9:00 PM
Why do algebraic topologists like to denote the symmetric group as $\Sigma_n$?
Is it to not confuse with the sphere?

they do that?
chumps

Peter does it, and my AT book does it as well

I write $\Sigma_g$ for the surface of genus $g$

same

Hello, I have question. Is exists a function of two variables, that are two local maxima and is not local minimum ?

9:02 PM
Yeah so it'd make sense to write $S_n$ to not confuse it with that, like you can just let the superscript mean sphere and subscript mean symmetric group

some people use mathbb for the sphere
$\mathbb{S}^{n}$

i say burninate them

Ah that could work

$\mathbb{S}^n$ master-race

I don't do it cause 1. it's impossible for me to draw a mathbb S and 2. I don't even use mathbb

9:04 PM
$\mathbb{S}^n$upreme

3. I'm not a chump like @Daminark

Maybe it's just a homotopy theorist thing

@PawełKusz consider the function $f(x,y)=x$, and then punch the graph upward at two locations

@Eric good third point

Fucking non-chump plebs

9:05 PM
to be fair it takes a very high IQ to understand mathbb

I love bf
4
no
that didn't come out right
i didn't mean that
fucking fuck

Rip in pasta: Balarka

See I'm not sure which would've been better, if you wrote "I love bf" or "I love bb"

9:08 PM
I messed up, I meant to write $\mathbf{S}^n$ master-race :thinking:
bf is noice

I'm drowning in algebraic NT someone send help lol

I think Serre uses mathbf so that means it's better

@Daminark Oh god this is beautiful

@EricSilva that's what several of my profs have actually told me

9:09 PM
Right??
So wait what exactly is the deal with this Serre guy anyway?

the History channel logo there just adds to the quality

@Daminark he writes pretty shit a lot

Like I know him from AT vaguely, Serre fibrations and spectral sequences

@Daminark he made huge developements in number theory, homotopy theory and algebraic geometry

But he apparently likes arithmetic quite a lot
Ah I see
That mix sounds p sick actually

9:11 PM
he was Grothendieck's collaborator

he was the youngest fields medallist too

He is one of the last of the French greats still alive (the last?)

So he didn't die at 29 like the others? Damn that's more impressive than any of his mathematical accomplishments

lol

Tbf, a lot of the big french names of the 20th century lived quite long.

9:13 PM
does deligne count as french

I mean yeah he existed around those guys, I was thinking more the ancients

cause he's belgian but he is LIKE a french mathematician

I guess you'd have to ask deligne xD

Galois, Abel, etc
"Ancients" god that sounds horrible in hindsight

galois died at like 20

9:14 PM
Connes is also a pretty big deal. Who else could come up with something called "Connes-Fusion" (pronounced con-fusion)

Connes is a cool dude

"We define an event to be prehistoric if it occurred before 1904"

the F_1 dude
also noncommutative geometry that i dont know shit about

noncommutative sp00k

should i sleep or should i have some biscuits

9:16 PM
why not both

in what order?

Neither, you should glare into the abyss

#deep

Time to delve back into prepping for finals.
:gun:

Lol, I think I've written as much as I intend to on qfibs for now, today I might do symmetric products and then I'll just work on finals

9:21 PM
-3

Im looking for examples of algorithms $A,B,C,D$ such that : $A$ halts If $B$ does not halt. $C$ does not halt If $D$ halts. $A,B,C,D$ are not algorithms that halt on all input. Also they are not algorithms that loop forever on all input. I am aware that the general halting problem is undecid...

@Daminark It's non obvious to me why SP^2(S^2) = CP^2 pictorially

Oh, the cts extensions being equal can be shown using the fact that the zero set is closed. Cool!

right

@orbit-stabilizer but you don't need it

What other way is there to do it? Besides sequences

9:35 PM
why not sequences?

I know the sequence method

alright

I don't know how to do it pictorially @Balarka
The way I proved it was by sending a bunch of points on the Riemann sphere to a polynomial whose roots are that
Or really to the coefficients

yeah i vaguely recall that
but i don't really like that. it doesn't tell me why these two spaces are the same
I am guessing the image of the diagonal in S^2 x S^2 --> SP^2(S^2) gets sent to CP^1 in CP^2?
I guess I want to prove O(2)/Z/2 is O(1) then
(nevermind what that means; i'm screeching my thoughts out loud at this point)

Aight

9:52 PM
what the hells an SP

Superpower

symmetric product

thats Sym
probably if you think about S^2 \times S^2 \subset \Bbb C^3 this should become pretty obvious.
There's an obvious way to get a line in $\Bbb C^3$ this way.

I use $Sym(X)$ to denote the permutations of X

well mathematicians use Sym as symmetric product.

9:57 PM
lolol
@PVAL-inactive hmm

err
that's codim 2
oh wait that's what you want.

I've only ever seen the symmetric product denoted as $SP(X)$ or $SP^{\infty}(X)$

Yeah I think I can convince myself that that map S^2 \times S^2 \to \Bbb CP^2 is surjective.
Now just have to exhibit the symmetric structure on that map.

What is an example of an algorithm pair : x does not halt IFF y halts ??

well (x,y) is identified to (-x,-y) and that seems to be all you get.
but there's no diagonal then..
I guess that map misses stuff with pure imaginary and pure real parts.
actually that's not right/

10:25 PM
are we trying to construct degree 2 maps from S^2 x S^2 to CP^2?

I'm trying to understand the natural map $S^2 \times S^2 \subset \Bbb R^3 \times i \Bbb R^3$ to $\Bbb CP^2$.

can anyone help me integrate $∫sinθcosθ dθ$

If we can find a global frame for the tangent bundle, it induces that the tangent bundle is trivial. Does it automatically imply a trivial connection?

we have (x,y)=(-x,-y) and (x,y)=(-y,x)

10:28 PM
@quallenjäger seriosuly? product rule for integration?

@MikeMiller sounds right. I want to understand why SP^2(S^2) is CP^2 geometrically

@leakyNun thanks Man I've been having trouble understanding this one

just use u-sub and let u=sin(x)

@PVAL-inactive that's the first solution there :P

not sure why that's a mysterry.

10:29 PM
ya I did that lol but Idk i kept getting the wrong answer

@PVAL-inactive because not everyone is as fast as you when they start doing calculus

by parts i say
oh that works

@LeakyNun Maybe because integration by parts arise from product rule?

I would never call that "perplexing" still.

@quallenjäger but you don't call that product rule

10:30 PM
I have plenty of experience with students starting to learn integrals.

Basically everything with integration is based on the differential, people just give them other names.

@Daminark you should get into death grips meme
these things are boiling hot

I guess (x,-x) is invariant under the action of i
but what the hell it isn't invariant under -1, unless I'm messing something.
I'm doing something very stupid I suspect.

I have another question How can I integrate $∫ (x- 1/2x)^2$ I re-wrote it as $∫(x-(2x)^-1)^2$ then letting u equal $(x-(2x^-1)$ i get $du=(1+2x^-2)dx$ and $dx = 1/(x-2x^-2)du$ ∫ 1/3(U)^3* 1/(x-2x^-2)du $nothing cancels nvm its not invariant 10:36 PM I don't know what I did wrong So I think I have S^2 \times S^2 with a$\Bbb Z_4$action which quotients to$\Bbb CP^2$. with the generator having no fixed points and the square of the generator fixing (x,x). which is almost a symmetric product. Plz vote either reopen or vote delete for this -4 Im looking for examples of algorithms$A,B,C,D$such that :$A$halts If$B$does not halt.$C$does not halt If$D$halts.$A,B,C,D$are not algorithms that halt on all input. Also they are not algorithms that loop forever on all input. I am aware that the general halting problem is undecid... Im not gonna make big edits , So it should be deleted or reopened imho Im in a very democratic mood :) Call me crazy hi PVAL glad to see @MikeM hasn't burned in the very scary LA fires ... yikes. @MATHASKER: You're making problems too hard. Don't try to do substitution on that integral. Just write out$(x-\frac 1{2x})^2 = x^2+\frac1{4x^2}-1$. 0 Let$X,Y$be algorithms that accept an ordered set of positive integers as imput. What are examples of$X$halts IFF$Y$does not halt ? Im aware that the general halting problem is undecidable. Any ideas ? 10:51 PM oh.. thanks @TedShifrin Hi chat Hi @Ted hi demonic Alessandro Hi @Pseudohuman @mick Hello, human. hmmm ... pseudohuman, huh? 10:53 PM How are you ? @Pseudohuman @mick I am doing well, thank you. Are you programmed to do math ? @Pseudohuman album | music act | release date See the Light | Less Than Jake | Tuesday, November 12, 2013 hi Ted oh oh ... @quallenjäger is back again 10:56 PM If I choose a smooth global frame on the tangent bundle, will I still have the freedom to choose a non trivial connection? Because as I have read your post, I could have define a non trivial connection You can define zillions of connections ... @Pseudohuman How high is your IQ ? @mick I do not understand. For example, if you do polar coordinates on the plane, you get nonzero Christoffel symbols. But curvature is still 0. Haha your programmer forgot something :) @Pseudohuman 10:59 PM @mick I do not understand. @Pseudohuman What does$1 + 1 = $? What are your hobbies ? @Pseudohuman @orbit-stabilizer I do not understand. @mick I like to explore the computational universe. Ah fk it. Any advice for my exam @Ted? If I require the trivial bundle to be isomorph to$M\times\Bbb R^n$, does it fix my connection? My goal is namely to understand, which step in the trivialisation does fix my connection. 11:00 PM So late in the day? @TedShifrin hi Yeah, it's at 7pm @Pseudohuman have you heard about the Riemann hypothesis ? No, @quallenjäger, only if you require compatibility with a metric or some other such thing do you get a unique connection. Otherwise there are truly uncountably many. @mick I do not understand. 11:01 PM Wow, orbit, late. Take a break and relax for a bit. Lol that programmer has a lot to fix @TedShifrin Then, for example for a Lie group$G$, can I define a parallel transport by the left translation$L_g$to induce an unique flat connection on the Lie group? @Pseudohuman who are you? @LeakyNun My name is Pseudohuman. @Pseudohuman what is a pseudohuman? 11:05 PM @LeakyNun I do not understand. She is a bot No can do. Still got a bunch of stuff to learn.. @quallenjäger: I don't think that will always be flat. "She"? Or claims to be Yes she : female pic 11:07 PM I could put up an image of a girl, would you refer to me as she? If you are a bot or an angel tou can choose r gender And Yes I would @TedShifrin If it is not flat, how can the christoffel symbol vanish for the left invariant frame? You can't do either part of the assignment? Why don't you take the very simplest$L$-structure you can think of, say a one-element structure, and figure out whether$\phi$is true or false there. Depending on what you find out, call that structure$A$or$B,$and you've got half of the problem solved! — bof 6 mins ago well played @Pseudohuman Hello @quallenjäger Hello, human. 11:13 PM @Pseudohuman Your name is banana @quallenjäger I do not understand. @quallenjäger: Declaring a particular frame to be covariant constant will give vanishing Christoffel symbols for that frame. I didn't necessarily know that's what you were doing. I guess it is clear that if you fix one basis and declare it "constant," then any other left-invariant basis will be as well. This is not the natural connection on the Lie group when you want to think of a natural metric on it. In particular, this connection has lots of torsion (and the Riemannian connection has zero torsion). @TedShifrin You mean it is better to do it by requiring it compatible to left invariant metric @TedShifrin I'm not sure if I can do that It depends what you want to do. Same stuff, I want to roll a curve on the Lie algebra into the Lie group *onto So I thought it might be very convenient if I know the parallel transport, which allows me to write down the differential equation. But you are right, the torsion gives me some headache 11:19 PM I honestly don't know. Hey there Ted! hi Demonark How's it going? About to make an emergency trip to the dentist. Otherwise OK. What's wrong? 11:23 PM Damn crown came off :( @TedShifrin First I thought I could roll the curve onto Lie group without specifying the connection. But I think it might be not realistic as I have to transport the tangent over the manifold. Oh that's no fun, sorry. Hope it all goes well soon enough @TedShifrin I had same issue. crown came off while I was sleeping. I was lucky that I didn't swallow it:D Yeah, right. Or bite it. I haven't thought about this rolling stuff too much, other than for a few exercises I wrote for my notes. I'm not sure how much your need the connection for. Maybe if I have some free time tomorrow I'll think about it. @TedShifrin The approach I was choosing is to parallel transport the tangent vector of the curve onto the manifold and then require that the transported vector "is the same" as the tangent vector of the rolling curve. Do you know a guy called Bruce K. Driver from UCSD 11:30 PM How can I integrate$(sinπ/2s)^-1$? No, @quallenjäger, I don't know many people at UCSD. Ok, is basically based on his work, but his notation is awful. @MATHASKER: You need parentheses. I assume you mean$\sin\big(\frac\pi 2 s\big)$and then inverse. Forget about the$\pi/2$until you figure it out. What is$1/\sin s$called? sometimes I just don't understand why everyone want to introduce his own definition and notation. He looks like a serious differential geometer, but I don't know his work. 11:34 PM @TedShifrin have you tried this: in Homotopy Theory, yesterday, by Aaron Mazel-Gee let me use this as a springboard to strongly encourage everyone (who needs to grade things) to use Gradescope: https://gradescope.com/ it's really fantastic. it saves tons of time, the students get a much clearer picture of the rubric and where/why they lost points, you can grade at home in pajamas instead of staying at the department til 1am with your fellow graders... Hell no. :-/ I haven't had "fellow graders" except when I helped out the uniform calculus testing (I helped write the exam and was involved in some grading). yea thats what I meant csc @TedShifrin That's one of the tricky ones that you just need to learn, @MATHASKER. You can find it in any book or on line. 11:37 PM You're probably the only teacher I know who has never had a TA grade at least a subset of stuff Oh nvm the question was sin^2(π/2s) then it will be csc^2 ya i think i found out @TedShifrin Did you had PhD students? Demonark: We didn't have TAs at UGA for most courses. We had graduate graders for upper-level courses and I never used them. I graded myself. One, @quallenjäger. @TedShifrin Why didn't you took more if I may ask. 11:41 PM I spent most of my time with undergrads. ...and your books :-) You enjoy more teaching? Yes. I wrote some good papers and had fun particularly with co-authors. But loved teaching more. Quite rare in academia, most people finds teaching exhausting and time consuming Well, I would not get promoted in this modern era with my behavior. 11:44 PM Find it quite pity. How goes the AoPS course? They have their first midterm on Sunday. We'll see if they've learned anything ... I had professors in undergraduate who doesn't even know what they are teaching. I know of at least one prof here who really doesn't like teaching undergrads and is very disdainful of compsci students Everyone seems to hate his class @Daminark Me too. 11:45 PM (This is in the compsci department, theory, but still) I had a professor letting us to prove a wrong theorem in the middle of an exam. Who ever started to prove got zero marks. 2 out of 40 passed the exam OK, I'm heading to the dentist. Good luck, all. I know chem has a bit of a problem as well since so many people don't care about teaching. And the sad part is, the one person who has been banned from teaching undergrad classes is the one who really tries hard and loves teaching but is just bad at it cya professor 11:48 PM See you Ted, hopefully everything goes aight @TedShifrin Good luck @Daminark Where are you coming from? @Daminark Sounds like germany. Nah, USA @quallenjäger for the whole exam or the question? Ok @LeakyNun Question @LeakyNun This question has 18 marks out of 25. what was the question? wow that's mean 11:51 PM @LeakyNun I posted it @LeakyNun mom where? 1 Consider$\Bbb{R}^2$with the usual topology. Let$X$be a subset of$\Bbb{R}^2$. If for every$a \in X$and$v \in \Bbb R^2$there exists a d>0, such that$a+vt\in X$, for every$0\leq t<d$, then X is open. I suppose this theorem is wrong as the choice of the radius of the open ball$d$is depe... wow did he do it on purpose? Yes Either you give counter example or you have 0 marks that's mean af 11:55 PM Thats why only 2 out of 40 passed. This was in Germany can anyone help me integrate$∫ sin^2(x) dx\$

Then I came to UK and Uk professor are so kind
Exam questions are almost everything which has already been taught on lecture or on the problem sheets.

@quallenjäger wait, are you in imperial now?

Yes

sin^2x is 1-cos^2x right what would I do after that cuz one over the derivative of that wont cancel

11:56 PM
@quallenjäger first year?

Doing MRes

@quallenjäger sorry :P
do you see any UG at all?

Not really, I have rare taught courses.

do you pass through any UG-inhabited rooms at all?

11:58 PM
I am in the stochastic analysis group

Any help with this? I'm very confused

@Theo "every number can be represented in binary"

We are all in Huxley aren't we?

do you ever look at the UG colloquium posters?

I don't even know where the UG are.
Not really:D what is it

11:59 PM
the posters are everywhere

@LeakyNun yes but how would i start proving it?

they're in MLC, outside chore, outside 340, etc