Mathematics - 2017-09-11 (page 1 of 4)

Kasmir Khaan

00:01

@TedShifrin Done =p

Ted Shifrin

Kasmir: Sorry for the delay. I broke a glass in the kitchen and had to clean it up. Now I need an extra cocktail!

Daminark

@Kasmir also look at Lang... chuckle

Kasmir Khaan

@TedShifrin Omg I hate when that happens ><

@Daminark dont get what you mean ? =p

Daminark

It was a joke, don't look at Lang

Kasmir Khaan

what is Lang ?

not story has been told before this

Is it something famous?

Now I know why Ted calls you denimork -.-

or daminork or something like that

Ted Shifrin

00:10

DEMONark.

Kasmir Khaan

Yes that :D

Daminark

That was angelic as far as jokes go!

Jenna Sloan

@mick No, but he has a degree in Finance and is the Information Services Director for Clay County, Minnesota.

Why do you ask?

Kasmir Khaan

@TedShifrin Thanks Ted Ill start reading now :D

Ted Shifrin

I'm sure it'll be just perfect, Kasmir :P

Kasmir Khaan

00:20

:D

Daminark

00:34

@Ted okay so Atiyah did some confusing functorial stuff to define all the vector bundle operations

Ted Shifrin

Well, unconfuzle yourself!

Daminark

Can I ask about the... "idea" behind the stuff?

Like, say if we take the direct sum of vector bundles

Vaguely, is it that you're doing things pointwise and retopologizing so that you still have a vector bundle?

Or hmm

Abcd

Number of solutions of the equation $$\sin(x)+\cos(x)= x^2-2x+\sqrt35$$ is:______

The answer that I got was zero (which is correct). My method was:

Maximum value of $$\sin(x)+\cos(x)= \sqrt2$$

and minimum value of the LHS function = $$\sqrt35 -1 \approx 5$$
Thus, it is impossible for the two graphs to intersect and there will be no solution.

I just want to know whether my approach was right or not coz reaching the answer isn't important but the solution is important.

Please ignore if the question wastes your time or is unnecessary busy work for you.

Daminark

If you're locally trivial maybe you can just say okay, near a point you have two neighborhoods on which you're the product bundle, so on the intersection it's still a product bundle where you've now got the direct sum of the stuff?

Balarka Sen

@Daminark Yeah so the non-functorial, transition-functions perspective, is the following

Suppose $E/B$ is a vector bundle

Daminark

00:46

But what if I don't wanna suppose that?

Lol jk move on :P

Balarka Sen

loool

Faust

taco

Jenna Sloan

🌮

Balarka Sen

Now, take two arbitrary trivializing neighborhods $U$ and $V$ on the base $B$; that means $E$ is a product over them.

Ted Shifrin

Demonark: You don't need to retopologize, really. You're just locally (on open sets) the direct sum of the two vector spaces.

Oh, never mind, Balarka is still awake.

Heya Faust :)

Jenna must be our local illustrator :)

Daminark

00:48

Yo @Faust and @Jenna!

Balarka Sen

You have trivializations $\rho_U : p^{-1}(U) \to U \times \Bbb R^n$ and $\rho_V : p^{-1}(V) \to V \times \Bbb R^n$. Suppose $U$ and $V$ intersect nontrivially; then these gives two different trivializations $p^{-1}(U \cap V) \to (U \cap V) \times \Bbb R^n$ of $B$ over $U \cap V$.

Faust

Hey @Daminark and @TedShifrin

Jenna Sloan

@Daminark 'sup

Ted Shifrin

Looks perfect to me, @Abcd.

Abcd

@TedShifrin Thank a lot for verifying Ted :)

Faust

00:49

I have a hw question that ask's to prove the following. sup(S ∪ T) = max{sup S,sup T} and inf(S ∪ T) = min{inf S, inf T}.

Ted Shifrin

Sure @Faust.

Faust

now in about a page i proved rather easily that

sup(S ∪ T) = max{sup S,sup T}

by showing enclosure in both ways

Ted Shifrin

It shouldn't be a page.

I would use up all the ink in the world like that :P

You mean inequalities both ways.

$\le$ is easy.

Faust

i have dysgraphia its not many words but i have to space them alot so people can read it and i showed both ways

Ted Shifrin

But $\ge$ takes a bit.

Balarka Sen

00:51

@Daminark Look at $\rho_{UV} = \rho_U \circ \rho_V^{-1} : (U \cap V) \times \Bbb R^n \to (U \cap V) \times \Bbb R^n$. This represents the "transition" from one trivialization to another, right?

Ted Shifrin

OK, Faust. I'm not trying to pick on you.

So what's your question?

Daminark

Looks like it @Balarka

Ted Shifrin

thinks to self: I thought Balarka never transitioned

3

Faust

anyway my question is can i use sup(S ∪ T) = max{sup S,sup T} and a logic argument on its truth value to prove that inf(S ∪ T) = min{inf S, inf T} has the same truth value and so is always true or false whenever sup(S ∪ T) = max{sup S,sup T} is true or false ?

basically i dont want to have to re-write everything out to prove it

Daminark

@JennaSloan chuckles at incoming glorious pun 'inf

Ted Shifrin

00:53

Ah. Here's the question, @Faust. Can you write $\inf S$ directly in terms of sup of something?

Balarka Sen

@Daminark Notice $\rho_{UV}(x, v) = (x, \varphi_{UV}(x, v))$ where $\varphi_{UV}(x, -) : \Bbb R^n \to \Bbb R^n$ is an isomorphism, because nothing happens to the first factor.

Faust

@TedShifrin its obviously the sup (-S)

but i dont know if that is a fact or something im assuming to be true

Balarka Sen

It's just transition from one local trivialization to other by some self-isomorphism of the fibers over each point.

Ted Shifrin

Almost.

Balarka Sen

The self-isomorphisms vary point-to-point.

Ted Shifrin

00:55

Demonark, shouldn't that have been inf'?

Faust

really?

hmm\

Ted Shifrin

Draw pictures or try an example, Faust.

Say the set is $(-2,3)$.

Daminark

ponders for a sec

Faust

ah so ic ant do it that way

Daminark

Maybe...

Ted Shifrin

00:56

Yes, yes, @Faust. You can. Just fix it!

Daminark

I was pondering Ted's thing, I'm still with you @Balarka

Faust

-sup (S) ?

Ted Shifrin

$-\sup(-S)$.

Faust

brain melting

Ted Shifrin

Pictures help.

Faust

00:58

whats that even mean in words/

Balarka Sen

@Daminark Gotcha. Then I can associate to any pair of (nontrivially intersecting) trivializing open sets $U, V$ on the base $B$ a transition function $\varphi_{UV} : U \cap V \to \text{GL}_n(\Bbb R)$, agreed?

Ted Shifrin

You turn it upside-down, but then you need to turn it upside-down again at the end.

Try my example.

Daminark

@Faust please don't let your brain melt, it gets all over the floor and someone's gotta clean that up

At least stand on a towel

Ted Shifrin

Demonark: Stop humo(u)ring and pay detention to Balarka.

Daminark

Agreed @Balarka

Balarka Sen

00:59

I am on detention with Daminark though

Faust

well the negative of the sup S is -3 and sup -S is 2 and the -sup -S is the correct inf of -2

Ted Shifrin

True enough, Balarka.

Faust

Its correct even though it makes my head hurt

Ted Shifrin

Well, it's the conjugation thing that shows up all over mathematics (change of basis, groups, etc.), Faust.

If you're going to change coordinates (flipping), you need to change back at the end.

Faust

$-\sup [-(S ∪ T)] = \inf (S \cup T) $

Ted Shifrin

01:01

\sup

\inf

but, yes.

I'd probably put another set of parentheses.

Faust

i still need to neagte the other side

Ted Shifrin

No.

You have it right.

Balarka Sen

@Daminark So, you can reconstruct the vector bundle $E/B$ just from a given chart on $B$ and the transition functions $\varphi_{UV}$ corresponding to each pair of open sets $U, V$ in the chart; just look at $\bigsqcup U \times \Bbb R^n/\sim$ where $\sim$ puts $(x, v)$ and $(y, v)$ in a equivalence class if they are related by a transition functions.

Faust

dont i need to show that inf(S ∪ T) = min{inf S, inf T}

Ted Shifrin

That should follow from this negation stuff.

Balarka Sen

01:03

Having the data of a vector bundle over a base is the exact same thing as having a trivializing chart and the data of the transition functions.

Ted Shifrin

Remember what happens to a max when you negate the set.

Faust

mm but it doesnt seem that coherient

Daminark

Why do we need the intersections to be non-trivial btw? If they only intersect at a point, that point is open so presumably if the trivializataion only occurs there we're still chill, right?

Ted Shifrin

Be patient, Faust.

Points aren't open. We're Hausdorff, Demonark.

I mean, unless you have isolated points.

Balarka Sen

@Daminark They can't intersect at a point; they have to intersect at an open set.

Ted Shifrin

01:04

Nontrivial means nonempty, anyhow.

Balarka Sen

Right.

Ted Shifrin

Sorry for interjecting, Balarka.

Balarka Sen

I mean technically even if they intersect trivially you can define the transition functions in a dumb way.

Daminark

Oh we're Hausdorff, alright

Eric Silva

Sup chat

Balarka Sen

01:04

@Ted No worries, I was about to say what you wrote :)

Ted Shifrin

Eric!! Hi :)

Eric Silva

What's goin down

Daminark

And I guess I was thinking of, okay we choose a point and two trivializing neighborhoods of it. But sure I gotchu now

Ted Shifrin

I had visitors for the weekend and taught my class, so now I'm having martinis :P

You doing OK, Eric? The people I know in FL who are posting on FB are doing fine.

Balarka Sen

@Eric I'm listening to a post-rock album and Daminark and I are goofing with vector bundles

how's it on your end

Faust

01:06

−sup[−(S∪T)]=inf(S∪T) = Negation of [min {inf S, inf T }]

Ted Shifrin

Slow down, Faust.

Eric Silva

@Ted my parents hunkered down in a safe place and now it's just riding it out but looks like everything is ok

Ted Shifrin

Let's work on the negative of $\sup(-(S\cup T))$.

Eric Silva

@Balarka what album

Ted Shifrin

I'm so glad for you and them, Eric.

whew

Eric Silva

01:07

Yeah I'm really relieved, just hope the house ends up being ok

Faust

ok

Balarka Sen

@Eric it's a fairly recent one, "In Silence We Yearn" by Oh Hiroshima

try this if you want to have a hear: youtube.com/watch?v=ZnvF5k4StzI

Eric Silva

Also what's the functorial point of view on vector bundles

Balarka Sen

prob my favorite out of the album

Faust

im confused i thought we already did that bit?

Eric Silva

01:08

I'll have a listen

Ted Shifrin

So we're taking the negative of $\max(\sup(-S),\sup(-T))$, right, @Faust? Did you write that?

Faust

no ah i see

Eric Silva

@Balarka this is p chill

Ted Shifrin

Actually we need $-(S\cup T) = (-S) \cup (-T)$. Is that clear?

Daminark

Let's say $T$ is a functor on vector spaces, you say it's continuous if $T:Hom(V,W)\to Hom(T(V),T(W))$ is

Faust

01:10

@TedShifrin why isnt it just $ S \cap T $

Daminark

So if $E$ is a vector bundle, define $T(E) = \bigcup_{x\in B} T(E_x)$

Ted Shifrin

Whoa. Where did that come from?

Balarka Sen

@Daminark anyway so, really, if you specify the transition functions (they have to satisfy $\varphi_{UU}(x) = I$ and $\varphi_{UV}(x)\varphi_{VW}(x) \varphi_{WU}(x) = I$ to consistently define a vector bundle - called the cocycle conditions) for a given open cover by charts on the base, that's the same thing as having a vector bundle.

Ted Shifrin

We're not doing set complement. We're flipping across the origin in $\Bbb R$.

Faust

ah

baka

Balarka Sen

01:11

@EricSilva i quite like it

Faust

ah i agree

Eric Silva

I always liked the vibe of post-rock

Good stuff for doing work to

Ted Shifrin

Not for me ... whatever the **** you're talking about :P

Faust

$-(S \cup T) = (-S) \cup (-T)$

i had to say it in words for it to make sense

Ted Shifrin

You can argue that that's correct, @Faust? Then proceed.

Balarka Sen

01:13

its almost a dead genre now though; good that this swedish band is coming up with new stuff

Ted Shifrin

Think about mirror images on $\Bbb R$.

Daminark

@Balarka so if you take the transition functions perhaps you can define vector bundle operations?

Balarka Sen

@TedShifrin you're a classical music person :P

Ted Shifrin

Well, I like folk from the 60's and 70's, too. :P

Balarka Sen

Ah me too

Eric Silva

01:14

IDT I dislike any particular kinds of music besides country music

Ted Shifrin

runs in horror

Daminark

@Eric but yeah so given $\phi:E\to F$ ($F$ is another vector bundle), you define $T(\phi)$ by $T(\phi_x) : T(E_x)\to T(F_x)$.

Ted Shifrin

although there's a newbie who made it as a "gay country singer" (and I contributed to his album) and I don't find him particularly country at all.

Faust

the negation of max{sup -S,sup -T}

Daminark

Now you want to give $T(E)$ and $T(F)$ topologies so that $T(\phi)$ is continuous

Balarka Sen

01:16

@Daminark Aha, exactly. Say $E/B$ and $E'/B$ are two vector bundles (say rank $n$ and $m$) and suppose their corresponding transition functions are $\varphi_{UV}$ and $\psi_{UV}$. The direct sum $E \oplus E'$ is defined out of the transition functions $\varphi_{UV} \oplus \psi_{UV}$ which sends each point $x \in U \cap V$ to the $(n+m) \times (n+m)$ matrix with two big diagonal blocks corresponding to $\varphi_{UV}(x)$ and $\psi_{UV}(x)$ and zero diagonal blocks off-diagonal

Faust

i need the negation of max { -sup (-S), -sup (-T) } dont i?

Ted Shifrin

No, you need $-\max(\sup(-S),\sup(-T))$, don't you?

Daminark

Oh that's slick @Balarka

Eric Silva

I just think countty is really boring

Balarka Sen

(Notice that $\varphi_{UV}(x)$ is an $n \times n$ matrix and $\psi_{UV}(x)$ is an $m \times m$ matrix)

Daminark

01:17

I'll be back in a few minutes

Ted Shifrin

I don't like the whining sound, Eric. But sometimes things are called country that I don't think are country.

Balarka Sen

@Daminark Cool, right?

Eric Silva

Does bluegrass count as country

Balarka Sen

It's just taking the isomorphisms $\Bbb R^n \to \Bbb R^n$ and $\Bbb R^m \to \Bbb R^m$ fiberwise and taking the "direct sum isomorphism" $\Bbb R^n \oplus \Bbb R^m \to \Bbb R^n \oplus \Bbb R^m$

Faust

I am trying to negate this statement? sup[−(S∪T)] = max {sup (-S), sup (-T)

Eric Silva

01:18

I don't think I like this functorial nonsense considering how concrete these objects really are

Faust

sorry im totally lost

Balarka Sen

@BalarkaSen I meant "zero matrix blocks"

@Eric Is this to Daminark?

Ted Shifrin

You're not negating a statement, @Faust. Just a real number.

Eric Silva

Yes @Balarka

Ted Shifrin

Eric, you and I are on one side, and they're on the other :P

Faust

01:20

@TedShifrin i dont understand that statement =P

Balarka Sen

Atiyah's construction unifies the construction of all the various vector bundles (direct sum/tensor product/hom/dual/etc etc etc)

useful for exposition, not sure if for pedagogy :P

Faust

i want to use that fact that sup[(S∪T)] = max {sup (S), sup (T) to prove the inf case

Eric Silva

I guess you don't really use this construction unless you already know what's happening

Balarka Sen

I found it pretty. I don't think I would tell it to anyone

@Eric yeah

Faust

why can't i just negate the statement sup[(S∪T)] = max {sup (S), sup (T)}

Ted Shifrin

01:22

Right, @Faust. So we need to take $-\max\{\sup(-S),\sup(-T)\}$. But we know $-\sup(-S) = \inf(S)$.

Don't say negate statement.

Eric Silva

I just think the concrete stuff with local trivialization a and transition maps is like... Really easy to visualize and not even a hassle to deal with

Ted Shifrin

That's a matter of logic.

Geometers like transition functions more than topologists, @EricSilva.

Eric Silva

I guess I make enough local computations that I want that picture

Balarka Sen

I like transition functions

Ted Shifrin

Good, @Balarka. You'll find them all through Riemann surfaces and line bundles.

Faust

01:24

Then we know that $-\sup[−(S∪T)] = \inf [(S) \cup (T)] $

Ted Shifrin

no minuses on the right, Faust (I think).

Balarka Sen

@EricSilva Is this country to you

Faust

oh

Eric Silva

@Balarka solid no

Balarka Sen

lol

Faust

01:25

your right

Eric Silva

Lol but I love bowie

Balarka Sen

Ron Davies was a country singer though

that's the guy who wrote the song

Eric Silva

I mean it just doesn't sound like country tho

Ted Shifrin

Not twangy or whiny enough? :D

Faust

so i know need to write out what -max{sup (-S), sup (-T)} is

Eric Silva

01:26

bc Bowie doesn't have a country sound

Balarka Sen

true

Ted Shifrin

Right, @Faust. What's $\max(-a,-b)$?

Faust

-b

Ted Shifrin

Grr.

Faust

sorry

Ted Shifrin

01:28

LOL

Faust

is that an interval?

Ted Shifrin

No, no, max of two numbers.

You used set brackets. My bad.

Balarka Sen

@Eric Is this country to you

Eric Silva

Similarly I think all along the watchtower performed by Hendrix vs Dylan are completely different genres

Ted Shifrin

What's $\max\{-a,-b\}$?

Eric Silva

01:29

Cause their sounds are suuuuuper different

Balarka Sen

right, I agree

Eric Silva

@Balarka this is definitely countrt

@Balarka this is definitely country

Balarka Sen

1000% country music

Eric Silva

It did the thing again where an edit turns into a new message

So weird

Ted Shifrin

Technology is appallingly unperfect.

Eric Silva

01:32

If bluegrass counts as country though then there's some country I can enjoy

Faust

@TedShifrin sorry i dont understand unless i know what they are i cant compare them as the result changes if one or both or none are negative?

Eric Silva

But it's weird bc I associate contry with certain styles of vocals but bluegrass doesn't have it

Daminark

Eh, the functors don't help pictorially but I think if you get used to them they're convenient if nothing else. Plus I'm in the void between being more pictorial than not and basically not being able to make any sense out of pictures so seeing both is useful

Ted Shifrin

That doesn't matter, @Faust. You should engage the min.

Faust

but the min a,b doesnt make sense

Ted Shifrin

01:33

thinks Demonark belongs in a deep void

Of course it does, @Faust. What do you mean?

Faust

let a=2 b=3

Daminark

But in one sweep everything you do to vector spaces can be done to bundles which is pretty nifty as far as things go

Faust

then the max is -2

but the min of a,b is 2

Ted Shifrin

What's $\max\{-2, 1\}$ and what's $\min\{2,-1\}$?

Well, there's a minus sign, isn't there? The same minus sign we had :)

Faust

ones 1 the other negative 1

Ted Shifrin

01:35

Nope. Try again.

Daminark

@TedShifrin I have done nothing to deserve this!

Ted Shifrin

OK, fine.

Faust

:p

Ted Shifrin

So ... $\max\{-a,-b\} = -\min\{a,b\}$. Isn't that convenient?

Eric Silva

I think math has a principle of conservation of difficulty, so you often pay the price for being too sleek

Ted Shifrin

01:36

slick? :D

Faust

yeah now i have all the pieces

Ted Shifrin

Although I love to be sleek, Eric :P

Yes, @Faust. You do. You'll get it all now.

Faust

but how do i write it out to get marks?

Eric Silva

Yeah slick that's the word

Faust

like what do i say that im doing?

Ted Shifrin

01:37

State carefully all the facts you're using. If something is non-obvious, prove it.

I don't know what your prof expects for level.

Faust

k ima write it out

Eric Silva

Although I think if sleek connotatively means everything is "pretty and shiny" and elegant then it still works

Ted Shifrin

Have fun. You'll do fine.

Faust

Thank you so much =)

Balarka Sen

@Eric what kind of music do you usually listen to

Ted Shifrin

01:37

I like being pretty and shiny, Eric :P

Balarka Sen

hip hop? rock? violent death metal?

Eric Silva

@BalarkaSen I don't listen to death metal lol

Balarka Sen

aw

Ted Shifrin

hits Balarka over the head and throws him in bed

Eric Silva

I listen to a lot of hip hop, rock, jazz, classical, Brazilian music, whatever

Balarka Sen

01:39

i see

Faust

Can anyone tell me if i finally got this negation correct?
https://math.stackexchange.com/questions/2424279/negation-of-logical-statement-in-graph-theory

Eric Silva

I don't think my tastes are really narrow

Balarka Sen

so you're not in the metal game at all

Eric Silva

I was when I was like 15

Ted Shifrin

Bye all :)

Faust

01:39

Bye Ted

Balarka Sen

See you, @Ted

Eric Silva

Bye @Ted!

Balarka Sen

I have found myself slowly drifting towards metal

Eric Silva

I am pretty into the Chicago hip hop scene lately @Balarka so there's that I guess

Faust

@BalarkaSen i didnt know you were a magnet!?

Eric Silva

01:41

I've been listening to a lot of chance, towkio, noname, Vic Mensa, bj the Chicago kid donnie trumpet etc lately

Balarka Sen

@Eric ah cool

tbh idt hip hop is my thing

Eric Silva

But that's just lately, I guess I listen to samba just as often because it's the kind of music I enjoy playing

I think hip hop is wide enough that finding some you enjoy is more a matter of finding the right ppl than unilaterally disliking the genre

Balarka Sen

well there's death grips loooool

Eric Silva

Although the same can be said for a lot of genres

Do you like them unironically

Balarka Sen

true

just their first mixtape

Eric Silva

01:44

I actually don't think they're bad

But I'm more into chill stuff I guess

Balarka Sen

they occasionally do come up with interesting sonics, true

Eric Silva

I also adore Kendrick

Balarka Sen

i have tried to listen to some of his things

good kid mAAd city eg

I can appreciate it, but I can't really say it's my thing

Eric Silva

I think to pimp a butterfly is one of my favorite albums of all time easily

Not a huge fan of DAMN. Tho

Balarka Sen

hm

Eric Silva

01:47

Tbf I grew up listening to a lot of Compton rappers so kdot is like, very much also a nostalgia trip for me in terms of sounds

Balarka Sen

yeah i can see why actually growing up listening to rap would create an appreciation for that

being from a rock slash folk background it's kinda hard for me

I guess I also like the underground far more than the mainstream for completely non musical reasons lol

Eric Silva

I mean underground stuff comes in all genres, I listen to a lot of underground stuff

Balarka Sen

i agree

Eric Silva

I play a lot of folk (being a mandolinist) so I love that kind of stuff

Balarka Sen

i have been recommended dalek recently

Daminark

01:55

I mostly do game soundtracks lmao

Balarka Sen

@Eric cool!

Eric Silva

I listen to those too @Daminark lol

Balarka Sen

@Daminark undertale soundtracks are amazing

Eric Silva

I agree with that^^

Daminark

Right? That and FTL are my favorite

Eric Silva

01:56

That shit is dope af

Balarka Sen

but yeah it's funny that my underground tastes has led me down to bizarre, bizarre shit

Eric Silva

That's kind of what you expect tho

I date a percussionist so I'm familiar with bizarre ass music lol

Balarka Sen

I have been listening a lot of the experimental Scott Walker albums recently

Tilt and Drift are my absolute favorite

Eric Silva

I went to a concert recently that was 3 hours of people tapping on desks

It was actually like super dope

Balarka Sen

heh

Transcript for

$Mathematics$ Mathematics