Kaj Hansen
lol @JulianRachman
 
Sayan Chattopadhyay
Sayan Chattopadhyay
@BalarkaSen is a prodigy
 
Kaj Hansen
Kaj Hansen
I have to choose 14 problems to do from the fourth set
 
Julian Rachman
Julian Rachman
I dont believe that he is 51
 
Sayan Chattopadhyay
Sayan Chattopadhyay
He is not 51
 
Kaj Hansen
Kaj Hansen
7:13 AM
@JulianRachman, Balarka is a teenager :P
 
Julian Rachman
Julian Rachman
hes 14 right?
 
Sayan Chattopadhyay
Sayan Chattopadhyay
Yes
 
Julian Rachman
Julian Rachman
I can tell because I would expect him to have over 20k rep at that age
wow......
 
Sayan Chattopadhyay
Sayan Chattopadhyay
@BalarkaSen so did u start topology after finishing sets and calculus
 
Balarka Sen
Balarka Sen
Of course @Sayan.
 
Sayan Chattopadhyay
Sayan Chattopadhyay
7:15 AM
Or u did something else
Wow.....
 
Julian Rachman
Julian Rachman
what????? I went the foundational way and went to real analysis
 
Balarka Sen
Balarka Sen
Oh, I don't mean I started topology right after studying sets and calculus, @Sayan.
 
Julian Rachman
Julian Rachman
Did you learn from simmons?
 
Sayan Chattopadhyay
Sayan Chattopadhyay
@JulianRachman u started real analysis
 
Balarka Sen
Balarka Sen
I started topology last year, after I studied algebra.
 
Julian Rachman
Julian Rachman
7:16 AM
and how did you get into alg top?
 
Balarka Sen
Balarka Sen
Yup, @Julian
 
Julian Rachman
Julian Rachman
@Sayan yes
 
Balarka Sen
Balarka Sen
@JulianRachman What d'you mean, how did I get into? I just started doing Munkres.
 
Julian Rachman
Julian Rachman
reallly? but the alg top is at the end though? you should have gone straight into Hatcher's book or something?
and how do you knw that you have enough of an understanding before getting into alg top
?
(so many q's)
 
Balarka Sen
Balarka Sen
So what if alg top is at the end? Hatcher is dense.
 
Sayan Chattopadhyay
Sayan Chattopadhyay
7:19 AM
@BalarkaSen then what did u do after calculus and set theory
 
Balarka Sen
Balarka Sen
@JulianRachman Er, I don't want to say this, but : you really need very less topology to understand algebraic topology :P Don't let that fact mess up your schedule though.
@SayanChattopadhyay I don't recall. And don't set me as an ideal either. A lot of what I know is very shaky.
 
Julian Rachman
Julian Rachman
@Balarka totally. I want to learn top to the fullest of my abilities. :)
But any specific onto where you thought that you knew enough to understand?
 
Balarka Sen
Balarka Sen
Good algorithm for knowing whether you know enough : Start reading --- if "I can't understand hell out of this" then end program --- if "this looks not that hard!" then continue reading.
 
Julian Rachman
Julian Rachman
And I know that i have to learn abstract algebra too so I was thinking about taking that after I finish topology.
 
Balarka Sen
Balarka Sen
Yes, sure.
 
Kaj Hansen
Kaj Hansen
7:25 AM
@BalarkaSen, does anyone do topology before calc?
 
Julian Rachman
Julian Rachman
so if you can understand, stop?
i dont think that is good philosophy... @Balarka
 
Balarka Sen
Balarka Sen
@KajHansen who knows. must be an unfortunate person.
 
Julian Rachman
Julian Rachman
@Kaj I would cry if I saw that
 
Kaj Hansen
Kaj Hansen
Then again, I don't see what the problem would be necessarily?
Or rather, it seems wrong but I cannot articulate why.
 
Balarka Sen
Balarka Sen
Well, actually I never studied analysis but it doesn't in the least affect my understanding of topology.
 
Julian Rachman
Julian Rachman
7:28 AM
@Balarka you should learn it
 
Balarka Sen
Balarka Sen
@KajHansen Well, he wouldn't understand a lot of examples.
 
Julian Rachman
Julian Rachman
it is like the why's and how's of calc
 
Balarka Sen
Balarka Sen
Analysis don't get into me. And I don't need it yet either.
 
Julian Rachman
Julian Rachman
So you study things only if you need it?
 
Balarka Sen
Balarka Sen
Kind of.
 
Julian Rachman
Julian Rachman
7:30 AM
I think that is wrong
study as much of everything as you can
 
Balarka Sen
Balarka Sen
Well, it didn't hurt, at least not yet.
I'd soon need to study analysis though. Have to restudy complex analysis and grasp sheaf cohomology.
 
Julian Rachman
Julian Rachman
You already studied grasp sheaf cohomology?
 
Balarka Sen
Balarka Sen
I already studied complex analysis :)
Sheaf cohomology I need to study after cohomology, to understand it geometrically, I guess.
 
Kaj Hansen
Kaj Hansen
Balarka is OP
 
Balarka Sen
Balarka Sen
That'd involve a lot of differential forms and etc. I don't know anything about it.
Probably I'd have to study differential geometry too. Ugh.
 
Julian Rachman
Julian Rachman
7:34 AM
I am not into geometry
I like abstraction and how it almost makes no sense
 
Balarka Sen
Balarka Sen
abstraction is stupid. I can't possibly do math without visualization and handwaving.
:P
 
Julian Rachman
Julian Rachman
but that is what math is!!!
 
Balarka Sen
Balarka Sen
it's not :)
 
Julian Rachman
Julian Rachman
we can essentially describe nothingness
the beauty and understanding of it comes from within the community that can
^^ that is my semi-deepest thought about mathematics
 
Balarka Sen
Balarka Sen
you've got a wrong idea about mathematics. category theory can't possibly be everything to math.
 
Julian Rachman
Julian Rachman
7:38 AM
it isnt
I didnt say it was everything
but abstraction can also be linked to almost anything
 
Balarka Sen
Balarka Sen
most of the people out there did abstraction not for abstraction's sake, but for some purpose.
anyway, as i said, visualization is an essential part of my understanding of mathematics.
 
Julian Rachman
Julian Rachman
well yes. I use visualization in topology almost 98% of the time
 
Balarka Sen
Balarka Sen
i'd have to interpret everything geometrically to even make sense of it.
and it's partially the reason i don't like analysis. the real geometric picture gets veiled out by the epsilons.
 
Julian Rachman
Julian Rachman
ah.... epsilon.....
:)
such a simple variable yet means so much and is used everywehre
 
Balarka Sen
Balarka Sen
We talked too much. I guess I should go now :P
 
infinitesimal
infinitesimal
7:44 AM
See you later pal.
:)
 
Julian Rachman
Julian Rachman
See ya @Balarka!
 
Mary Star
Mary Star
8:42 AM
Hello!! Does someone of you have an idea for this:
0
Q: Write the force as a sum

Mary StarWe suppose that a force $\overrightarrow{F}$ (for example, the gravity) is applied vertically downwards to an object that is placed at a plane which has an angle of $45^{\circ}$ with the horizantal direction. Express this force as a sum of a force that acts parallel to the plan and of a force t...

calculus physics vector-analysis vectors
??
 
infinitesimal
infinitesimal
The incline plane is one of the oldest questions in physics classes :-)
 
infinitesimal
infinitesimal
8:58 AM
2
Q: Determining the direction of friction

farmerjoeI had a question which has always been of slight confusion to me. Say you are dealing with your typical block-on-an-angled-plane setup. You have a block with mass m that is initially at rest. The block sits on a surface at angle θ to the ground, has coefficient of kinetic friction μk, and a co...

newtonian-mechanics friction
 
Mary Star
Mary Star
So is it $\overrightarrow{F}=F \left ( \cos 45^{\circ} \overrightarrow{i} + \sin 45^{\circ} \overrightarrow{j} \right )=F \left ( \frac{1}{\sqrt{2}} \overrightarrow{i}+\frac{1}{\sqrt{2}} \overrightarrow{j} \right )$ ?? @infinitesimal
 
infinitesimal
infinitesimal
 
Axoren
Axoren
9:17 AM
Anyone on right now that's willing to look into a question?
 
infinitesimal
infinitesimal
Askaway :-)
 
Axoren
Axoren
0
Q: A discrepancy in the total number of conjunctive normal forms and the number of distinct boolean functions.

AxorenConsider a set of truth literals $C$. The set $\{\text T, \text F\}^{C}$ is the set of all boolean functions over $C$. This comes from the notation $\mathcal{Y}^\mathcal{X}$ which is the set of all functions $f : \mathcal{X} \to \mathcal{Y}$. The set of all conjugations will be a subset of $\{...

functions boolean-algebra
Why aren't the numbers the same ; _ ;
There are more conjunctive normal forms than there are distinct boolean functions, but there shouldn't be.
I can't figure out what I've done wrong when calculating the stuff
And I'm too tired to keep banging my head against it
I can't think of any CNFs that don't represent a boolean function.
So the only possibility is that there's a boolean function with 2+ CNFs that represent it.
 
Mary Star
Mary Star
$||\overrightarrow{F}_{perpendicular}||=\frac{1}{\sqrt{2}}||\overrightarrow{F}|| \Rightarrow ||\overrightarrow{F}_{perpendicular}||^2=\frac{1}{2}||\overrightarrow{F}||^2$
$||\overrightarrow{F}_{parallel}||=\frac{1}{\sqrt{2}}||\overrightarrow{F}|| \Rightarrow ||\overrightarrow{F}_{parallel}||^2=\frac{1}{2}||\overrightarrow{F}||^2 $

When we add these two relations we get:

$||\overrightarrow{F}_{perpendicular}||^2+||\overrightarrow{F}_{parallel}||^2=||\overrightarrow{F}||^2$

Is this correct??
Or have I understood it wrong?? @infinitesimal
 
infinitesimal
infinitesimal
9:44 AM
Looks ok to me.
F = mg
 
Axoren
Axoren
@infinitesimal What do you think about my question?
 
infinitesimal
infinitesimal
+1
 
Axoren
Axoren
No, I mean do you have any input on it?
I appreciate the +1, however.
 
infinitesimal
infinitesimal
Nope, sorry.
 
Axoren
Axoren
Alright, thanks.
 
infinitesimal
infinitesimal
9:51 AM
@MaryStar the vertically downward force is the sum of the other two forces, right?
Where the other two forces are the perpendicular force and the parallel force.
$F = F_{perp} + F_{para}$
 
 
2 hours later…
newbie
newbie
12:18 PM
hi guys, how can i show that 2^(loglog n)>log_2 n ?
 
Moses
Moses
12:31 PM
Hi @DanielFischer
 
Daniel Fischer
Daniel Fischer
Hi @Moses.
 
Moses
Moses
12:50 PM
@DanielFischer Do you know why it is that when considering a integrand of the form $\sqrt{a^{2}+x^{2}}$, it is valid to use the substitution $x = a\sin \theta$, how is that not possibley changing the value of the integral?
 
Daniel Fischer
Daniel Fischer
@Moses When you do that substitution, the $dx$ becomes $d(a\sin\theta) = a \cos\theta\,d\theta$, you introduce a weight to keep the integral unchanged. In terms of the fundamental theorem of calculus, it is basically the chain rule. However, for $\sqrt{a^2 + x^2}$, a hyperbolic substitution like $x = a \sinh t$ looks more promising, one would use the trigonometric substitution for $\sqrt{a^2 - x^2}$ more likely.
 
Moses
Moses
@DanielFischer So what are the restrictions of substitutions? You can say let $x$ be anything as long as you change $dx$ accordingly?
 
Daniel Fischer
Daniel Fischer
@Moses It should be invertible and differentiable, though both can be relaxed a little. So for substitutions like $x = a\sin \theta$, you typically restrict the domain of $\theta$ to be a subset of $[-\pi/2,\pi/2]$, since $\sin$ is invertible there.
You can also restrict to subsets of $\left[ \bigl(k-\frac{1}{2}\bigr)\pi,\bigl(k+\frac{1}{2}\bigr)\pi\right]$ for integers $k\neq 0$, where for odd $k$ you need to take care of the fact that $\sin$ is monotonically decreasing there, so the substitution inverts the orientation.
 
Moses
Moses
1:10 PM
@DanielFischer So you can only substitute $x$ with a differentiable, surjective and injective function. I understand why it needs to be differentiable, since you require $dx = ...$ but why does it have to be injective and surjective?
 
teadawg1337
teadawg1337
Ugh... I just can't make any sense of these results I've found while working with polylogs...
 
Daniel Fischer
Daniel Fischer
@Moses It needn't necessarily be bijective. For example you have $$\int_{-1}^1 f(x)\,dx = \int_{-\pi/2}^{5\pi/2} f(\sin\theta)\cos\theta\,d\theta$$ for all integrable $f$, even though $\sin$ isn't injective on $[-\pi/2,5\pi/2]$. That's because the integral over $[\pi/2,5\pi/2$ cancels itself out since $\sin$ symmetrically goes from $1$ to $-1$ and back there. If you look at the FTC formulation, with $F' = f$, you have
$$\int_a^b f(x)\,dx = F(b) - F(a) = F(u(\beta)) - F(u(\alpha)) = \int_\alpha^\beta (F\circ u)'(t)\,dt = \int_\alpha^\beta f(u(t))\cdot u'(t)\,dt$$ for all continuously differentiable $u$ with $a = u(\alpha),\; b = u(\beta)$ such that $f(u(t))$ is defined for all $t\in [\alpha,\beta]$, whether $u$ is injective or it wobbles to and fro a little between the end points.
 
Moses
Moses
@DanielFischer Yeah I see, but in most cases you would consider a bijective differentiable function as a substitution?
 
Daniel Fischer
Daniel Fischer
You need surjectivity - that means the image of $u$ must cover the entire interval over which you integrate - or you'll leave out some values of $f$ in the composition $f\circ u$. Non-injectivity is cancelled out since the derivative has different signs when $u$ is in- or decreasing.
@Moses Most of the time, it's simpler to use a bijective substitution. And when you go to higher dimensions, non-bijective substitutions become complicated.
 
Moses
Moses
1:33 PM
@DanielFischer As an example, the substitution $x = a \tan \theta$ for integrand $\sqrt{a^{2} + x^{2}}$ is only applicable for $\theta \in [\frac{-k\pi}{2}, \frac{k\pi}{2}]$, where $k \in \mathbb{N}$? so that the differentiability requirement is satisfied.
 
Balarka Sen
Balarka Sen
@DanielFischer Hatcher wants me to prove that every map $S^n \to S^n$ can be homotoped to have a fixed point. Instincts tell me to rotate the sphere so that $f(x)$ moves all the way to $x$. But this looks suspiciously easy, especially for an exercise set on a chapter on homology. Am I messing up?
 
Daniel Fischer
Daniel Fischer
@Moses $\tan$ has poles in $\bigl(k+\frac{1}{2}\bigr)\pi$, so you can only take $\Bigl(\bigl(k-\frac{1}{2}\bigr)\pi,\bigl(k+\frac{1}{2}\bigr)\pi\Bigr)$ or subsets of that. Unless you have a special situation and know very well what you do.
@BalarkaSen That's how I would do it too. There are of course brazillions of other ways.
 
Moses
Moses
@DanielFischer I made a mistake, I meant $(\frac{-k\pi}{2}, \frac{k\pi}{2})$.
 
Daniel Fischer
Daniel Fischer
@BalarkaSen Of course, only for $n > 0$.
 
Balarka Sen
Balarka Sen
Phew. For a wild moment, I though I forgot the basics.
Yeah, $S^0$ is disconnected :)
 
Daniel Fischer
Daniel Fischer
1:38 PM
Totally
 
Moses
Moses
I think thats still including asymptotes.
 
Balarka Sen
Balarka Sen
2:02 PM
@DanielFischer Ex. 12 "[...] on the other hand, show via covering spaces that any map S^2 --> S^1 \times S^1 is nullhomotopic" huh? what about the map that takes the sphere, fixes one hemisphere, grabs the other hemisphere and pulls it round, identifying it to the fixed hemisphere? (i.e., the thickened loop inside the torus)
oh wait.
i am thinking about the solid torus. duh.
 
Daniel Fischer
Daniel Fischer
@BalarkaSen The map should be continuous. Does Hatcher have a convention that only continuous maps shall be considered?
 
Balarka Sen
Balarka Sen
yeah, he assumes that.
[gah, it's obvious that it is nullhomotopic. pull it up to the universal cover $\Bbb R^2$. pft]
 
abel
abel
2:19 PM
@DanielFischer, how did you get to be so good in math?
 
Daniel Fischer
Daniel Fischer
Practice, I guess. Decades of reading and playing.
4
 
infinitesimal
infinitesimal
playing with the ideas?
 
abel
abel
how do you go about that? i mean in practical terms. what is it you do. stuff like that
 
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