Mathematics - 2016-08-12

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aste123

00:03

Hello

Can someone help me with this question?

0

Q: How to find that $\exists !xP(x) \equiv \exists x(P(x) \land \forall y(P(y) \rightarrow y=x))$ when only LHS is given?

$$\exists !xP(x) \equiv \exists x(P(x) \land \forall y(P(y) \rightarrow y=x))$$ By writing down what each of the side means in English sentence, I can show that both of them are are the same thing. LHS: There exists unique value of x such that it satisfies P(x) RHS: There exists some values x ...

logic propositional-calculus predicate-logic first-order-logic quantifiers

7 hours later…

Jake1234

07:29

Hello, quick question in category theory - could someone give me an example of two different right inverses for one morphism?

arctic tern

consider the only map {1,2}->{a}. one right inverse is a->1, the other is a->2.

or consider the x-axis coordinate projection map $\Bbb R^2\to \Bbb R$. then if $f:\Bbb R\to\Bbb R$ is any function, $x\mapsto (x,f(x))$ is a section.

in Set, the right inverses of a surjection p:X->Y may be constructed by picking one element from each fiber

so if, say, we pick $s(y)$ in every fiber $p^{-1}(y)$, then $y\mapsto s(y)$ defines a function $Y\to X$ which is right inverse to $p:X\to Y$

Jake1234

of course, thanks

4 hours later…

Lionel Ricci

11:25

Anyone know a good reference for the "standard topology" on the space of smooth functions on a manifold (locally compact)?

1 hour later…

0celo7

12:28

@LionelRicci Hirsch.

Balarka Sen

12:56

The dirac delta function is pretty nice.

Lionel Ricci

13:16

@0celo7 I suppose you'r thinking about the book Differential Topology; i'll look into it, thanks!

0celo7

13:26

@LionelRicci It's a standard reference on function spaces on manifolds

user227867

14:16

Hello @robjohn I am back!

user227867

14:27

Heya @arctictern

arctic tern

yo

robjohn

@Jasper and green!

user227867

@robjohn My new youtube channel youtube.com/channel/UCnKEDPuP0in8tzAXsq15Gfw

arctic tern

huh.

user227867

14:42

I am going to take a nap.

SoumyoB

15:16

so today's whole evening I spent on just trying to understand the definition of fiber bundle

good lawd mathematicians know how to complicate things

Huy

no

SoumyoB

my head is literally spinning trying to process the definition

Balarka Sen

what about it?

it should be relatively simple to understand if you have the right background

SoumyoB

I don't think I have the right background haha

Huy

why is that funny

SoumyoB

15:19

I just read the definition of manifold for the first time today

I just knew basic topology

Balarka Sen

@SoumyoB oh, then you shouldn't expect to understand it

SoumyoB

Idk man I'm finding strange things funny today Ig I'm going crazy

Huy

ok

SoumyoB

what does embedding on a fiber mean

Balarka Sen

context?

SoumyoB

15:22

my prof has actually given me a reading work on a certain paper that presents an algorithm that determines in a finite number of steps whether a curve is simple

he had told me to just kind of 'gloss over' it, stating the reason that I would in fact not be able to make much sense of it

I hate it when I have to gloss over things without thoroughly understanding it

Einer

If you hate it and have enough time at your hands, read everything you need to know to thoroughly understand it.

SoumyoB

oh and the context would be manifolds, I guess @BalarkaSen

that's what I've been trying to do @Einer

Balarka Sen

No, I meant, what's the statement where you see "embedding on a fiber"?

Huy

I know such an algorithm

plot the curve and then check for self-intersections

only 2 steps

Einer

I mean... It's not uncommon that reading a given paper takes months, if you have to catch up on all the background first

SoumyoB

15:26

@BalarkaSen it's as follows:

'Suppose that M is smooth (a connected compact 2-manifold), say $C^\infty $. A regular curve $\gamma $ is

*A regular curve $\gamma $ in $M$ is the image of a $C^1$ map $f:S^1\rightarrow M$ such that the tangent map $Tf: TS^1 \rightarrow TM$ is an embedding on each fiber

Balarka Sen

OK. It means it's a map which is a fiber-wise embedding. Restrict to each fiber and it takes a fiber inside that fiber over the image point.

That is to say, $Tf$ embeds $T_xS^1$ in $T_{f(x)}M$.

@Huy checking for self intersections can get complicated quickly

Huy

@BalarkaSen: not if you have functioning eyes

(the algorithm was not meant seriously btw)

Balarka Sen

I can give you a fractal-like curve or something.

SoumyoB

@BalarkaSen a fiber is the preimage of a point in the base space right?

Balarka Sen

I think things would be simpler if one simplicially approximates the curve first. Then it's just a matter of checking if a graph in a manifold is a cycle.

@SoumyoB Yes.

In the case of $TM$, it's just the tangent space.

SoumyoB

15:35

wait I haven't read up on tangent space

yet

Huy

dud

SoumyoB

so fiber is supposed to be a subset of the total space?

Balarka Sen

yes, by definition

SoumyoB

just checking, en.wikipedia.org/wiki/Fiber_bundle#Formal_definition

Balarka Sen

you should read up on basic smooth manifold theory first before trying to understand these things though, in my opinion. $TM$ is by definition the tangent bundle: fibers are tangent spaces.

SoumyoB

15:37

$F$ is nowhere mentioned to be a subset of $E$

@BalarkaSen

Balarka Sen

because it defines fiber bundles in a more general context

SoumyoB

do you have any books you would recommend for this?

Balarka Sen

fibers sit inside $E$: they are simply homeomorphic to a specific $F$.

SoumyoB

not homeomorphic to $U \times F$ where $U$ is the neighbourhood?

Balarka Sen

no, read again

$\pi^{-1}(U)$ is homeomorphic to $U \times F$, not a fiber.

$\pi^{-1}(p)$ is homeomorphic to $F$

@SoumyoB for differential topology? guillemin-pollack. but you need to know multivariable calculus.

SoumyoB

15:41

oh thanks I'll take a look

wow it's available online, thank the good lords

@BalarkaSen in mean time could you just give me a crude layman's definition of what is a tangent map and embedding in each fiber?

Balarka Sen

before tangent map, you need to know what tangent space to a manifold is

SoumyoB

so what is a tangent space then? I just need a layman's intro

thanks by the way for your incredible patience though dude

blms.oxfordjournals.org/content/1/3/310.full.pdf

this is the paper in case anyone is interested

Balarka Sen

take a surface (aka 2-manifold) $S$ inside $\Bbb R^3$. tangent space of $S$ at $p$, denoted as $T_pS$ is exactly what it should mean: the plane which approximates $S$ at $p$ better than any other plane.

aka, the best linear approximation

SoumyoB

ohh that makes sense

Balarka Sen

that is to say, it's the tangent plane. for 1-manifolds (curves) it's the tangent line to the curve

SoumyoB

15:52

so basically the slope expressed as a vector?

Balarka Sen

a tangent line is not a vector...

it's where the tangent vectors live

it's a vector space. "a line".

SoumyoB

ok got it

Balarka Sen

for higher dimensional manifolds it's just a generalization

SoumyoB

and so what would be a tangent map then?

Balarka Sen

it's just the derivative

SoumyoB

15:54

I see

thanks, I'll try to make do with this definition for the while

Balarka Sen

do you know what derivative of a map $\Bbb R^m \to \Bbb R^n$ mean though?

SoumyoB

be right back after dinner

SoumyoB

16:08

@BalarkaSen you mean the calculus derivative right?

I mean the limit?

Balarka Sen

One needs to be careful with that.

SoumyoB

um are you talking about partial derivative or the Jacobian?

Balarka Sen

The Jacobian.

SoumyoB

well I do know the definition of Jacobian

Balarka Sen

That's what's happening here.

SoumyoB

16:14

@BalarkaSen are you a professor? Just curious

Balarka Sen

No

SoumyoB

I'm a fifth year math major undergraduate by the way

so are you a post doc?

I mean pursuing post doc?

Balarka Sen

Nope.

arctic tern

he's retired, but likes to tell people he's in high school

(:

SoumyoB

lol really

Balarka Sen

16:15

hehe, I like that

SoumyoB

sorry but you just seem so knowledgeable, dude

Balarka Sen

you should interact with more people here then, if you think so

SoumyoB

indeed, I've come to discover this place is a talking library

Balarka Sen

i gotta go

3 hours later…

0celo7

19:48

@BalarkaSen You can represent a 1-dim vector space by a nonzero vector living in it, though

1 hour later…

user227867

20:52

@0celo7 When did math objects start to have life?

0celo7

@Jasper Genesis

user227867

I watched a movie called 'The book of revelation' and it was about a man being abducted and raped by three women.

0celo7

Yeah?

user227867

I still don't know why the movie is called so.

user227867

21:09

Seems there are not many people in chat today.

Huy

wrong

user227867

I miss Chris's Sis. Seems she has disappeared.

user 1618033

@Jasper hhhhhhheeeeeeeeeyyyy!!!!

@Jasper How are you doing? Long time no see! :-)

user227867

@user1618033 GREAT CHRIS!!!

user 1618033

@Jasper LOL

@Jasper You're finally back! :-)

user227867

21:13

@user1618033 By the way, is your book out already?

user 1618033

@Jasper No, but the book is in a stage where things do not depend on me anymore. It could have been if I had some more luck.

user227867

@user1618033 I see. I think that means it will be out after all the copy editing is done...

user 1618033

@Jasper There are some issues that do not depend on the normal publishing flow. I can't say more at the moment.

Except that all the work related to me is almost done.

user227867

OK. I suspect not many people in the world can understand your work, which is why it is delayed, hehe.

user 1618033

@Jasper how are you doing these days? Some (breaking)news about you?

user227867

21:18

@user1618033 Still trying to get better. I hope to start studying math on the first day of 2017. I have made some singing videos on youtube. Feel free to comment and subscribe! It is here: youtube.com/channel/UCnKEDPuP0in8tzAXsq15Gfw LOL!

user 1618033

@Jasper AWESOME!!!!!!

0celo7

Still stuck on Ricci flow

my quality of life has plummeted

I cannot think about anything else

user 1618033

@Jasper Well, you can become a singer before doing a mathematician!!

user227867

@user1618033 Are you getting closer to starting your math degree? Is that still the plan?

Hippalectryon

@Jasper We all ask the same questions :P

user 1618033

21:21

@Jasper First I wanna see my book out, I'm burning to see this step accomplished! I worked for so long and so hard, it's very important for my mood, spirit to see things done.

@Hippalectryon :D

@Jasper the story around my book is so weird that I tell you that almost no one would believe it! So crazy events that delayed the publishing of the book, and it's not related to me.

user227867

@user1618033 I see. I believe you. So many crazy things happened to me, which is why I am crazy.

user 1618033

Hope things will improve soon, and things will return back to normal.

@Jasper Well, you sing so nicely, really!

user227867

@user1618033 Thanks. If you have girl friends who want to date me after watching them, let me know!

user 1618033

@Jasper Well, maybe a bit of craziness is good for all of us, at least to challenge ourselves and become better and get better results, but I don't think you are different from us.

@Jasper hehe, OK!!! :D

user227867

Very sad that there is terrorism almost every week somewhere.

user 1618033

21:28

Indeed.

user227867

ISIS, Taliban, Al Qaeda, etc.

user227867

Luckily, no terrorism yet where I live.

user 1618033

Things were difficult in Europe lately.

@Jasper it would be awesome if you uploaded some math too. Singing and doing math at the same time.

:D

user227867

@user1618033 Hehe, I don't have any theorems unlike you. I am only a banana.

user227867

math.stonybrook.edu/~aknapp/download.html please star this. Let the math community know about these free math books!

3

user 1618033

21:37

@Jasper Cool.

@Jasper Do you still have plans to leave for US?

user227867

@user1618033 Yes.

user 1618033

@Jasper When?

user227867

@user1618033 Long time. Need to get well, study, take exam, etc, hehe. Just a plan at the moment.

user 1618033

@Jasper How is your OCD these days?

user227867

@user1618033 Since your plan is taking a long time too, maybe we will meet in the US, lol.

user 1618033

21:42

@Jasper hehe, I don't see myself going in US.

user227867

@user1618033 I have resolved some parts of it, but there are some hard things about the parts left which I need to resolve in the remaining months of this year.

user 1618033

@Jasper that would be great

user227867

@user1618033 Maybe after your book is published, all the top universities in the world will invite you to study there.

user 1618033

@Jasper I don't think anyone will invite me anywhere, and I never thought of that. Having the book out is enough for me.

Hippalectryon

@Jasper It's unlikely, but it probably will make publishing other works much easier

user 1618033

21:56

@Jasper Then I'm very different from any mathematician I met so far, my perception on mathematics is completely different compared with what I heard and learned from around.

My place couldn't be in any of these possible uni, and then I don't do mathematics, I paint mathematics, mathematics also has a spiritual side in my view, a place with feelings, emotions.

Anyway.

user227867

22:27

@user1618033 I see. You must be sleeping. Good night!

mick

23:57

Bye

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