Mathematics - 2015-04-27 (page 2 of 3)

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11:00 AM

I learned about sets after statistics as a way of generalizing the concrete

Without logic you can't even venture in stats

Yes, but formal logic isn't necessary

If there are 8 red marbles and 2 green marbles

Stats is just what it is...yes its good but the main beauty of maths comes in logical proof

the probability of drawing a red marble is 8/10

I don't need set theory for that, it just makes the task more burdensom

How do define probability

11:01 AM

I don't need to define it

Define it

I just need to calculate what needs to be calcualted so that I can achieve whatever goal i am after

That's is being mechanical

For instance, if i'm at the casino and I play this ball game, I can calculate whether the bet is fair

I don't need to knwo about set theory to survive

Sure, abstract mathematics is beautiful

but statistics is practical

and enables us to further ourselves and our society

sure abstract mathematics does too, but to a lesser degree than statistics

It is mechanical

11:03 AM

Statistics help doctors know what medicines to give

they help insurance companies know what premiums to charge

They help governments know how to plan their budgets

Define probability if you say that's stats is everything

It helps the social world go round

Define

Definitions is the scope of abstraction

We don't care about definitions we care about doing good for the world

Probability doesn't need to be defined if you ahve a self consistent system

According to you you woke up in the morning and found stats thats rubbish

11:05 AM

I instinctively knew that the chance was 8/10

this is truth

You have to start from grains then only you can go to a large place like stats

maybe most people can't naturally grasp this

nope

Kids start learning how to add, before they know about group theory

even though group theory is a higher level abstraction than arithmatic

that's number theory

you learn the concrete first

then you learn to abstract

iwriteonbananas

@DiscipleofBarney too bad :(

11:06 AM

that is why when you have a tough question, the teacher will often say, start with a simple problem first, then abstract it

rather than say, abstract first then apply

If so please define probability

Why?

I don't need to define probabilty to do probability

If you don't know definitions and don't understand them you are just being mechanical

But mechanical is enough to be productive in society

and the money is in being productive

But its not good enough to do maths

11:08 AM

not in defining productivity

yet i've managed to do maths for years, are you saying statistics isn't maths?

are you saying statisticians aren't mathematicians?

I am saying stats is maths but everything co.es for abstractions and definitions

abstractions and definitions jsut get in the way though

they are just things to decipher in order to get to the meat of application

they are jsut a way of communicating the concrete

it is the concrete that is important

And you are being mechanical

yes, and being mechanical = being productive, which is where the money is

the money is in being productive, not in defining what productivity is

If you don't know how to pour the concrete concrete is rubbish

11:10 AM

pouring the concrete is application

And that way of pouring is abstraction

abstraction is talking about the moleculare structure of concrete

Comes from abstraction

which isn't necessary to pour

Halo Jessika K

If you don't know the molecular level then when it destroys while you were pouring it is your fault you didn't care the definitions

11:12 AM

you don't need to know molecular level to know if it is dangerous

you juave to check if it has been dangerous in the past

hey guys

or test it on animals

hello zed111

I was just explaining why statistics is the most useful branch of mathematics

over abstract mathematics

11:45 AM

Is anyone here good in Chemistry?

Ah, no time.

yes i am

12:08 PM

rational homotopy groups seem interesting.

12:32 PM

@Bal do you know any jokes about the rational homotopy groups

12:50 PM

why in the world do you think there would be any jokes on that?

1:07 PM

Morning, @JulianRachman.

1:28 PM

Does $wt(e)$ mean weight of the edge in this answer: math.stackexchange.com/a/1253723/22544

1:44 PM

user image

@Mats: Your dendrites have dandruff :P

@TedShifrin I see. Did this joke work in english? In Swedish we say "rötter", which can mean both roots of a tree and roots of a polynomial.

Yes, roots are both in English. I was just playing with the picture and playing with hairs versus nerve cells. :P

ok

2:31 PM

Zeros of polynomials with 0,1 coefficients: retro.seals.ch/digbib/view?pid=ensmat-001:1993:39::566

3:06 PM

@Huy how's your day?

@JulianRachman: A bit busy at first. My toilet was clogged, so I had to get it fixed and then I went to buy some household items and had lunch. In the afternoon, I played a bit of GTA and then just relaxed. I think I will start studying a bit again tomorrow.

What about you, @JulianRachman?

Nothing much at the current moment. Just getting ready for school. @Huy

@JulianRachman: What are you studying these days?

@Huy just Simmons Topology For now. I want to get into some category theory but I want to get through Simmons first. And you?

@JulianRachman: Trying to get a bit better at Differential Geometry. My exam last time went horribly.

3:15 PM

@Huy lol. There are always those times...

@JulianRachman: Yeah, it was a bit unfortunate because I focussed to much on GR and too little on DG. But I'll try to make up for it this time.

@Huy @TedShifrin Everyone! I need your guys' opinion on something if that is fine.

If it's not about maths, I might be able to help.

@Huy haha. Ya. Stay of the GTA. It is ruining your mind.

It's actually a good way to relax, for me.

3:18 PM

Everyone: what are your thoughts on starting something like this (codeforamerica.org) but for the outreach of mathematics and for the love of doing it? like starting linked brigades of communities. (just a thought :) )
I thought of this after looking over some things on Facebook.

@JulianRachman: Can you give a quick TL;DR for codeforamerica.org?

Or like this: globalbrigades.org
a movement for mathematics

What exactly is meant by a movement, @JulianRachman?

It is like a network of communities that come together to forward a political movement. But what I want to do it develop "brigades of people" in different communities to forward the outreach of mathematics and have everyone learn the true beauty of it.

quick question, is there rules for when you aren't allowed to rewrite a function as a product and use integration by parts?

for example, in my textbook, they rewrite sin(x)^n to => sin(x)*sin(x)^(n-1) before applying integration by parts

3:26 PM

Hello @Julian.

Hello @BalarkaSen.

3:50 PM

@JulianRachman: How would you want to achieve that?

4:04 PM

Hi!!!!!@BalarkaSen

@JulianRachman I think something like that already exists. A site with math problems you can work on and learn from. Is that what you had in mind?

morning

morning, @mike

hi @remember

Morning @MikeMiller

4:20 PM

Evening, @MikeMiller.

@Huy expanding mathematics to the people that can't and get younger kids more interested in math.

@Jeff no

Common Core is ruining people's love and interest in mathematics

Well could also like make mathematical models of things.

@JulianRachman: What's Common Core?

4:35 PM

new mathematics curriculum in the US; more conceptual than before (we now expect the students to do more than memorize times tables)

Ya what is it?

Always US. zzz

@Huy listen to @Mike

Hi @TedShifrin

hello @Ted.

gosh, i am exceptionally tired today.

4:39 PM

I agree with @Mike's statement. Parents who don't know what they're talking about are claiming all sorts of horrible things.

hi @Balarka, mr eyeglasses

Hello @RоryMcCune

Hallo @Alessandro @Huy @TedShifrin

Guten Tag, @evinda.

@JulianRachman: Problems in the US fortunately aren't mine yet.

Was gibt es so neues? @TedShifrin

highschool education has always been a hell in this part of the continent.

4:40 PM

Real hell

Hi @TedShifrin

the new curriculum in here is an even fresher hell. thankfully it's the next batch who will suffer.

Ich hab' meine letzte Klasse gelehrt, @evinda.

hi @Remember

Your school @BalarkaSen

@Balarka: Remember that a curriculum must be designed for the common student, not for kids like you and others around here.

Und jetzt? @TedShifrin

4:42 PM

Bald fahre ich ab, @evinda :P

Here in my school of you know more than the teacher its your fault and you have to get out of the class....hell

I had a few teachers in high school who weren't very fond of me, @Remember ... and plenty of students who aren't, either :P

@TedShifrin Wohin geht's?

California ...

@TedShifrin i don't think there is anything special about me and the other kids around here. i am sure if one could set up a good curriculum, then almost everyone would have about the same intellectual progress as us.

4:44 PM

Schön :) @TedShifrin

You're very wrong, @Balarka, I'm sorry.

Vielen Dank, @evinda :)

morning @Ted

need to go 4 miles to submit a passport app... not so happy

good night, @Mike

that's better than having to go 400 miles, @Mike

@Ted: I'd rather 400 in Nebraska than 4 in LA

Probably faster, too

Finally a buzzfeed post about math :)

buzzfeed.com/alyssaweaver1994/…

4:53 PM

Well, I don't think you'd find Univ Nebraska up to your UCLA standard.

You planning to flee the country?

For two weeks, yes

Cool ... where? (Hmm, I wonder when my passport expires.)

vancouver

mine expired just last year... I didn't renew it, because when was I gonna need a passport? :P

Oh, you'll probably need it more than you think ... lots of international conferences.

right... that was the joke. I just wasn't in on that joke last year.

4:56 PM

Such things are usually easier to renew rather than to start from scratch. Except you have to get a picture taken ...

I have all the relevant docs + picture, and my old passport

It should be just a matter of giving it to them and leaving.

You can certainly send it registered mail.

No, I can't. I got my passport when I was under 16, and that's not acceptable for mail-requesting one.

Ohhh ...

Trust me, I did my research on how not to have to go 4 miles out of my way.

4:57 PM

Poor dear. 4 miles.

...in LA...

You can walk it in an hour.

You know @TedShifrin people in my school fear math as if they are facing death and that's the problem with Indian teaching the people just believe in solving questions mechanically and that's a huge problem

Sure, @Remember, but it turns out that most people dislike thinking and puzzling way more than they like doing mechanics.

You haven't tried to walk in LA in a long time if you think that. It's just not true anymore. Having to wait to walk at a streetlight every few feet is bad, but trying to cross freeways is much, much harder.

I had originally intended to get an apt a couple miles away and walk it but after trying it's just not feasible.

5:01 PM

Yeah, crossing freeways sucks.

brb

But we Indians over here don't have anything special everyone can become like us @TedShifrin... You just need to improve the teaching processes in our country and for that we great teachers like you

@TedShifrin: How hard is Nash embedding, to understand? Or to follow the original proof?

I've never read the original paper, @Huy. It's some serious PDE, starting with the implicit function theorem in function spaces.

@Remember: If you think the high school education, on average, is any better in the US, I'm pretty sure you're wrong.

it's not conceptually very appealing IMO o

5:05 PM

@TedShifrin: What proof do you do with your students?

It's pretty amusing, the Nash embedding theorem.

I have never done a proof, @Huy. I saw one in a graduate PDE course I took in graduate school.

Why like that US is far more developed than India for sure @TedShifrin

I see.

Don't be so sure, @Remember.

5:06 PM

Riemannian manifolds, unlike smooth manifolds, should not be thought as submanifolds of $\Bbb R^n$; that they actually are is an amusing fact

Even for smooth manifolds, sometimes it gets in the way to try to think of them as subset of Euclidean space.

iwriteonbananas

5:21 PM

hey balarka, ted, mike

hello

hi bananas

iwriteonbananas

@BalarkaSen there is an equivalence of categories between $Cov_X$ and $Funct( \Pi(X), Sets )$

what is Cov_X, again?

iwriteonbananas

where $Cov_X$ is the category of coverings of $X$

the other category is the category whose objects are functors from the fundamental groupoid to sets, and whose morphisms are natural transformations

5:23 PM

okay. that's just a fancy way to say something simple.

iwriteonbananas

so all covering theory information is contained in some category theory stuff

Pfeh.

meh.

iwriteonbananas

haha

No, it's not - how do you prove that equivalence of categories?

iwriteonbananas

5:25 PM

i've constructed a functor $Cov_X \to Funct(...)$

the next problem asks me to construct one in the other direction

and the next problem asks me to show that these two functors form an equivalence of categories

i am not prepared to think of it as anything other than a fancy language, bananas.

In this case, @BalarkaSen, it's not anything other.

i haven't stumbled upon anything which seriously uses category theory as a tool.

Except for the numerous times I've told you about something that does, but I'm not wasting my time with this discussion again.

iwriteonbananas

the functor $Funct(...) \to Cov_X$ constructs a covering from a functor $\Pi(X) \to Sets$

i think thats pretty kewl

5:28 PM

so far it's been about "instead of proving these ten thousand similar-looking facts, you blah categorify blah functor and blah done"

iwriteonbananas

@BalarkaSen lolol yeah i kinda have that imperssion too

@MikeMiller You told me about some knot homology thing, I think, but you haven't elaborated.

Hello @barakmanos :)

TQFTs, things geometric topologistts think about because they're very natural manifold invariants, have benefitted from an $\infty$-categorical viewpoint; see Lurie's cobordism hypothesis paper. Or try to do what this paper does - solve a classical differential topology problem - without the tools of what you would call categorical homotopy theory.

The notion of the Fukaya category has been substantially influential in symplectic geometry.

Disciple of Barney

@BalarkaSen Disparaging cat theory again...

5:42 PM

One doesn't have to have a taste for something. I probably wouldn't like to work with $\infty$-toposes or whatever. But claiming that very categorical things haven't been used in other fields is willful ignorance; and looking down on a field just because it's not your taste in math is rude.

Disciple of Barney

@MikeMiller Anyone you are hopeing to see or listen to at the pim converence?

Nobody in particular. One reason I'm going is to get a broad overview of what's going on in the field.

And by nobody in particular I mean I'm excited to see just about everyone.

@MikeMiller i never claimed so. i have only said that i have never stumbled upon anything which uses cat theory.

i'd personally love to know about the applications, but 'till then, i'd be ignorant towards it.

;)

i'm checking out your keywords, not sure if i'd understand anything :p

Well, now that you're aware of such things, you don't need to say that anymore! :)

true, i guess.

Disciple of Barney

5:51 PM

@MikeMiller This week there was a This American Life about how hard it is to change peoples minds, and they mentioned studies that show that even when showing evidence the other just digs in to their belief deeper...

(it was actually about the rare times that people do change their mind)

I sometimes wish I had a commute so I had an excuse to listen to NPR

iwriteonbananas

what's NPR?

hell, definition of a topological field theory in Lurie's paper neither has anything to do with field theory nor with topology, apparently. (i understood it! :p)

not sure why i should care about a functor from n-cobordisms to vector spaces over k, yet.

I don't understand how the field thwory shows up. I read Atiyah's original paper and didn't see it there either.

Disciple of Barney

National Public Radio (in the USA)

iwriteonbananas

5:57 PM

ok, i thought it was nepalese rupee

The point is that you get an invariant of spaces that's natural under cobordisms instead of maps. This gives you, for instance, a way to decompose manifolds into simpler manifolds and calculate the invariants that way (like a sphere into two discs).

This should be an inherently valuable idea. But Lurie's examples help give it form.

hmm

but i guess it's not much different from treating k-vector space invariants of manifolds itself since every closed orientable manifold can be thought of as a bordism from the empty manifold to itself.

If you do that, how do you decompose manifolds like in my sphere example?

The whole value is that you can define numerical invariants that are computable by cutting your manifold into simple pieces.

(as you just noticed, we're frequently interested in what the value is on a closed $(k+1)$-fold...)

sorry, eating right now. i'll figure out what this stuff is about just after it. it sure sounds good.

i am probably missing something

6:27 PM

This is a sentence from Hungerford: "The only roots of x^2 - 2 over Q are \sqrt{2} and -\sqrt{􀀘2}
= (x - 􀀘l)(x + 􀀘2).

What does this mean?

@BalarkaSen

@MikeMiller

Cleary x^2-2 has no rational root

Hey everyone

Disciple of Barney

I think it means with coefficients over $\Bbb Q$, rather than a finite field for example @zed111

okay thanks @DiscipleofBarney

@Nickolas What's up?

@evinda hey

@evinda studying for a test on mathematica....

6:37 PM

@Nickolas When will the examination be?

@evinda tomorrow

@Nickolas And have you still a lot to learn?

@evinda no, just a revision

@Nickolas Nice :-)

@Nickolas Any news? :)

@evinda bought two new books, Mathematical Analysis 1 and 2 by Zorich. Great books on calculus!!!!

6:49 PM

@Nickolas So do you like them more than the ones that are offered from Eudoxos?

@Nickolas: We used them for our analysis course as well. They are pretty good. =)

@evinda yes. I also borrowed from the library Calulus from Apostol, great book as well!

@Huy great to hear I made a nice addition to my library :)

@Nickolas Nice :)

@evinda what about you

@Nickolas I don't often read from books... I study the stuff from my lecture notes...

7:01 PM

@evinda well I'm sort of bibliomaniac... I have books on my shelves that I've not read yet

@Nickolas: I hardly have books on my shelves that I've already read.

@Nickolas Aha

@Huy I spend most of my pocket money on books :P

@Nickolas: I spend it on IKEA furniture.

hahaha

7:06 PM

I wish I was joking. Whenever I go to IKEA, I find so many practical, seemingly inexpensive things and it's always very hard to resist.

ok, i'm back.

@Huy: I need to furnish a new apartment soon. I'm hoping I don't die from massive IKEA costs.

@MikeMiller right, one can decompose a manifold $M$ as $M_1 \cup M_2$, and then the corresponding invariant would be $Z(M_1) \cdot Z(M_2)$, via functoriality.

i get it.

Could someone help me out for a sec: Are constant functions Riemann-integrable over [0,inf[?

I'd say yes, but a task I'm dealing with implies the opposite.

@MikeMiller: Do you have other household things already, like stuff for kitchen, vacuum cleaner, etc?

7:18 PM

No. The furnishing is only one part.

@MikeMiller: When I moved out, I didn't realise how much all that "other stuff" would cost. I wasn't very happy when I realised how much money I spent in the first month.

@MikeMiller: I wish I could go to IKEA with you. Not gonna lie, I love going to IKEA. And buying things in IKEA. ^w^

whoa, wait, @Mike. but that only holds for disjoint unions (i mean $\otimes$ where i write $\cdot$, btdubs)

@Balarka Decompose a smooth manifold into two submanifolds, whose boundaries agree. Call this $M_1, M_2$. Then $Z(M)$ decomposes (this is functorial!) into the composition of $Z(M_1)$ and $Z(M_2)$.

i don't know what you mean when you say composition of vector spaces

The cobordisms are maps between vector spaces!!

7:26 PM

blergh, yes. i was thinking of bordisms.

? bordism, annoyingly, is synonymous with cobordism

but that in mind I dunno what you mean

ok, this is getting confusing. cob(n) is a category of (n-1)-manifolds with morphisms being bordism. the functor is cob(n) --> vec(K) for some field K. that is, associating manifolds with K-vector spaces, not?

you're associating morphisms of cob(n) to maps between K-vector spaces.

yes

but $Z(M)$ is $Z$ applied to $M$, an object of cob(n) (manifold), not a morphism...?

give me until I'm in my office

I meant $M$ an $(n+1)$-fold, aka cobordism between empty manifold, aka morphism $k \to k$ aka element of $k$

that's what I mean by "numerical invariant" of $(n+1)$-folds

7:34 PM

... that saves the day. what a lame notation.

everyone else seems to have gotten it fine ;)

no need for the silent treatment, just giving you a hard time

:p

7:55 PM

i like Lurie's paper. it requires just about zero prerequisites (except basic undegrad stuff, 'course) to read it. i think i'm gonna have a skim through.

it gets harder later, but it's the best intro to (and justification for...) $\infty$-cats I know

never read the whole thing, just the first couple chapters or so

okay.

thanks for the paper, though!

sure sure

in practice geometric topologists don't work with fully extended TQFTs very much I don't think, the most common ones are 3+1 and 4+1 dimensional; as you'll see it doesn't seem particularly computationally feasible as you try to fully extend something that's past, say, 2 dimensions

but I find it conceptually/theoretically very satisfying.

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