Mathematics - 2018-04-03 (page 3 of 3)

Ted Shifrin

22:17

@Balarka: Good to know you still be alive.

anakhronizein

@TedShifrin if I have a smooth vector bundle of rank n, does it matter if I take the exterior powers higher than n? Or is the exterior bundle "maxed out" at the nth exterior power?

I feel like this is a dumb question, but the notes I am reading define the exterior bundle of a vector bundle to just be the direct sums of the powers up to the rank, n.

Daminark

@anakhronizein so, is the exterior bundle going to be, take the direct sum of exterior powers of the fibers and make that into another vector bundle?

If so, I'm guessing the reason for that is merely because in vector spaces, the exterior powers higher than the dimension are all 0

anakhronizein

Yeah, that's what I was wondering.

But the notation doesn't seem to give any hint at that.

Instead it looks just like $\Lambda^0 E\oplus\dotsc\oplus\Lambda^n E$ where $E\to M$ is the vector bundle.

$E$ is not naturally a vector space

Daminark

Huh, what does $\Lambda^kE$ represent?

anakhronizein

So I would assume it would have to be exterior products on the fibres.

Daminark

22:30

Oh oh that's what the notation isn't suggesting

That feels like the most likely scenario, all the other linear algebra operations I've seen translated to vector bundles are fiberwise (in particular, the tensor product), and this would explain why they stop at $n$

anakhronizein

I thought the tensor product was done via the transition maps.

user193319

0

Q: Vitali Covering Lemma Proof

Why may we assume that each interval in $\mathcal{F}$ is contained in $\mathcal{O}$? What warrants this reduction? Why is statement (4) true? If $x \in E - \bigcup_{k=1}^n I_k$, then $x \in E$ and $x \notin I_k$ for every $k=1,...,n$. Given some $\epsilon > 0$, there exists $I \in \mathcal{...

Daminark

Well, maybe it's constructed like so to preserve continuity but the fibers end up being the tensor products

anakhronizein

So just $E_1\otimes E_2 = \sqcup_{p\in M} E_{1,p}\otimes E_{2,p}$?

So $\Lambda^kE := \sqcup_{p\in M}\Lambda^k E_p$, analogously.

0celo7

22:46

It maxes out at n

the higher order exterior products are just the zero space

anakhronizein

@0celo7 is that the proper definition there?

0celo7

@anakhronizein as a set, yes, but it's worth it to write down what the local trivializations of this bundle should be

anakhronizein

Thanks, will do!

0celo7

If you feel like going completely abstract, Lang has a program where he extends any "natural/functorial" operator on vector spaces to vector bundles in one shot

I think it's just a very abstract version of what I just told you to do

But it also works for exact sequences which is useful

anakhronizein

Yeah I was actually just going to sit down to do it when I thought if I could just apply $\Lambda^k$ to the bundle transition maps.

Yeah, that makes it trivial, doesn't it.

Thanks again!

skullpatrol

22:55

\o @Semiclassical

Semiclassical

o/

skullpatrol

how's it going?

Akiva Weinberger

@PVAL-inactive Hey, you mentioned you're colorblind

PVAL-inactive

@Akiva yeah

Akiva Weinberger

Is one of these color schemes easier to understand than the other?

PVAL-inactive

22:58

the one with black

Akiva Weinberger

Arright, that's what I thought

I'll keep that in mind if I ever need to color code something

(like knot tricolorability in that example)

PVAL-inactive

that example isn't too bad

skullpatrol

I'd prefer to be called color challenged :P

PVAL-inactive

as you usually don't need to refer to the "green arcs"

or the "red arcs"

In cases where people have arguments and then actually refer to colored pieces of their diagrams it becomes a lot more confusing

Akiva Weinberger

I heard that the Portal video games used blue and orange colored portals because it's a good color scheme for colorblind people. (The color is important for the mechanic)

Actually, nearly everything in those games is blue and orange. The three types of paint in Portal 2 are blue, orange, and white. (They do different things in the game so it's important to know which is which)

Eric Silva

23:05

they have purple and red too tho

Akiva Weinberger

You mean the co-op campaign?

Eric Silva

yup

PVAL-inactive

yeah i'm pretty sure i found it difficult between the orange and red when playing that game.

Akiva Weinberger

I think one player had dark blue and light blue portals, and the other had red and orange portals

PVAL-inactive

isnt orange bouncy

and red is slippery

Eric Silva

23:07

yeah shortboi i think was the blues and tallthinboi was the red orange

Akiva Weinberger

Blue paint was bouncy and orange paint was speedy

PVAL-inactive

oh well to me orange paint was red

Akiva Weinberger

If the paint colors were hard to tell apart for you then that ruins what I just said about them being chosen for colorblind people

PVAL-inactive

well i last played that game in 2011

so I'm not sure my memory can be trusted

Eric Silva

holy crap that game came out seven years ago

Akiva Weinberger

23:11

Can you tell that the left two things are different

(That picture has blue, orange, white, and water)

PVAL-inactive

yeah but to me the orange looks like red.

Akiva Weinberger

They're very similar anyway so it makes sense

Can I ask one last question?

How would you classify your current user icon

PVAL-inactive

blue?

It's either blue or purple

Akiva Weinberger

It's purple

PVAL-inactive

grats

Akiva Weinberger

23:18

Oh I should have asked about peanut butter and the Statue of Liberty but I guess I've bothered you enough

Zee

I don’t buy your color blind , color blind people have zero problems detecting colors when they stand alone , the issue is 100% when you put two colors next to each other

PVAL-inactive

Are you being sarcastic?

Akiva Weinberger

No, he's being drunk

Zee

Apparently there is more than one kind ...

I had a cousin who was color blind , he didn’t even know it until one day we put orange shirt next to a green one and he could not tell the difference

Akiva Weinberger

I heard a horrible story

'cause apparently it's pretty common for people not to realize that they're colorblind until well into adulthood

There was this guy whose childhood dream was to be a pilot

Essentially all he ever wanted to do

And you have to be all sorts of fit to be a pilot, perfect vision for example

And on the day he was trying out, he found out he was colorblind.

Like, when they gave him a colorblindness test.

Zee

23:25

I heard a worse story ,this guy wanted to be a pilot all his life , he had a perfect vision and was allowed in but then he dropped out couse it’s too much work

Akiva Weinberger

He had never known before that. And his dream was essentially shattered that instant

Eric Silva

dont they make glasses that can correct certain kinds of color blindness now

Zee

Being a pilot is a terrible life

Akiva Weinberger

Yeah, they have to taylor them to the indivual's eyes @EricSilva

I don't think they'd allow him to be a pilot even with them, assuming they were able to fix his colorblindness anyway

Zee

I would imagine these glasses are for very specific and uncommon types of color blindness

Unless you mean something high tech with chips and electronics

Eric Silva

23:29

for a second i was confused and then realized you meant tailor @Akiva, but THEN i thought that using "taylor" to describe something that is designed to "fit" is brilliant

Akiva Weinberger

You can look them up, I don't remember the details of how they work

skullpatrol

seeing the world through rose colored glasses

Akiva Weinberger

@EricSilva Oh, I didn't realize it was spelled the other way. And also I don't get the joke

Eric Silva

cuz that's what taylor expansions do

Akiva Weinberger

Oh I see

Zee

23:30

That’s a terrible line of reasoning

Akiva Weinberger

23:48

A meme (45)

Secret

Hmm, seems if I had stayed up I will be joining an interesting discussion above

When will we be able to distort the spacetime geometry of earth so much that time zones no longer exist

skullpatrol

when the earth falls into a blackhole :P

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$Mathematics$ Mathematics